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Bernhard Schölkopf

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preprint2026arXiv

Adaptive Inverted-Index Routing for Granular Mixtures-of-Experts

Mixture-of-experts (MoE) models enable scalable transformer architectures by activating only a subset of experts per token. Recent evidence suggests that performance improves with increasingly granular experts, i.e., many small experts instead of a few large ones. However, this regime substantially increases routing cost, which can dominate computation. We introduce adaptive inverted-index routing for MoE (AIR-MoE), an inverted-index-inspired routing architecture based on vector quantization (VQ). In a first stage, AIR-MoE performs coarse shortlisting by assigning tokens to VQ codewords to construct a candidate set of experts. In a second stage, fine scoring computes exact routing scores restricted to this shortlist. This two-stage procedure approximates true top-k routing while avoiding full expert scoring and, in contrast to prior work, imposing no structural constraints on expert parameters. AIR-MoE serves as a drop-in replacement for standard routers and requires no modifications to the model architecture or loss function. We further provide a lower bound on the mass recall achieved by AIR-MoE that yields insights into its inner workings. Empirically, we demonstrate that AIR-MoE achieves improved performance compared to existing routing approaches in granular MoE settings.

preprint2026arXiv

Are Language Models Efficient Reasoners? A Perspective from Logic Programming

Modern language models (LMs) exhibit strong deductive reasoning capabilities, yet standard evaluations emphasize correctness while overlooking a key aspect of reasoning: efficiency. In real-world reasoning scenarios, much of the available information is irrelevant, and effective deductive inference requires identifying and ignoring such distractions. We propose a framework for assessing LM reasoning efficiency through the lens of logic programming, introducing a simple method to align proofs written in natural language -- as generated by an LM -- with shortest proofs found by executing the logic program. Efficiency is quantified by measuring how well a model avoids unnecessary inference. Empirically, we construct a dataset of math word problems injected with various number of irrelevant axioms that vary in semantic overlap with the goal theorem. We find that current LMs show marked accuracy declines under such conditions -- even with minimal, domain-consistent distractions -- and the proofs they generate frequently exhibit detours through irrelevant inferences.

preprint2026arXiv

Imagining and building wise machines: The centrality of AI metacognition

Although AI has become increasingly smart, its wisdom has not kept pace. In this article, we examine what is known about human wisdom and sketch a vision of its AI counterpart. We analyze human wisdom as a set of strategies for solving intractable problems-those outside the scope of analytic techniques-including both object-level strategies like heuristics [for managing problems] and metacognitive strategies like intellectual humility, perspective-taking, or context-adaptability [for managing object-level strategies]. We argue that AI systems particularly struggle with metacognition; improved metacognition would lead to AI more robust to novel environments, explainable to users, cooperative with others, and safer in risking fewer misaligned goals with human users. We discuss how wise AI might be benchmarked, trained, and implemented.

preprint2026arXiv

Learning POMDP World Models from Observations with Language-Model Priors

Whether navigating a building, operating a robot, or playing a game, an agent that acts effectively in an environment must first learn an internal model of how that environment works. Partially-observable Markov decision processes (POMDPs) provide a flexible modeling class for such internal world models, but learning them from observation-action trajectories alone is challenging and typically requires extensive environment interaction. We ask whether language-model priors can reduce costly interaction by leveraging prior knowledge, and introduce \emph{Pinductor} (POMDP-inductor): an LLM proposes candidate POMDP models from a few observation-action trajectories and iteratively refines them to optimize a belief-based likelihood score. Despite using strictly less information, \emph{Pinductor} matches the performance and sample efficiency of LLM-based POMDP learning methods that assume privileged access to the hidden state, while significantly surpassing the sample efficiency of tabular POMDP baselines. Further results show that performance scales with LLM capability and degrades gracefully as semantic information about the environment is withheld. Together, these results position language-model priors as a practical tool for sample-efficient world-model learning under partial observability, and a step toward generalist agents in real-world environments. Code is available at https://github.com/atomresearch/pinductor.

preprint2026arXiv

Mechanism Design Is Not Enough: Prosocial Agents for Cooperative AI

Ensuring that AI agents behave safely and beneficially when interacting with other parties has emerged as one of the central challenges of modern AI safety. While mechanism design, as the theory of designing rules to align individual and collective objectives, can incentivize cooperative behavior, it is still an open question whether it alone is sufficient to maximize LLM agents' social welfare. This work proves that the answer is negative: drawing from incomplete contract theory, we formally show that when contracts cannot distinguish all relevant future contingencies, there is a strictly positive welfare loss that no realistic mechanism can eliminate. We show that prosocial agents, who weigh others' welfare alongside their own, can close this gap and achieve outcomes that are socially superior and individually beneficial. Experimentally, we show that in multi-agent resource-allocation environments and canonical social dilemmas where agents are powered by large language models, prosociality is beneficial. The implication for AI safety is clear: to enable cooperative interactions at scale, designing adequate mechanisms is not sufficient; agents must be built to be intrinsically prosocial.

preprint2026arXiv

Neural Posterior Estimation of Terrain Parameters from Radar Sounder Data

Radar sounders are electromagnetic instruments that can probe deep into the subsurface of Earth and other planetary bodies by processing the echo of transmitted radar waves. Conventional approaches for analyzing such data rely on approximate assumptions and often produce point estimates that ignore parameter correlations as well as galactic and measurement noise. We propose a simulation-based inference approach to terrain parameter inversion from radar sounder data, where synthetic observations from a GPU-based simulator are used to train a neural network-based density estimator for neural posterior estimation (NPE). By explicitly conditioning on reference surface assumptions, the proposed framework allows systematic evaluation of posterior robustness to reference surface variability. We demonstrate that our NPE model is well calibrated on simulated data and transferable to real Mars radar profiles, where we analyze terrain parameters using literature-informed reference values.

preprint2026arXiv

Riemannian Networks over Full-Rank Correlation Matrices

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

preprint2026arXiv

TabPFN-3: Technical Report

Tabular data underpins most high-value prediction problems in science and industry, and TabPFN has driven the foundation model revolution for this modality. Designed with feedback from our users, TabPFN-3 builds on this foundation to scale state-of-the-art performance to datasets with 1M training rows and substantially reduce training and inference time. Pretrained exclusively on synthetic data from our prior, TabPFN-3 dramatically pushes the frontier of tabular prediction and brings substantial gains on time series, relational, and tabular-text data. On the standard tabular benchmark TabArena, a forward pass of TabPFN-3 outperforms all other models, including tuned and ensembled baselines, by a significant margin, and pareto-dominates the speed/performance frontier. On more diverse datasets, TabPFN-3 ranks first on datasets with many classes, and beats 8-hour-tuned gradient-boosted-tree baselines on datasets up to 1M training rows and 200 features. TabPFN-3 introduces test-time compute scaling to tabular foundation models. Our API offering TabPFN-3-Plus (Thinking) exploits this to beat all non-TabPFN models by over 200 Elo on TabArena, rising to 420 Elo on the largest data subset, and outperforms AutoGluon 1.5 extreme while being 10x faster, without using LLMs, real data, internet search or any other model besides TabPFN. TabPFN-3 extends the capabilities of our models, enabling SOTA prediction on relational data (new SOTA foundation model on RelBenchV1) and tabular-text data (SOTA on TabSTAR via TabPFN-3-Plus); and improves existing integrations: a specialized checkpoint, TabPFN-TS-3, ranks 2nd on the time-series benchmark fev-bench, and SHAP-value computation is up to 120x faster. TabPFN-3 achieves this performance while being up to 20x faster than TabPFN-2.5. In addition, a reduced KV cache and row-chunking scale to 1M rows on one H100 with fast inference speed.

preprint2026arXiv

Trustworthy AI Suffers from Invariance Conflicts and Causality is The Solution

As artificial intelligence (AI), including machine learning (ML) models and foundation models (FMs), is increasingly deployed in high-stakes domains, ensuring their trustworthiness has become a central challenge. However, the core trustworthy AI objectives, such as fairness, robustness, privacy, and explainability, are hard to achieve simultaneously, especially while preserving utility. This position paper argues that causality is necessary to understand and balance trade-offs in performance and multiple objectives of trustworthy AI. We ground our arguments in re-interpreting trustworthy AI trade-offs as incompatible invariance requirements under different changes to the data-generating process. We then illustrate that causality provides a unifying framework for understanding how trade-offs in trustworthy AI arise, and how they can be softened or resolved through selective invariance. This perspective applies to both classical ML models and large-scale FMs. Our paper discusses how causal assumptions may be applied explicitly or implicitly in modern large-scale systems. Finally, we outline open challenges and opportunities for using causality to build more trustworthy AI.

preprint2023arXiv

AutoML Two-Sample Test

Two-sample tests are important in statistics and machine learning, both as tools for scientific discovery as well as to detect distribution shifts. This led to the development of many sophisticated test procedures going beyond the standard supervised learning frameworks, whose usage can require specialized knowledge about two-sample testing. We use a simple test that takes the mean discrepancy of a witness function as the test statistic and prove that minimizing a squared loss leads to a witness with optimal testing power. This allows us to leverage recent advancements in AutoML. Without any user input about the problems at hand, and using the same method for all our experiments, our AutoML two-sample test achieves competitive performance on a diverse distribution shift benchmark as well as on challenging two-sample testing problems. We provide an implementation of the AutoML two-sample test in the Python package autotst.

preprint2023arXiv

Exploring the Latent Space of Autoencoders with Interventional Assays

Autoencoders exhibit impressive abilities to embed the data manifold into a low-dimensional latent space, making them a staple of representation learning methods. However, without explicit supervision, which is often unavailable, the representation is usually uninterpretable, making analysis and principled progress challenging. We propose a framework, called latent responses, which exploits the locally contractive behavior exhibited by variational autoencoders to explore the learned manifold. More specifically, we develop tools to probe the representation using interventions in the latent space to quantify the relationships between latent variables. We extend the notion of disentanglement to take the learned generative process into account and consequently avoid the limitations of existing metrics that may rely on spurious correlations. Our analyses underscore the importance of studying the causal structure of the representation to improve performance on downstream tasks such as generation, interpolation, and inference of the factors of variation.

preprint2022arXiv

A machine learning route between band mapping and band structure

Electronic band structure (BS) and crystal structure are the two complementary identifiers of solid state materials. While convenient instruments and reconstruction algorithms have made large, empirical, crystal structure databases possible, extracting quasiparticle dispersion (closely related to BS) from photoemission band mapping data is currently limited by the available computational methods. To cope with the growing size and scale of photoemission data, we develop a pipeline including probabilistic machine learning and the associated data processing, optimization and evaluation methods for band structure reconstruction, leveraging theoretical calculations. The pipeline reconstructs all 14 valence bands of a semiconductor and shows excellent performance on benchmarks and other materials datasets. The reconstruction uncovers previously inaccessible momentum-space structural information on both global and local scales, while realizing a path towards integration with materials science databases. Our approach illustrates the potential of combining machine learning and domain knowledge for scalable feature extraction in multidimensional data.

preprint2022arXiv

A Witness Two-Sample Test

The Maximum Mean Discrepancy (MMD) has been the state-of-the-art nonparametric test for tackling the two-sample problem. Its statistic is given by the difference in expectations of the witness function, a real-valued function defined as a weighted sum of kernel evaluations on a set of basis points. Typically the kernel is optimized on a training set, and hypothesis testing is performed on a separate test set to avoid overfitting (i.e., control type-I error). That is, the test set is used to simultaneously estimate the expectations and define the basis points, while the training set only serves to select the kernel and is discarded. In this work, we propose to use the training data to also define the weights and the basis points for better data efficiency. We show that 1) the new test is consistent and has a well-controlled type-I error; 2) the optimal witness function is given by a precision-weighted mean in the reproducing kernel Hilbert space associated with the kernel; and 3) the test power of the proposed test is comparable or exceeds that of the MMD and other modern tests, as verified empirically on challenging synthetic and real problems (e.g., Higgs data).

preprint2022arXiv

Action-Sufficient State Representation Learning for Control with Structural Constraints

Perceived signals in real-world scenarios are usually high-dimensional and noisy, and finding and using their representation that contains essential and sufficient information required by downstream decision-making tasks will help improve computational efficiency and generalization ability in the tasks. In this paper, we focus on partially observable environments and propose to learn a minimal set of state representations that capture sufficient information for decision-making, termed \textit{Action-Sufficient state Representations} (ASRs). We build a generative environment model for the structural relationships among variables in the system and present a principled way to characterize ASRs based on structural constraints and the goal of maximizing cumulative reward in policy learning. We then develop a structured sequential Variational Auto-Encoder to estimate the environment model and extract ASRs. Our empirical results on CarRacing and VizDoom demonstrate a clear advantage of learning and using ASRs for policy learning. Moreover, the estimated environment model and ASRs allow learning behaviors from imagined outcomes in the compact latent space to improve sample efficiency.

preprint2022arXiv

Adversarially Robust Kernel Smoothing

We propose a scalable robust learning algorithm combining kernel smoothing and robust optimization. Our method is motivated by the convex analysis perspective of distributionally robust optimization based on probability metrics, such as the Wasserstein distance and the maximum mean discrepancy. We adapt the integral operator using supremal convolution in convex analysis to form a novel function majorant used for enforcing robustness. Our method is simple in form and applies to general loss functions and machine learning models. Exploiting a connection with optimal transport, we prove theoretical guarantees for certified robustness under distribution shift. Furthermore, we report experiments with general machine learning models, such as deep neural networks, to demonstrate competitive performance with the state-of-the-art certifiable robust learning algorithms based on the Wasserstein distance.

preprint2022arXiv

Causal Inference Through the Structural Causal Marginal Problem

We introduce an approach to counterfactual inference based on merging information from multiple datasets. We consider a causal reformulation of the statistical marginal problem: given a collection of marginal structural causal models (SCMs) over distinct but overlapping sets of variables, determine the set of joint SCMs that are counterfactually consistent with the marginal ones. We formalise this approach for categorical SCMs using the response function formulation and show that it reduces the space of allowed marginal and joint SCMs. Our work thus highlights a new mode of falsifiability through additional variables, in contrast to the statistical one via additional data.

preprint2022arXiv

CausalAdv: Adversarial Robustness through the Lens of Causality

The adversarial vulnerability of deep neural networks has attracted significant attention in machine learning. As causal reasoning has an instinct for modelling distribution change, it is essential to incorporate causality into analyzing this specific type of distribution change induced by adversarial attacks. However, causal formulations of the intuition of adversarial attacks and the development of robust DNNs are still lacking in the literature. To bridge this gap, we construct a causal graph to model the generation process of adversarial examples and define the adversarial distribution to formalize the intuition of adversarial attacks. From the causal perspective, we study the distinction between the natural and adversarial distribution and conclude that the origin of adversarial vulnerability is the focus of models on spurious correlations. Inspired by the causal understanding, we propose the Causal inspired Adversarial distribution alignment method, CausalAdv, to eliminate the difference between natural and adversarial distributions by considering spurious correlations. Extensive experiments demonstrate the efficacy of the proposed method. Our work is the first attempt towards using causality to understand and mitigate the adversarial vulnerability.

preprint2022arXiv

Compositional Multi-Object Reinforcement Learning with Linear Relation Networks

Although reinforcement learning has seen remarkable progress over the last years, solving robust dexterous object-manipulation tasks in multi-object settings remains a challenge. In this paper, we focus on models that can learn manipulation tasks in fixed multi-object settings and extrapolate this skill zero-shot without any drop in performance when the number of objects changes. We consider the generic task of bringing a specific cube out of a set to a goal position. We find that previous approaches, which primarily leverage attention and graph neural network-based architectures, do not generalize their skills when the number of input objects changes while scaling as $K^2$. We propose an alternative plug-and-play module based on relational inductive biases to overcome these limitations. Besides exceeding performances in their training environment, we show that our approach, which scales linearly in $K$, allows agents to extrapolate and generalize zero-shot to any new object number.

preprint2022arXiv

From Statistical to Causal Learning

We describe basic ideas underlying research to build and understand artificially intelligent systems: from symbolic approaches via statistical learning to interventional models relying on concepts of causality. Some of the hard open problems of machine learning and AI are intrinsically related to causality, and progress may require advances in our understanding of how to model and infer causality from data.

preprint2022arXiv

Function Classes for Identifiable Nonlinear Independent Component Analysis

Unsupervised learning of latent variable models (LVMs) is widely used to represent data in machine learning. When such models reflect the ground truth factors and the mechanisms mapping them to observations, there is reason to expect that they allow generalization in downstream tasks. It is however well known that such identifiability guaranties are typically not achievable without putting constraints on the model class. This is notably the case for nonlinear Independent Component Analysis, in which the LVM maps statistically independent variables to observations via a deterministic nonlinear function. Several families of spurious solutions fitting perfectly the data, but that do not correspond to the ground truth factors can be constructed in generic settings. However, recent work suggests that constraining the function class of such models may promote identifiability. Specifically, function classes with constraints on their partial derivatives, gathered in the Jacobian matrix, have been proposed, such as orthogonal coordinate transformations (OCT), which impose orthogonality of the Jacobian columns. In the present work, we prove that a subclass of these transformations, conformal maps, is identifiable and provide novel theoretical results suggesting that OCTs have properties that prevent families of spurious solutions to spoil identifiability in a generic setting.

preprint2022arXiv

Generalization and Robustness Implications in Object-Centric Learning

The idea behind object-centric representation learning is that natural scenes can better be modeled as compositions of objects and their relations as opposed to distributed representations. This inductive bias can be injected into neural networks to potentially improve systematic generalization and performance of downstream tasks in scenes with multiple objects. In this paper, we train state-of-the-art unsupervised models on five common multi-object datasets and evaluate segmentation metrics and downstream object property prediction. In addition, we study generalization and robustness by investigating the settings where either a single object is out of distribution -- e.g., having an unseen color, texture, or shape -- or global properties of the scene are altered -- e.g., by occlusions, cropping, or increasing the number of objects. From our experimental study, we find object-centric representations to be useful for downstream tasks and generally robust to most distribution shifts affecting objects. However, when the distribution shift affects the input in a less structured manner, robustness in terms of segmentation and downstream task performance may vary significantly across models and distribution shifts.

preprint2022arXiv

Independent mechanism analysis, a new concept?

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation.

preprint2022arXiv

Learning Neural Causal Models with Active Interventions

Discovering causal structures from data is a challenging inference problem of fundamental importance in all areas of science. The appealing properties of neural networks have recently led to a surge of interest in differentiable neural network-based methods for learning causal structures from data. So far, differentiable causal discovery has focused on static datasets of observational or fixed interventional origin. In this work, we introduce an active intervention targeting (AIT) method which enables a quick identification of the underlying causal structure of the data-generating process. Our method significantly reduces the required number of interactions compared with random intervention targeting and is applicable for both discrete and continuous optimization formulations of learning the underlying directed acyclic graph (DAG) from data. We examine the proposed method across multiple frameworks in a wide range of settings and demonstrate superior performance on multiple benchmarks from simulated to real-world data.

preprint2022arXiv

Leveling Down in Computer Vision: Pareto Inefficiencies in Fair Deep Classifiers

Algorithmic fairness is frequently motivated in terms of a trade-off in which overall performance is decreased so as to improve performance on disadvantaged groups where the algorithm would otherwise be less accurate. Contrary to this, we find that applying existing fairness approaches to computer vision improve fairness by degrading the performance of classifiers across all groups (with increased degradation on the best performing groups). Extending the bias-variance decomposition for classification to fairness, we theoretically explain why the majority of fairness classifiers designed for low capacity models should not be used in settings involving high-capacity models, a scenario common to computer vision. We corroborate this analysis with extensive experimental support that shows that many of the fairness heuristics used in computer vision also degrade performance on the most disadvantaged groups. Building on these insights, we propose an adaptive augmentation strategy that, uniquely, of all methods tested, improves performance for the disadvantaged groups.

preprint2022arXiv

Maximum Mean Discrepancy Distributionally Robust Nonlinear Chance-Constrained Optimization with Finite-Sample Guarantee

This paper is motivated by addressing open questions in distributionally robust chance-constrained programs (DRCCP) using the popular Wasserstein ambiguity sets. Specifically, the computational techniques for those programs typically place restrictive assumptions on the constraint functions and the size of the Wasserstein ambiguity sets is often set using costly cross-validation (CV) procedures or conservative measure concentration bounds. In contrast, we propose a practical DRCCP algorithm using kernel maximum mean discrepancy (MMD) ambiguity sets, which we term MMD-DRCCP, to treat general nonlinear constraints without using ad-hoc reformulation techniques. MMD-DRCCP can handle general nonlinear and non-convex constraints with a proven finite-sample constraint satisfaction guarantee of a dimension-independent $\mathcal{O}(\frac{1}{\sqrt{N}})$ rate, achievable by a practical algorithm. We further propose an efficient bootstrap scheme for constructing sharp MMD ambiguity sets in practice without resorting to CV. Our algorithm is validated numerically on a portfolio optimization problem and a tube-based distributionally robust model predictive control problem with non-convex constraints.

preprint2022arXiv

On Pitfalls of Identifiability in Unsupervised Learning. A Note on: "Desiderata for Representation Learning: A Causal Perspective"

Model identifiability is a desirable property in the context of unsupervised representation learning. In absence thereof, different models may be observationally indistinguishable while yielding representations that are nontrivially related to one another, thus making the recovery of a ground truth generative model fundamentally impossible, as often shown through suitably constructed counterexamples. In this note, we discuss one such construction, illustrating a potential failure case of an identifiability result presented in "Desiderata for Representation Learning: A Causal Perspective" by Wang & Jordan (2021). The construction is based on the theory of nonlinear independent component analysis. We comment on implications of this and other counterexamples for identifiable representation learning.

preprint2022arXiv

On the Adversarial Robustness of Causal Algorithmic Recourse

Algorithmic recourse seeks to provide actionable recommendations for individuals to overcome unfavorable classification outcomes from automated decision-making systems. Recourse recommendations should ideally be robust to reasonably small uncertainty in the features of the individual seeking recourse. In this work, we formulate the adversarially robust recourse problem and show that recourse methods that offer minimally costly recourse fail to be robust. We then present methods for generating adversarially robust recourse for linear and for differentiable classifiers. Finally, we show that regularizing the decision-making classifier to behave locally linearly and to rely more strongly on actionable features facilitates the existence of adversarially robust recourse.

preprint2022arXiv

On the Fairness of Causal Algorithmic Recourse

Algorithmic fairness is typically studied from the perspective of predictions. Instead, here we investigate fairness from the perspective of recourse actions suggested to individuals to remedy an unfavourable classification. We propose two new fairness criteria at the group and individual level, which -- unlike prior work on equalising the average group-wise distance from the decision boundary -- explicitly account for causal relationships between features, thereby capturing downstream effects of recourse actions performed in the physical world. We explore how our criteria relate to others, such as counterfactual fairness, and show that fairness of recourse is complementary to fairness of prediction. We study theoretically and empirically how to enforce fair causal recourse by altering the classifier and perform a case study on the Adult dataset. Finally, we discuss whether fairness violations in the data generating process revealed by our criteria may be better addressed by societal interventions as opposed to constraints on the classifier.

preprint2022arXiv

Original or Translated? A Causal Analysis of the Impact of Translationese on Machine Translation Performance

Human-translated text displays distinct features from naturally written text in the same language. This phenomena, known as translationese, has been argued to confound the machine translation (MT) evaluation. Yet, we find that existing work on translationese neglects some important factors and the conclusions are mostly correlational but not causal. In this work, we collect CausalMT, a dataset where the MT training data are also labeled with the human translation directions. We inspect two critical factors, the train-test direction match (whether the human translation directions in the training and test sets are aligned), and data-model direction match (whether the model learns in the same direction as the human translation direction in the dataset). We show that these two factors have a large causal effect on the MT performance, in addition to the test-model direction mismatch highlighted by existing work on the impact of translationese. In light of our findings, we provide a set of suggestions for MT training and evaluation. Our code and data are at https://github.com/EdisonNi-hku/CausalMT

preprint2022arXiv

Phenomenology of Double Descent in Finite-Width Neural Networks

`Double descent' delineates the generalization behaviour of models depending on the regime they belong to: under- or over-parameterized. The current theoretical understanding behind the occurrence of this phenomenon is primarily based on linear and kernel regression models -- with informal parallels to neural networks via the Neural Tangent Kernel. Therefore such analyses do not adequately capture the mechanisms behind double descent in finite-width neural networks, as well as, disregard crucial components -- such as the choice of the loss function. We address these shortcomings by leveraging influence functions in order to derive suitable expressions of the population loss and its lower bound, while imposing minimal assumptions on the form of the parametric model. Our derived bounds bear an intimate connection with the spectrum of the Hessian at the optimum, and importantly, exhibit a double descent behaviour at the interpolation threshold. Building on our analysis, we further investigate how the loss function affects double descent -- and thus uncover interesting properties of neural networks and their Hessian spectra near the interpolation threshold.

preprint2022arXiv

Physical Derivatives: Computing policy gradients by physical forward-propagation

Model-free and model-based reinforcement learning are two ends of a spectrum. Learning a good policy without a dynamic model can be prohibitively expensive. Learning the dynamic model of a system can reduce the cost of learning the policy, but it can also introduce bias if it is not accurate. We propose a middle ground where instead of the transition model, the sensitivity of the trajectories with respect to the perturbation of the parameters is learned. This allows us to predict the local behavior of the physical system around a set of nominal policies without knowing the actual model. We assay our method on a custom-built physical robot in extensive experiments and show the feasibility of the approach in practice. We investigate potential challenges when applying our method to physical systems and propose solutions to each of them.

preprint2022arXiv

Probing the Robustness of Independent Mechanism Analysis for Representation Learning

One aim of representation learning is to recover the original latent code that generated the data, a task which requires additional information or inductive biases. A recently proposed approach termed Independent Mechanism Analysis (IMA) postulates that each latent source should influence the observed mixtures independently, complementing standard nonlinear independent component analysis, and taking inspiration from the principle of independent causal mechanisms. While it was shown in theory and experiments that IMA helps recovering the true latents, the method's performance was so far only characterized when the modeling assumptions are exactly satisfied. Here, we test the method's robustness to violations of the underlying assumptions. We find that the benefits of IMA-based regularization for recovering the true sources extend to mixing functions with various degrees of violation of the IMA principle, while standard regularizers do not provide the same merits. Moreover, we show that unregularized maximum likelihood recovers mixing functions which systematically deviate from the IMA principle, and provide an argument elucidating the benefits of IMA-based regularization.

preprint2022arXiv

Real Robot Challenge: A Robotics Competition in the Cloud

Dexterous manipulation remains an open problem in robotics. To coordinate efforts of the research community towards tackling this problem, we propose a shared benchmark. We designed and built robotic platforms that are hosted at MPI for Intelligent Systems and can be accessed remotely. Each platform consists of three robotic fingers that are capable of dexterous object manipulation. Users are able to control the platforms remotely by submitting code that is executed automatically, akin to a computational cluster. Using this setup, i) we host robotics competitions, where teams from anywhere in the world access our platforms to tackle challenging tasks ii) we publish the datasets collected during these competitions (consisting of hundreds of robot hours), and iii) we give researchers access to these platforms for their own projects.

preprint2022arXiv

Regret Bounds for Gaussian-Process Optimization in Large Domains

The goal of this paper is to characterize Gaussian-Process optimization in the setting where the function domain is large relative to the number of admissible function evaluations, i.e., where it is impossible to find the global optimum. We provide upper bounds on the suboptimality (Bayesian simple regret) of the solution found by optimization strategies that are closely related to the widely used expected improvement (EI) and upper confidence bound (UCB) algorithms. These regret bounds illuminate the relationship between the number of evaluations, the domain size (i.e. cardinality of finite domains / Lipschitz constant of the covariance function in continuous domains), and the optimality of the retrieved function value. In particular, we show that even when the number of evaluations is far too small to find the global optimum, we can find nontrivial function values (e.g. values that achieve a certain ratio with the optimal value).

preprint2022arXiv

Resampling Base Distributions of Normalizing Flows

Normalizing flows are a popular class of models for approximating probability distributions. However, their invertible nature limits their ability to model target distributions whose support have a complex topological structure, such as Boltzmann distributions. Several procedures have been proposed to solve this problem but many of them sacrifice invertibility and, thereby, tractability of the log-likelihood as well as other desirable properties. To address these limitations, we introduce a base distribution for normalizing flows based on learned rejection sampling, allowing the resulting normalizing flow to model complicated distributions without giving up bijectivity. Furthermore, we develop suitable learning algorithms using both maximizing the log-likelihood and the optimization of the Kullback-Leibler divergence, and apply them to various sample problems, i.e. approximating 2D densities, density estimation of tabular data, image generation, and modeling Boltzmann distributions. In these experiments our method is competitive with or outperforms the baselines.

preprint2022arXiv

Score matching enables causal discovery of nonlinear additive noise models

This paper demonstrates how to recover causal graphs from the score of the data distribution in non-linear additive (Gaussian) noise models. Using score matching algorithms as a building block, we show how to design a new generation of scalable causal discovery methods. To showcase our approach, we also propose a new efficient method for approximating the score's Jacobian, enabling to recover the causal graph. Empirically, we find that the new algorithm, called SCORE, is competitive with state-of-the-art causal discovery methods while being significantly faster.

preprint2022arXiv

Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style

Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice.

preprint2022arXiv

Source-Free Adaptation to Measurement Shift via Bottom-Up Feature Restoration

Source-free domain adaptation (SFDA) aims to adapt a model trained on labelled data in a source domain to unlabelled data in a target domain without access to the source-domain data during adaptation. Existing methods for SFDA leverage entropy-minimization techniques which: (i) apply only to classification; (ii) destroy model calibration; and (iii) rely on the source model achieving a good level of feature-space class-separation in the target domain. We address these issues for a particularly pervasive type of domain shift called measurement shift which can be resolved by restoring the source features rather than extracting new ones. In particular, we propose Feature Restoration (FR) wherein we: (i) store a lightweight and flexible approximation of the feature distribution under the source data; and (ii) adapt the feature-extractor such that the approximate feature distribution under the target data realigns with that saved on the source. We additionally propose a bottom-up training scheme which boosts performance, which we call Bottom-Up Feature Restoration (BUFR). On real and synthetic data, we demonstrate that BUFR outperforms existing SFDA methods in terms of accuracy, calibration, and data efficiency, while being less reliant on the performance of the source model in the target domain.

preprint2022arXiv

Structural Causal 3D Reconstruction

This paper considers the problem of unsupervised 3D object reconstruction from in-the-wild single-view images. Due to ambiguity and intrinsic ill-posedness, this problem is inherently difficult to solve and therefore requires strong regularization to achieve disentanglement of different latent factors. Unlike existing works that introduce explicit regularizations into objective functions, we look into a different space for implicit regularization -- the structure of latent space. Specifically, we restrict the structure of latent space to capture a topological causal ordering of latent factors (i.e., representing causal dependency as a directed acyclic graph). We first show that different causal orderings matter for 3D reconstruction, and then explore several approaches to find a task-dependent causal factor ordering. Our experiments demonstrate that the latent space structure indeed serves as an implicit regularization and introduces an inductive bias beneficial for reconstruction.

preprint2022arXiv

The Role of Pretrained Representations for the OOD Generalization of Reinforcement Learning Agents

Building sample-efficient agents that generalize out-of-distribution (OOD) in real-world settings remains a fundamental unsolved problem on the path towards achieving higher-level cognition. One particularly promising approach is to begin with low-dimensional, pretrained representations of our world, which should facilitate efficient downstream learning and generalization. By training 240 representations and over 10,000 reinforcement learning (RL) policies on a simulated robotic setup, we evaluate to what extent different properties of pretrained VAE-based representations affect the OOD generalization of downstream agents. We observe that many agents are surprisingly robust to realistic distribution shifts, including the challenging sim-to-real case. In addition, we find that the generalization performance of a simple downstream proxy task reliably predicts the generalization performance of our RL agents under a wide range of OOD settings. Such proxy tasks can thus be used to select pretrained representations that will lead to agents that generalize.

preprint2022arXiv

The unpopular Package: a Data-driven Approach to De-trend TESS Full Frame Image Light Curves

The majority of observed pixels on the Transiting Exoplanet Survey Satellite (TESS) are delivered in the form of full frame images (FFI). However, the FFIs contain systematic effects such as pointing jitter and scattered light from the Earth and Moon that must be removed before downstream analysis. We present unpopular, an open-source Python package to de-trend TESS FFI light curves based on the causal pixel model method. Under the assumption that shared flux variations across multiple distant pixels are likely to be systematics, unpopular removes these common (i.e., popular) trends by modeling the systematics in a given pixel's light curve as a linear combination of light curves from many other distant pixels. To prevent overfitting we employ ridge regression and a train-and-test framework where the data points being de-trended are separated from those used to obtain the model coefficients. We also allow for simultaneous fitting with a polynomial model to capture any long-term astrophysical trends. We validate our method by de-trending different sources (e.g., supernova, tidal disruption event, exoplanet-hosting star, fast rotating star) and comparing our light curves to those obtained by other pipelines when appropriate. We also show that unpopular is able to preserve sector-length astrophysical signals, allowing for the extraction of multi-sector light curves from the FFI data. The unpopular source code and tutorials are freely available online.

preprint2022arXiv

Towards Total Recall in Industrial Anomaly Detection

Being able to spot defective parts is a critical component in large-scale industrial manufacturing. A particular challenge that we address in this work is the cold-start problem: fit a model using nominal (non-defective) example images only. While handcrafted solutions per class are possible, the goal is to build systems that work well simultaneously on many different tasks automatically. The best performing approaches combine embeddings from ImageNet models with an outlier detection model. In this paper, we extend on this line of work and propose \textbf{PatchCore}, which uses a maximally representative memory bank of nominal patch-features. PatchCore offers competitive inference times while achieving state-of-the-art performance for both detection and localization. On the challenging, widely used MVTec AD benchmark PatchCore achieves an image-level anomaly detection AUROC score of up to $99.6\%$, more than halving the error compared to the next best competitor. We further report competitive results on two additional datasets and also find competitive results in the few samples regime.\freefootnote{$^*$ Work done during a research internship at Amazon AWS.} Code: github.com/amazon-research/patchcore-inspection.

preprint2022arXiv

Variational Causal Dynamics: Discovering Modular World Models from Interventions

Latent world models allow agents to reason about complex environments with high-dimensional observations. However, adapting to new environments and effectively leveraging previous knowledge remain significant challenges. We present variational causal dynamics (VCD), a structured world model that exploits the invariance of causal mechanisms across environments to achieve fast and modular adaptation. By causally factorising a transition model, VCD is able to identify reusable components across different environments. This is achieved by combining causal discovery and variational inference to learn a latent representation and transition model jointly in an unsupervised manner. Specifically, we optimise the evidence lower bound jointly over a representation model and a transition model structured as a causal graphical model. In evaluations on simulated environments with state and image observations, we show that VCD is able to successfully identify causal variables, and to discover consistent causal structures across different environments. Moreover, given a small number of observations in a previously unseen, intervened environment, VCD is able to identify the sparse changes in the dynamics and to adapt efficiently. In doing so, VCD significantly extends the capabilities of the current state-of-the-art in latent world models while also comparing favourably in terms of prediction accuracy.

preprint2022arXiv

Visual Representation Learning Does Not Generalize Strongly Within the Same Domain

An important component for generalization in machine learning is to uncover underlying latent factors of variation as well as the mechanism through which each factor acts in the world. In this paper, we test whether 17 unsupervised, weakly supervised, and fully supervised representation learning approaches correctly infer the generative factors of variation in simple datasets (dSprites, Shapes3D, MPI3D) from controlled environments, and on our contributed CelebGlow dataset. In contrast to prior robustness work that introduces novel factors of variation during test time, such as blur or other (un)structured noise, we here recompose, interpolate, or extrapolate only existing factors of variation from the training data set (e.g., small and medium-sized objects during training and large objects during testing). Models that learn the correct mechanism should be able to generalize to this benchmark. In total, we train and test 2000+ models and observe that all of them struggle to learn the underlying mechanism regardless of supervision signal and architectural bias. Moreover, the generalization capabilities of all tested models drop significantly as we move from artificial datasets towards more realistic real-world datasets. Despite their inability to identify the correct mechanism, the models are quite modular as their ability to infer other in-distribution factors remains fairly stable, providing only a single factor is out-of-distribution. These results point to an important yet understudied problem of learning mechanistic models of observations that can facilitate generalization.

preprint2021arXiv

A prior-based approximate latent Riemannian metric

Stochastic generative models enable us to capture the geometric structure of a data manifold lying in a high dimensional space through a Riemannian metric in the latent space. However, its practical use is rather limited mainly due to inevitable complexity. In this work we propose a surrogate conformal Riemannian metric in the latent space of a generative model that is simple, efficient and robust. This metric is based on a learnable prior that we propose to learn using a basic energy-based model. We theoretically analyze the behavior of the proposed metric and show that it is sensible to use in practice. We demonstrate experimentally the efficiency and robustness, as well as the behavior of the new approximate metric. Also, we show the applicability of the proposed methodology for data analysis in the life sciences.

preprint2021arXiv

A survey of algorithmic recourse: definitions, formulations, solutions, and prospects

Machine learning is increasingly used to inform decision-making in sensitive situations where decisions have consequential effects on individuals' lives. In these settings, in addition to requiring models to be accurate and robust, socially relevant values such as fairness, privacy, accountability, and explainability play an important role for the adoption and impact of said technologies. In this work, we focus on algorithmic recourse, which is concerned with providing explanations and recommendations to individuals who are unfavourably treated by automated decision-making systems. We first perform an extensive literature review, and align the efforts of many authors by presenting unified definitions, formulations, and solutions to recourse. Then, we provide an overview of the prospective research directions towards which the community may engage, challenging existing assumptions and making explicit connections to other ethical challenges such as security, privacy, and fairness.

preprint2021arXiv

A theory of independent mechanisms for extrapolation in generative models

Generative models can be trained to emulate complex empirical data, but are they useful to make predictions in the context of previously unobserved environments? An intuitive idea to promote such extrapolation capabilities is to have the architecture of such model reflect a causal graph of the true data generating process, such that one can intervene on each node independently of the others. However, the nodes of this graph are usually unobserved, leading to overparameterization and lack of identifiability of the causal structure. We develop a theoretical framework to address this challenging situation by defining a weaker form of identifiability, based on the principle of independence of mechanisms. We demonstrate on toy examples that classical stochastic gradient descent can hinder the model's extrapolation capabilities, suggesting independence of mechanisms should be enforced explicitly during training. Experiments on deep generative models trained on real world data support these insights and illustrate how the extrapolation capabilities of such models can be leveraged.

preprint2021arXiv

Assaying Large-scale Testing Models to Interpret COVID-19 Case Numbers

Large-scale testing is considered key to assess the state of the current COVID-19 pandemic. Yet, the link between the reported case numbers and the true state of the pandemic remains elusive. We develop mathematical models based on competing hypotheses regarding this link, thereby providing different prevalence estimates based on case numbers, and validate them by predicting SARS-CoV-2-attributed death rate trajectories. Assuming that individuals were tested based solely on a predefined risk of being infectious implies the absolute case numbers reflect the prevalence, but turned out to be a poor predictor, consistently overestimating growth rates at the beginning of two COVID-19 epidemic waves. In contrast, assuming that testing capacity is fully exploited performs better. This leads to using the percent-positive rate as a more robust indicator of epidemic dynamics, however we find it is subject to a saturation phenomenon that needs to be accounted for as the number of tests becomes larger.

preprint2021arXiv

Spatial Context Awareness for Unsupervised Change Detection in Optical Satellite Images

Detecting changes on the ground in multitemporal Earth observation data is one of the key problems in remote sensing. In this paper, we introduce Sibling Regression for Optical Change detection (SiROC), an unsupervised method for change detection in optical satellite images with medium and high resolution. SiROC is a spatial context-based method that models a pixel as a linear combination of its distant neighbors. It uses this model to analyze differences in the pixel and its spatial context-based predictions in subsequent time periods for change detection. We combine this spatial context-based change detection with ensembling over mutually exclusive neighborhoods and transitioning from pixel to object-level changes with morphological operations. SiROC achieves competitive performance for change detection with medium-resolution Sentinel-2 and high-resolution Planetscope imagery on four datasets. Besides accurate predictions without the need for training, SiROC also provides a well-calibrated uncertainty of its predictions. This makes the method especially useful in conjunction with deep-learning based methods for applications such as pseudo-labeling.

preprint2021arXiv

Towards Causal Representation Learning

The two fields of machine learning and graphical causality arose and developed separately. However, there is now cross-pollination and increasing interest in both fields to benefit from the advances of the other. In the present paper, we review fundamental concepts of causal inference and relate them to crucial open problems of machine learning, including transfer and generalization, thereby assaying how causality can contribute to modern machine learning research. This also applies in the opposite direction: we note that most work in causality starts from the premise that the causal variables are given. A central problem for AI and causality is, thus, causal representation learning, the discovery of high-level causal variables from low-level observations. Finally, we delineate some implications of causality for machine learning and propose key research areas at the intersection of both communities.

preprint2021arXiv

TriFinger: An Open-Source Robot for Learning Dexterity

Dexterous object manipulation remains an open problem in robotics, despite the rapid progress in machine learning during the past decade. We argue that a hindrance is the high cost of experimentation on real systems, in terms of both time and money. We address this problem by proposing an open-source robotic platform which can safely operate without human supervision. The hardware is inexpensive (about \SI{5000}[\$]{}) yet highly dynamic, robust, and capable of complex interaction with external objects. The software operates at 1-kilohertz and performs safety checks to prevent the hardware from breaking. The easy-to-use front-end (in C++ and Python) is suitable for real-time control as well as deep reinforcement learning. In addition, the software framework is largely robot-agnostic and can hence be used independently of the hardware proposed herein. Finally, we illustrate the potential of the proposed platform through a number of experiments, including real-time optimal control, deep reinforcement learning from scratch, throwing, and writing.

preprint2020arXiv

A Commentary on the Unsupervised Learning of Disentangled Representations

The goal of the unsupervised learning of disentangled representations is to separate the independent explanatory factors of variation in the data without access to supervision. In this paper, we summarize the results of Locatello et al., 2019, and focus on their implications for practitioners. We discuss the theoretical result showing that the unsupervised learning of disentangled representations is fundamentally impossible without inductive biases and the practical challenges it entails. Finally, we comment on our experimental findings, highlighting the limitations of state-of-the-art approaches and directions for future research.

preprint2020arXiv

A Kernel Mean Embedding Approach to Reducing Conservativeness in Stochastic Programming and Control

We apply kernel mean embedding methods to sample-based stochastic optimization and control. Specifically, we use the reduced-set expansion method as a way to discard sampled scenarios. The effect of such constraint removal is improved optimality and decreased conservativeness. This is achieved by solving a distributional-distance-regularized optimization problem. We demonstrated this optimization formulation is well-motivated in theory, computationally tractable and effective in numerical algorithms.

preprint2020arXiv

A New Distribution-Free Concept for Representing, Comparing, and Propagating Uncertainty in Dynamical Systems with Kernel Probabilistic Programming

This work presents the concept of kernel mean embedding and kernel probabilistic programming in the context of stochastic systems. We propose formulations to represent, compare, and propagate uncertainties for fairly general stochastic dynamics in a distribution-free manner. The new tools enjoy sound theory rooted in functional analysis and wide applicability as demonstrated in distinct numerical examples. The implication of this new concept is a new mode of thinking about the statistical nature of uncertainty in dynamical systems.

preprint2020arXiv

Adaptation and Robust Learning of Probabilistic Movement Primitives

Probabilistic representations of movement primitives open important new possibilities for machine learning in robotics. These representations are able to capture the variability of the demonstrations from a teacher as a probability distribution over trajectories, providing a sensible region of exploration and the ability to adapt to changes in the robot environment. However, to be able to capture variability and correlations between different joints, a probabilistic movement primitive requires the estimation of a larger number of parameters compared to their deterministic counterparts, that focus on modeling only the mean behavior. In this paper, we make use of prior distributions over the parameters of a probabilistic movement primitive to make robust estimates of the parameters with few training instances. In addition, we introduce general purpose operators to adapt movement primitives in joint and task space. The proposed training method and adaptation operators are tested in a coffee preparation and in robot table tennis task. In the coffee preparation task we evaluate the generalization performance to changes in the location of the coffee grinder and brewing chamber in a target area, achieving the desired behavior after only two demonstrations. In the table tennis task we evaluate the hit and return rates, outperforming previous approaches while using fewer task specific heuristics.

preprint2020arXiv

Bayesian Online Prediction of Change Points

Online detection of instantaneous changes in the generative process of a data sequence generally focuses on retrospective inference of such change points without considering their future occurrences. We extend the Bayesian Online Change Point Detection algorithm to also infer the number of time steps until the next change point (i.e., the residual time). This enables to handle observation models which depend on the total segment duration, which is useful to model data sequences with temporal scaling. The resulting inference algorithm for segment detection can be deployed in an online fashion, and we illustrate applications to synthetic and to two medical real-world data sets.

preprint2020arXiv

Causal Discovery from Heterogeneous/Nonstationary Data with Independent Changes

It is commonplace to encounter heterogeneous or nonstationary data, of which the underlying generating process changes across domains or over time. Such a distribution shift feature presents both challenges and opportunities for causal discovery. In this paper, we develop a framework for causal discovery from such data, called Constraint-based causal Discovery from heterogeneous/NOnstationary Data (CD-NOD), to find causal skeleton and directions and estimate the properties of mechanism changes. First, we propose an enhanced constraint-based procedure to detect variables whose local mechanisms change and recover the skeleton of the causal structure over observed variables. Second, we present a method to determine causal orientations by making use of independent changes in the data distribution implied by the underlying causal model, benefiting from information carried by changing distributions. After learning the causal structure, next, we investigate how to efficiently estimate the "driving force" of the nonstationarity of a causal mechanism. That is, we aim to extract from data a low-dimensional representation of changes. The proposed methods are nonparametric, with no hard restrictions on data distributions and causal mechanisms, and do not rely on window segmentation. Furthermore, we find that data heterogeneity benefits causal structure identification even with particular types of confounders. Finally, we show the connection between heterogeneity/nonstationarity and soft intervention in causal discovery. Experimental results on various synthetic and real-world data sets (task-fMRI and stock market data) are presented to demonstrate the efficacy of the proposed methods.

preprint2020arXiv

Crackovid: Optimizing Group Testing

We study the problem usually referred to as group testing in the context of COVID-19. Given $n$ samples taken from patients, how should we select mixtures of samples to be tested, so as to maximize information and minimize the number of tests? We consider both adaptive and non-adaptive strategies, and take a Bayesian approach with a prior both for infection of patients and test errors. We start by proposing a mathematically principled objective, grounded in information theory. We then optimize non-adaptive optimization strategies using genetic algorithms, and leverage the mathematical framework of adaptive sub-modularity to obtain theoretical guarantees for the greedy-adaptive method.

preprint2020arXiv

Disentangling Factors of Variation Using Few Labels

Learning disentangled representations is considered a cornerstone problem in representation learning. Recently, Locatello et al. (2019) demonstrated that unsupervised disentanglement learning without inductive biases is theoretically impossible and that existing inductive biases and unsupervised methods do not allow to consistently learn disentangled representations. However, in many practical settings, one might have access to a limited amount of supervision, for example through manual labeling of (some) factors of variation in a few training examples. In this paper, we investigate the impact of such supervision on state-of-the-art disentanglement methods and perform a large scale study, training over 52000 models under well-defined and reproducible experimental conditions. We observe that a small number of labeled examples (0.01--0.5\% of the data set), with potentially imprecise and incomplete labels, is sufficient to perform model selection on state-of-the-art unsupervised models. Further, we investigate the benefit of incorporating supervision into the training process. Overall, we empirically validate that with little and imprecise supervision it is possible to reliably learn disentangled representations.

preprint2020arXiv

From Variational to Deterministic Autoencoders

Variational Autoencoders (VAEs) provide a theoretically-backed and popular framework for deep generative models. However, learning a VAE from data poses still unanswered theoretical questions and considerable practical challenges. In this work, we propose an alternative framework for generative modeling that is simpler, easier to train, and deterministic, yet has many of the advantages of VAEs. We observe that sampling a stochastic encoder in a Gaussian VAE can be interpreted as simply injecting noise into the input of a deterministic decoder. We investigate how substituting this kind of stochasticity, with other explicit and implicit regularization schemes, can lead to an equally smooth and meaningful latent space without forcing it to conform to an arbitrarily chosen prior. To retrieve a generative mechanism to sample new data, we introduce an ex-post density estimation step that can be readily applied also to existing VAEs, improving their sample quality. We show, in a rigorous empirical study, that the proposed regularized deterministic autoencoders are able to generate samples that are comparable to, or better than, those of VAEs and more powerful alternatives when applied to images as well as to structured data such as molecules. \footnote{An implementation is available at: \url{https://github.com/ParthaEth/Regularized_autoencoders-RAE-}}

preprint2020arXiv

Geometrically Enriched Latent Spaces

A common assumption in generative models is that the generator immerses the latent space into a Euclidean ambient space. Instead, we consider the ambient space to be a Riemannian manifold, which allows for encoding domain knowledge through the associated Riemannian metric. Shortest paths can then be defined accordingly in the latent space to both follow the learned manifold and respect the ambient geometry. Through careful design of the ambient metric we can ensure that shortest paths are well-behaved even for deterministic generators that otherwise would exhibit a misleading bias. Experimentally we show that our approach improves interpretability of learned representations both using stochastic and deterministic generators.

preprint2020arXiv

Learning Dynamical Systems using Local Stability Priors

A coupled computational approach to simultaneously learn a vector field and the region of attraction of an equilibrium point from generated trajectories of the system is proposed. The nonlinear identification leverages the local stability information as a prior on the system, effectively endowing the estimate with this important structural property. In addition, the knowledge of the region of attraction plays an experiment design role by informing the selection of initial conditions from which trajectories are generated and by enabling the use of a Lyapunov function of the system as a regularization term. Numerical results show that the proposed method allows efficient sampling and provides an accurate estimate of the dynamics in an inner approximation of its region of attraction.

preprint2020arXiv

Learning Neural Causal Models from Unknown Interventions

Promising results have driven a recent surge of interest in continuous optimization methods for Bayesian network structure learning from observational data. However, there are theoretical limitations on the identifiability of underlying structures obtained from observational data alone. Interventional data provides much richer information about the underlying data-generating process. However, the extension and application of methods designed for observational data to include interventions is not straightforward and remains an open problem. In this paper we provide a general framework based on continuous optimization and neural networks to create models for the combination of observational and interventional data. The proposed method is even applicable in the challenging and realistic case that the identity of the intervened upon variable is unknown. We examine the proposed method in the setting of graph recovery both de novo and from a partially-known edge set. We establish strong benchmark results on several structure learning tasks, including structure recovery of both synthetic graphs as well as standard graphs from the Bayesian Network Repository.

preprint2020arXiv

Learning to Play Table Tennis From Scratch using Muscular Robots

Dynamic tasks like table tennis are relatively easy to learn for humans but pose significant challenges to robots. Such tasks require accurate control of fast movements and precise timing in the presence of imprecise state estimation of the flying ball and the robot. Reinforcement Learning (RL) has shown promise in learning of complex control tasks from data. However, applying step-based RL to dynamic tasks on real systems is safety-critical as RL requires exploring and failing safely for millions of time steps in high-speed regimes. In this paper, we demonstrate that safe learning of table tennis using model-free Reinforcement Learning can be achieved by using robot arms driven by pneumatic artificial muscles (PAMs). Softness and back-drivability properties of PAMs prevent the system from leaving the safe region of its state space. In this manner, RL empowers the robot to return and smash real balls with 5 m\s and 12m\s on average to a desired landing point. Our setup allows the agent to learn this safety-critical task (i) without safety constraints in the algorithm, (ii) while maximizing the speed of returned balls directly in the reward function (iii) using a stochastic policy that acts directly on the low-level controls of the real system and (iv) trains for thousands of trials (v) from scratch without any prior knowledge. Additionally, we present HYSR, a practical hybrid sim and real training that avoids playing real balls during training by randomly replaying recorded ball trajectories in simulation and applying actions to the real robot. This work is the first to (a) fail-safe learn of a safety-critical dynamic task using anthropomorphic robot arms, (b) learn a precision-demanding problem with a PAM-driven system despite the control challenges and (c) train robots to play table tennis without real balls. Videos and datasets are available at muscularTT.embodied.ml.

preprint2020arXiv

MYND: Unsupervised Evaluation of Novel BCI Control Strategies on Consumer Hardware

Neurophysiological studies are typically conducted in laboratories with limited ecological validity, scalability, and generalizability of findings. This is a significant challenge for the development of brain-computer interfaces (BCIs), which ultimately need to function in unsupervised settings on consumer-grade hardware. We introduce MYND: A framework that couples consumer-grade recording hardware with an easy-to-use application for the unsupervised evaluation of BCI control strategies. Subjects are guided through experiment selection, hardware fitting, recording, and data upload in order to self-administer multi-day studies that include neurophysiological recordings and questionnaires. As a use case, we evaluate two BCI control strategies ("Positive memories" and "Music imagery") in a realistic scenario by combining MYND with a four-channel electroencephalogram (EEG). Thirty subjects recorded 70.4 hours of EEG data with the system at home. The median headset fitting time was 25.9 seconds, and a median signal quality of 90.2% was retained during recordings.Neural activity in both control strategies could be decoded with an average offline accuracy of 68.5% and 64.0% across all days. The repeated unsupervised execution of the same strategy affected performance, which could be tackled by implementing feedback to let subjects switch between strategies or devise new strategies with the platform.

preprint2020arXiv

Orthogonal Structure Search for Efficient Causal Discovery from Observational Data

The problem of inferring the direct causal parents of a response variable among a large set of explanatory variables is of high practical importance in many disciplines. Recent work exploits stability of regression coefficients or invariance properties of models across different experimental conditions for reconstructing the full causal graph. These approaches generally do not scale well with the number of the explanatory variables and are difficult to extend to nonlinear relationships. Contrary to existing work, we propose an approach which even works for observational data alone, while still offering theoretical guarantees including the case of partially nonlinear relationships. Our algorithm requires only one estimation for each variable and in our experiments we apply our causal discovery algorithm even to large graphs, demonstrating significant improvements compared to well established approaches.

preprint2020arXiv

S2RMs: Spatially Structured Recurrent Modules

Capturing the structure of a data-generating process by means of appropriate inductive biases can help in learning models that generalize well and are robust to changes in the input distribution. While methods that harness spatial and temporal structures find broad application, recent work has demonstrated the potential of models that leverage sparse and modular structure using an ensemble of sparingly interacting modules. In this work, we take a step towards dynamic models that are capable of simultaneously exploiting both modular and spatiotemporal structures. We accomplish this by abstracting the modeled dynamical system as a collection of autonomous but sparsely interacting sub-systems. The sub-systems interact according to a topology that is learned, but also informed by the spatial structure of the underlying real-world system. This results in a class of models that are well suited for modeling the dynamics of systems that only offer local views into their state, along with corresponding spatial locations of those views. On the tasks of video prediction from cropped frames and multi-agent world modeling from partial observations in the challenging Starcraft2 domain, we find our models to be more robust to the number of available views and better capable of generalization to novel tasks without additional training, even when compared against strong baselines that perform equally well or better on the training distribution.

preprint2020arXiv

Semi-Supervised Learning, Causality and the Conditional Cluster Assumption

While the success of semi-supervised learning (SSL) is still not fully understood, Schölkopf et al. (2012) have established a link to the principle of independent causal mechanisms. They conclude that SSL should be impossible when predicting a target variable from its causes, but possible when predicting it from its effects. Since both these cases are somewhat restrictive, we extend their work by considering classification using cause and effect features at the same time, such as predicting disease from both risk factors and symptoms. While standard SSL exploits information contained in the marginal distribution of all inputs (to improve the estimate of the conditional distribution of the target given inputs), we argue that in our more general setting we should use information in the conditional distribution of effect features given causal features. We explore how this insight generalises the previous understanding, and how it relates to and can be exploited algorithmically for SSL.

preprint2020arXiv

SLEIPNIR: Deterministic and Provably Accurate Feature Expansion for Gaussian Process Regression with Derivatives

Gaussian processes are an important regression tool with excellent analytic properties which allow for direct integration of derivative observations. However, vanilla GP methods scale cubically in the amount of observations. In this work, we propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We then prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior. To furthermore illustrate the practical applicability of our method, we then apply it to ODIN, a recently developed algorithm for ODE parameter inference. In an extensive experiments section, all results are empirically validated, demonstrating the speed, accuracy, and practical applicability of this approach.

preprint2020arXiv

Testing Goodness of Fit of Conditional Density Models with Kernels

We propose two nonparametric statistical tests of goodness of fit for conditional distributions: given a conditional probability density function $p(y|x)$ and a joint sample, decide whether the sample is drawn from $p(y|x)r_x(x)$ for some density $r_x$. Our tests, formulated with a Stein operator, can be applied to any differentiable conditional density model, and require no knowledge of the normalizing constant. We show that 1) our tests are consistent against any fixed alternative conditional model; 2) the statistics can be estimated easily, requiring no density estimation as an intermediate step; and 3) our second test offers an interpretable test result providing insight on where the conditional model does not fit well in the domain of the covariate. We demonstrate the interpretability of our test on a task of modeling the distribution of New York City's taxi drop-off location given a pick-up point. To our knowledge, our work is the first to propose such conditional goodness-of-fit tests that simultaneously have all these desirable properties.

preprint2020arXiv

Towards causal generative scene models via competition of experts

Learning how to model complex scenes in a modular way with recombinable components is a pre-requisite for higher-order reasoning and acting in the physical world. However, current generative models lack the ability to capture the inherently compositional and layered nature of visual scenes. While recent work has made progress towards unsupervised learning of object-based scene representations, most models still maintain a global representation space (i.e., objects are not explicitly separated), and cannot generate scenes with novel object arrangement and depth ordering. Here, we present an alternative approach which uses an inductive bias encouraging modularity by training an ensemble of generative models (experts). During training, experts compete for explaining parts of a scene, and thus specialise on different object classes, with objects being identified as parts that re-occur across multiple scenes. Our model allows for controllable sampling of individual objects and recombination of experts in physically plausible ways. In contrast to other methods, depth layering and occlusion are handled correctly, moving this approach closer to a causal generative scene model. Experiments on simple toy data qualitatively demonstrate the conceptual advantages of the proposed approach.

preprint2020arXiv

Worst-Case Risk Quantification under Distributional Ambiguity using Kernel Mean Embedding in Moment Problem

In order to anticipate rare and impactful events, we propose to quantify the worst-case risk under distributional ambiguity using a recent development in kernel methods -- the kernel mean embedding. Specifically, we formulate the generalized moment problem whose ambiguity set (i.e., the moment constraint) is described by constraints in the associated reproducing kernel Hilbert space in a nonparametric manner. We then present the tractable approximation and its theoretical justification. As a concrete application, we numerically test the proposed method in characterizing the worst-case constraint violation probability in the context of a constrained stochastic control system.

preprint2019arXiv

Multidimensional Contrast Limited Adaptive Histogram Equalization

Contrast enhancement is an important preprocessing technique for improving the performance of downstream tasks in image processing and computer vision. Among the existing approaches based on nonlinear histogram transformations, contrast limited adaptive histogram equalization (CLAHE) is a popular choice for dealing with 2D images obtained in natural and scientific settings. The recent hardware upgrade in data acquisition systems results in significant increase in data complexity, including their sizes and dimensions. Measurements of densely sampled data higher than three dimensions, usually composed of 3D data as a function of external parameters, are becoming commonplace in various applications in the natural sciences and engineering. The initial understanding of these complex multidimensional datasets often requires human intervention through visual examination, which may be hampered by the varying levels of contrast permeating through the dimensions. We show both qualitatively and quantitatively that using our multidimensional extension of CLAHE (MCLAHE) simultaneously on all dimensions of the datasets allows better visualization and discernment of multidimensional image features, as demonstrated using cases from 4D photoemission spectroscopy and fluorescence microscopy. Our implementation of multidimensional CLAHE in Tensorflow is publicly accessible and supports parallelization with multiple CPUs and various other hardware accelerators, including GPUs.

preprint2019arXiv

Reliable Real Time Ball Tracking for Robot Table Tennis

Robot table tennis systems require a vision system that can track the ball position with low latency and high sampling rate. Altering the ball to simplify the tracking using for instance infrared coating changes the physics of the ball trajectory. As a result, table tennis systems use custom tracking systems to track the ball based on heuristic algorithms respecting the real time constrains applied to RGB images captured with a set of cameras. However, these heuristic algorithms often report erroneous ball positions, and the table tennis policies typically need to incorporate additional heuristics to detect and possibly correct outliers. In this paper, we propose a vision system for object detection and tracking that focus on reliability while providing real time performance. Our assumption is that by using multiple cameras, we can find and discard the errors obtained in the object detection phase by checking for consistency with the positions reported by other cameras. We provide an open source implementation of the proposed tracking system to simplify future research in robot table tennis or related tracking applications with strong real time requirements. We evaluate the proposed system thoroughly in simulation and in the real system, outperforming previous work. Furthermore, we show that the accuracy and robustness of the proposed system increases as more cameras are added. Finally, we evaluate the table tennis playing performance of an existing method in the real robot using the proposed vision system. We measure a slight increase in performance compared to a previous vision system even after removing all the heuristics previously present to filter out erroneous ball observations.

preprint2017arXiv

Causal Consistency of Structural Equation Models

Complex systems can be modelled at various levels of detail. Ideally, causal models of the same system should be consistent with one another in the sense that they agree in their predictions of the effects of interventions. We formalise this notion of consistency in the case of Structural Equation Models (SEMs) by introducing exact transformations between SEMs. This provides a general language to consider, for instance, the different levels of description in the following three scenarios: (a) models with large numbers of variables versus models in which the `irrelevant' or unobservable variables have been marginalised out; (b) micro-level models versus macro-level models in which the macro-variables are aggregate features of the micro-variables; (c) dynamical time series models versus models of their stationary behaviour. Our analysis stresses the importance of well specified interventions in the causal modelling process and sheds light on the interpretation of cyclic SEMs.

preprint2016arXiv

A Causal, Data-Driven Approach to Modeling the Kepler Data

Astronomical observations are affected by several kinds of noise, each with its own causal source; there is photon noise, stochastic source variability, and residuals coming from imperfect calibration of the detector or telescope. The precision of NASA Kepler photometry for exoplanet science---the most precise photometric measurements of stars ever made---appears to be limited by unknown or untracked variations in spacecraft pointing and temperature, and unmodeled stellar variability. Here we present the Causal Pixel Model (CPM) for Kepler data, a data-driven model intended to capture variability but preserve transit signals. The CPM works at the pixel level so that it can capture very fine-grained information about the variation of the spacecraft. The CPM predicts each target pixel value from a large number of pixels of other stars sharing the instrument variabilities while not containing any information on possible transits in the target star. In addition, we use the target star's future and past (auto-regression). By appropriately separating, for each data point, the data into training and test sets, we ensure that information about any transit will be perfectly isolated from the model. The method has four hyper-parameters (the number of predictor stars, the auto-regressive window size, and two L2-regularization amplitudes for model components), which we set by cross-validation. We determine a generic set of hyper-parameters that works well for most of the stars and apply the method to a corresponding set of target stars. We find that we can consistently outperform (for the purposes of exoplanet detection) the Kepler Pre-search Data Conditioning (PDC) method for exoplanet discovery.

preprint2016arXiv

Depth Estimation Through a Generative Model of Light Field Synthesis

Light field photography captures rich structural information that may facilitate a number of traditional image processing and computer vision tasks. A crucial ingredient in such endeavors is accurate depth recovery. We present a novel framework that allows the recovery of a high quality continuous depth map from light field data. To this end we propose a generative model of a light field that is fully parametrized by its corresponding depth map. The model allows for the integration of powerful regularization techniques such as a non-local means prior, facilitating accurate depth map estimation.

preprint2016arXiv

Discovery and Visualization of Nonstationary Causal Models

It is commonplace to encounter nonstationary data, of which the underlying generating process may change over time or across domains. The nonstationarity presents both challenges and opportunities for causal discovery. In this paper we propose a principled framework to handle nonstationarity, and develop some methods to address three important questions. First, we propose an enhanced constraint-based method to detect variables whose local mechanisms are nonstationary and recover the skeleton of the causal structure over observed variables. Second, we present a way to determine some causal directions by taking advantage of information carried by changing distributions. Third, we develop a method for visualizing the nonstationarity of causal modules. Experimental results on various synthetic and real-world data sets are presented to demonstrate the efficacy of our methods.

preprint2016arXiv

Kernel Mean Shrinkage Estimators

A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in RKHSs. Given a finite sample, an empirical average has been used commonly as a standard estimator of the true kernel mean. Despite a widespread use of this estimator, we show that it can be improved thanks to the well-known Stein phenomenon. We propose a new family of estimators called kernel mean shrinkage estimators (KMSEs), which benefit from both theoretical justifications and good empirical performance. The results demonstrate that the proposed estimators outperform the standard one, especially in a "large d, small n" paradigm.

preprint2016arXiv

Kernel-based Tests for Joint Independence

We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.

preprint2016arXiv

Recovery of non-linear cause-effect relationships from linearly mixed neuroimaging data

Causal inference concerns the identification of cause-effect relationships between variables. However, often only linear combinations of variables constitute meaningful causal variables. For example, recovering the signal of a cortical source from electroencephalography requires a well-tuned combination of signals recorded at multiple electrodes. We recently introduced the MERLiN (Mixture Effect Recovery in Linear Networks) algorithm that is able to recover, from an observed linear mixture, a causal variable that is a linear effect of another given variable. Here we relax the assumption of this cause-effect relationship being linear and present an extended algorithm that can pick up non-linear cause-effect relationships. Thus, the main contribution is an algorithm (and ready to use code) that has broader applicability and allows for a richer model class. Furthermore, a comparative analysis indicates that the assumption of linear cause-effect relationships is not restrictive in analysing electroencephalographic data.

preprint2016arXiv

Screening Rules for Convex Problems

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather information on the optimal solution. In the second step, this information is used to produce screening rules, i.e. safely identifying unimportant weight variables of the optimal solution. Our general framework leads to a large variety of useful existing as well as new screening rules for many applications. For example, we provide new screening rules for general simplex and $L_1$-constrained problems, Elastic Net, squared-loss Support Vector Machines, minimum enclosing ball, as well as structured norm regularized problems, such as group lasso.

preprint2016arXiv

The Arrow of Time in Multivariate Time Series

We prove that a time series satisfying a (linear) multivariate autoregressive moving average (VARMA) model satisfies the same model assumption in the reversed time direction, too, if all innovations are normally distributed. This reversibility breaks down if the innovations are non-Gaussian. This means that under the assumption of a VARMA process with non-Gaussian noise, the arrow of time becomes detectable. Our work thereby provides a theoretic justification of an algorithm that has been used for inferring the direction of video snippets. We present a slightly modified practical algorithm that estimates the time direction for a given sample and prove its consistency. We further investigate how the performance of the algorithm depends on sample size, number of dimensions of the time series and the order of the process. An application to real world data from economics shows that considering multivariate processes instead of univariate processes can be beneficial for estimating the time direction. Our result extends earlier work on univariate time series. It relates to the concept of causal inference, where recent methods exploit non-Gaussianity of the error terms for causal structure learning.

preprint2016arXiv

The population of long-period transiting exoplanets

The Kepler Mission has discovered thousands of exoplanets and revolutionized our understanding of their population. This large, homogeneous catalog of discoveries has enabled rigorous studies of the occurrence rate of exoplanets and planetary systems as a function of their physical properties. However, transit surveys like Kepler are most sensitive to planets with orbital periods much shorter than the orbital periods of Jupiter and Saturn, the most massive planets in our Solar System. To address this deficiency, we perform a fully automated search for long-period exoplanets with only one or two transits in the archival Kepler light curves. When applied to the $\sim 40,000$ brightest Sun-like target stars, this search produces 16 long-period exoplanet candidates. Of these candidates, 6 are novel discoveries and 5 are in systems with inner short-period transiting planets. Since our method involves no human intervention, we empirically characterize the detection efficiency of our search. Based on these results, we measure the average occurrence rate of exoplanets smaller than Jupiter with orbital periods in the range 2-25 years to be $2.0\pm0.7$ planets per Sun-like star.

preprint2016arXiv

Unifying distillation and privileged information

Distillation (Hinton et al., 2015) and privileged information (Vapnik & Izmailov, 2015) are two techniques that enable machines to learn from other machines. This paper unifies these two techniques into generalized distillation, a framework to learn from multiple machines and data representations. We provide theoretical and causal insight about the inner workings of generalized distillation, extend it to unsupervised, semisupervised and multitask learning scenarios, and illustrate its efficacy on a variety of numerical simulations on both synthetic and real-world data.

preprint2015arXiv

A systematic search for transiting planets in the K2 data

Photometry of stars from the K2 extension of NASA's Kepler mission is afflicted by systematic effects caused by small (few-pixel) drifts in the telescope pointing and other spacecraft issues. We present a method for searching K2 light curves for evidence of exoplanets by simultaneously fitting for these systematics and the transit signals of interest. This method is more computationally expensive than standard search algorithms but we demonstrate that it can be efficiently implemented and used to discover transit signals. We apply this method to the full Campaign 1 dataset and report a list of 36 planet candidates transiting 31 stars, along with an analysis of the pipeline performance and detection efficiency based on artificial signal injections and recoveries. For all planet candidates, we present posterior distributions on the properties of each system based strictly on the transit observables.

preprint2015arXiv

Causal and anti-causal learning in pattern recognition for neuroimaging

Pattern recognition in neuroimaging distinguishes between two types of models: encoding- and decoding models. This distinction is based on the insight that brain state features, that are found to be relevant in an experimental paradigm, carry a different meaning in encoding- than in decoding models. In this paper, we argue that this distinction is not sufficient: Relevant features in encoding- and decoding models carry a different meaning depending on whether they represent causal- or anti-causal relations. We provide a theoretical justification for this argument and conclude that causal inference is essential for interpretation in neuroimaging.

preprint2015arXiv

Causal Inference by Identification of Vector Autoregressive Processes with Hidden Components

A widely applied approach to causal inference from a non-experimental time series $X$, often referred to as "(linear) Granger causal analysis", is to regress present on past and interpret the regression matrix $\hat{B}$ causally. However, if there is an unmeasured time series $Z$ that influences $X$, then this approach can lead to wrong causal conclusions, i.e., distinct from those one would draw if one had additional information such as $Z$. In this paper we take a different approach: We assume that $X$ together with some hidden $Z$ forms a first order vector autoregressive (VAR) process with transition matrix $A$, and argue why it is more valid to interpret $A$ causally instead of $\hat{B}$. Then we examine under which conditions the most important parts of $A$ are identifiable or almost identifiable from only $X$. Essentially, sufficient conditions are (1) non-Gaussian, independent noise or (2) no influence from $X$ to $Z$. We present two estimation algorithms that are tailored towards conditions (1) and (2), respectively, and evaluate them on synthetic and real-world data. We discuss how to check the model using $X$.

preprint2015arXiv

Causal interpretation rules for encoding and decoding models in neuroimaging

Causal terminology is often introduced in the interpretation of encoding and decoding models trained on neuroimaging data. In this article, we investigate which causal statements are warranted and which ones are not supported by empirical evidence. We argue that the distinction between encoding and decoding models is not sufficient for this purpose: relevant features in encoding and decoding models carry a different meaning in stimulus- and in response-based experimental paradigms. We show that only encoding models in the stimulus-based setting support unambiguous causal interpretations. By combining encoding and decoding models trained on the same data, however, we obtain insights into causal relations beyond those that are implied by each individual model type. We illustrate the empirical relevance of our theoretical findings on EEG data recorded during a visuo-motor learning task.

preprint2015arXiv

Computing Functions of Random Variables via Reproducing Kernel Hilbert Space Representations

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations which can be applied to points drawn from the respective distributions. We refer to our approach as {\em kernel probabilistic programming}. We illustrate it on synthetic data, and show how it can be used for nonparametric structural equation models, with an application to causal inference.

preprint2015arXiv

Decoding index finger position from EEG using random forests

While invasively recorded brain activity is known to provide detailed information on motor commands, it is an open question at what level of detail information about positions of body parts can be decoded from non-invasively acquired signals. In this work it is shown that index finger positions can be differentiated from non-invasive electroencephalographic (EEG) recordings in healthy human subjects. Using a leave-one-subject-out cross-validation procedure, a random forest distinguished different index finger positions on a numerical keyboard above chance-level accuracy. Among the different spectral features investigated, high $β$-power (20-30 Hz) over contralateral sensorimotor cortex carried most information about finger position. Thus, these findings indicate that finger position is in principle decodable from non-invasive features of brain activity that generalize across individuals.

preprint2015arXiv

Distinguishing Cause from Effect Based on Exogeneity

Recent developments in structural equation modeling have produced several methods that can usually distinguish cause from effect in the two-variable case. For that purpose, however, one has to impose substantial structural constraints or smoothness assumptions on the functional causal models. In this paper, we consider the problem of determining the causal direction from a related but different point of view, and propose a new framework for causal direction determination. We show that it is possible to perform causal inference based on the condition that the cause is "exogenous" for the parameters involved in the generating process from the cause to the effect. In this way, we avoid the structural constraints required by the SEM-based approaches. In particular, we exploit nonparametric methods to estimate marginal and conditional distributions, and propose a bootstrap-based approach to test for the exogeneity condition; the testing results indicate the causal direction between two variables. The proposed method is validated on both synthetic and real data.

preprint2015arXiv

Distinguishing cause from effect using observational data: methods and benchmarks

The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X, Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: Additive Noise Methods (ANM) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 datasets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the additive-noise method originally proposed by Hoyer et al. (2009), which obtains an accuracy of 63+-10 % and an AUC of 0.74+-0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.

preprint2015arXiv

Removing systematic errors for exoplanet search via latent causes

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application.

preprint2015arXiv

Towards a Learning Theory of Cause-Effect Inference

We pose causal inference as the problem of learning to classify probability distributions. In particular, we assume access to a collection $\{(S_i,l_i)\}_{i=1}^n$, where each $S_i$ is a sample drawn from the probability distribution of $X_i \times Y_i$, and $l_i$ is a binary label indicating whether "$X_i \to Y_i$" or "$X_i \leftarrow Y_i$". Given these data, we build a causal inference rule in two steps. First, we featurize each $S_i$ using the kernel mean embedding associated with some characteristic kernel. Second, we train a binary classifier on such embeddings to distinguish between causal directions. We present generalization bounds showing the statistical consistency and learning rates of the proposed approach, and provide a simple implementation that achieves state-of-the-art cause-effect inference. Furthermore, we extend our ideas to infer causal relationships between more than two variables.

preprint2015arXiv

Transfer Learning in Brain-Computer Interfaces

The performance of brain-computer interfaces (BCIs) improves with the amount of available training data, the statistical distribution of this data, however, varies across subjects as well as across sessions within individual subjects, limiting the transferability of training data or trained models between them. In this article, we review current transfer learning techniques in BCIs that exploit shared structure between training data of multiple subjects and/or sessions to increase performance. We then present a framework for transfer learning in the context of BCIs that can be applied to any arbitrary feature space, as well as a novel regression estimation method that is specifically designed for the structure of a system based on the electroencephalogram (EEG). We demonstrate the utility of our framework and method on subject-to-subject transfer in a motor-imagery paradigm as well as on session-to-session transfer in one patient diagnosed with amyotrophic lateral sclerosis (ALS), showing that it is able to outperform other comparable methods on an identical dataset.

preprint2014arXiv

Causal Discovery with Continuous Additive Noise Models

We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identifiable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for finitely many samples, RESIT (Regression with Subsequent Independence Test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation.

preprint2014arXiv

Consistency of Causal Inference under the Additive Noise Model

We analyze a family of methods for statistical causal inference from sample under the so-called Additive Noise Model. While most work on the subject has concentrated on establishing the soundness of the Additive Noise Model, the statistical consistency of the resulting inference methods has received little attention. We derive general conditions under which the given family of inference methods consistently infers the causal direction in a nonparametric setting.

preprint2014arXiv

Justifying Information-Geometric Causal Inference

Information Geometric Causal Inference (IGCI) is a new approach to distinguish between cause and effect for two variables. It is based on an independence assumption between input distribution and causal mechanism that can be phrased in terms of orthogonality in information space. We describe two intuitive reinterpretations of this approach that makes IGCI more accessible to a broader audience. Moreover, we show that the described independence is related to the hypothesis that unsupervised learning and semi-supervised learning only works for predicting the cause from the effect and not vice versa.

preprint2014arXiv

Kernel Mean Estimation via Spectral Filtering

The problem of estimating the kernel mean in a reproducing kernel Hilbert space (RKHS) is central to kernel methods in that it is used by classical approaches (e.g., when centering a kernel PCA matrix), and it also forms the core inference step of modern kernel methods (e.g., kernel-based non-parametric tests) that rely on embedding probability distributions in RKHSs. Muandet et al. (2014) has shown that shrinkage can help in constructing "better" estimators of the kernel mean than the empirical estimator. The present paper studies the consistency and admissibility of the estimators in Muandet et al. (2014), and proposes a wider class of shrinkage estimators that improve upon the empirical estimator by considering appropriate basis functions. Using the kernel PCA basis, we show that some of these estimators can be constructed using spectral filtering algorithms which are shown to be consistent under some technical assumptions. Our theoretical analysis also reveals a fundamental connection to the kernel-based supervised learning framework. The proposed estimators are simple to implement and perform well in practice.

preprint2014arXiv

Learning to Deblur

We describe a learning-based approach to blind image deconvolution. It uses a deep layered architecture, parts of which are borrowed from recent work on neural network learning, and parts of which incorporate computations that are specific to image deconvolution. The system is trained end-to-end on a set of artificially generated training examples, enabling competitive performance in blind deconvolution, both with respect to quality and runtime.

preprint2014arXiv

Quantifying causal influences

Many methods for causal inference generate directed acyclic graphs (DAGs) that formalize causal relations between $n$ variables. Given the joint distribution on all these variables, the DAG contains all information about how intervening on one variable changes the distribution of the other $n-1$ variables. However, quantifying the causal influence of one variable on another one remains a nontrivial question. Here we propose a set of natural, intuitive postulates that a measure of causal strength should satisfy. We then introduce a communication scenario, where edges in a DAG play the role of channels that can be locally corrupted by interventions. Causal strength is then the relative entropy distance between the old and the new distribution. Many other measures of causal strength have been proposed, including average causal effect, transfer entropy, directed information, and information flow. We explain how they fail to satisfy the postulates on simple DAGs of $\leq3$ nodes. Finally, we investigate the behavior of our measure on time-series, supporting our claims with experiments on simulated data.

preprint2014arXiv

Randomized Nonlinear Component Analysis

Classical methods such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques are only able to reveal linear relationships in data. Although nonlinear variants of PCA and CCA have been proposed, these are computationally prohibitive in the large scale. In a separate strand of recent research, randomized methods have been proposed to construct features that help reveal nonlinear patterns in data. For basic tasks such as regression or classification, random features exhibit little or no loss in performance, while achieving drastic savings in computational requirements. In this paper we leverage randomness to design scalable new variants of nonlinear PCA and CCA; our ideas extend to key multivariate analysis tools such as spectral clustering or LDA. We demonstrate our algorithms through experiments on real-world data, on which we compare against the state-of-the-art. A simple R implementation of the presented algorithms is provided.

preprint2013arXiv

Domain Generalization via Invariant Feature Representation

This paper investigates domain generalization: How to take knowledge acquired from an arbitrary number of related domains and apply it to previously unseen domains? We propose Domain-Invariant Component Analysis (DICA), a kernel-based optimization algorithm that learns an invariant transformation by minimizing the dissimilarity across domains, whilst preserving the functional relationship between input and output variables. A learning-theoretic analysis shows that reducing dissimilarity improves the expected generalization ability of classifiers on new domains, motivating the proposed algorithm. Experimental results on synthetic and real-world datasets demonstrate that DICA successfully learns invariant features and improves classifier performance in practice.

preprint2013arXiv

From Ordinary Differential Equations to Structural Causal Models: the deterministic case

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of Structural Causal Models, especially for cyclic models.

preprint2013arXiv

Kernel Mean Estimation and Stein's Effect

A mean function in reproducing kernel Hilbert space, or a kernel mean, is an important part of many applications ranging from kernel principal component analysis to Hilbert-space embedding of distributions. Given finite samples, an empirical average is the standard estimate for the true kernel mean. We show that this estimator can be improved via a well-known phenomenon in statistics called Stein's phenomenon. After consideration, our theoretical analysis reveals the existence of a wide class of estimators that are better than the standard. Focusing on a subset of this class, we propose efficient shrinkage estimators for the kernel mean. Empirical evaluations on several benchmark applications clearly demonstrate that the proposed estimators outperform the standard kernel mean estimator.

preprint2013arXiv

Learning from Distributions via Support Measure Machines

This paper presents a kernel-based discriminative learning framework on probability measures. Rather than relying on large collections of vectorial training examples, our framework learns using a collection of probability distributions that have been constructed to meaningfully represent training data. By representing these probability distributions as mean embeddings in the reproducing kernel Hilbert space (RKHS), we are able to apply many standard kernel-based learning techniques in straightforward fashion. To accomplish this, we construct a generalization of the support vector machine (SVM) called a support measure machine (SMM). Our analyses of SMMs provides several insights into their relationship to traditional SVMs. Based on such insights, we propose a flexible SVM (Flex-SVM) that places different kernel functions on each training example. Experimental results on both synthetic and real-world data demonstrate the effectiveness of our proposed framework.

preprint2013arXiv

Maximizing Kepler science return per telemetered pixel: Detailed models of the focal plane in the two-wheel era

Kepler's immense photometric precision to date was maintained through satellite stability and precise pointing. In this white paper, we argue that image modeling--fitting the Kepler-downlinked raw pixel data--can vastly improve the precision of Kepler in pointing-degraded two-wheel mode. We argue that a non-trivial modeling effort may permit continuance of photometry at 10-ppm-level precision. We demonstrate some baby steps towards precise models in both data-driven (flexible) and physics-driven (interpretably parameterized) modes. We demonstrate that the expected drift or jitter in positions in the two-weel era will help with constraining calibration parameters. In particular, we show that we can infer the device flat-field at higher than pixel resolution; that is, we can infer pixel-to-pixel variations in intra-pixel sensitivity. These results are relevant to almost any scientific goal for the repurposed mission; image modeling ought to be a part of any two-wheel repurpose for the satellite. We make other recommendations for Kepler operations, but fundamentally advocate that the project stick with its core mission of finding and characterizing Earth analogs. [abridged]

preprint2013arXiv

On a link between kernel mean maps and Fraunhofer diffraction, with an application to super-resolution beyond the diffraction limit

We establish a link between Fourier optics and a recent construction from the machine learning community termed the kernel mean map. Using the Fraunhofer approximation, it identifies the kernel with the squared Fourier transform of the aperture. This allows us to use results about the invertibility of the kernel mean map to provide a statement about the invertibility of Fraunhofer diffraction, showing that imaging processes with arbitrarily small apertures can in principle be invertible, i.e., do not lose information, provided the objects to be imaged satisfy a generic condition. A real world experiment shows that we can super-resolve beyond the Rayleigh limit.

preprint2013arXiv

One-Class Support Measure Machines for Group Anomaly Detection

We propose one-class support measure machines (OCSMMs) for group anomaly detection which aims at recognizing anomalous aggregate behaviors of data points. The OCSMMs generalize well-known one-class support vector machines (OCSVMs) to a space of probability measures. By formulating the problem as quantile estimation on distributions, we can establish an interesting connection to the OCSVMs and variable kernel density estimators (VKDEs) over the input space on which the distributions are defined, bridging the gap between large-margin methods and kernel density estimators. In particular, we show that various types of VKDEs can be considered as solutions to a class of regularization problems studied in this paper. Experiments on Sloan Digital Sky Survey dataset and High Energy Particle Physics dataset demonstrate the benefits of the proposed framework in real-world applications.

preprint2013arXiv

Semi-Supervised Domain Adaptation with Non-Parametric Copulas

A new framework based on the theory of copulas is proposed to address semi- supervised domain adaptation problems. The presented method factorizes any multivariate density into a product of marginal distributions and bivariate cop- ula functions. Therefore, changes in each of these factors can be detected and corrected to adapt a density model accross different learning domains. Impor- tantly, we introduce a novel vine copula model, which allows for this factorization in a non-parametric manner. Experimental results on regression problems with real-world data illustrate the efficacy of the proposed approach when compared to state-of-the-art techniques.

preprint2013arXiv

Spatial statistics, image analysis and percolation theory

We develop a novel method for detection of signals and reconstruction of images in the presence of random noise. The method uses results from percolation theory. We specifically address the problem of detection of multiple objects of unknown shapes in the case of nonparametric noise. The noise density is unknown and can be heavy-tailed. The objects of interest have unknown varying intensities. No boundary shape constraints are imposed on the objects, only a set of weak bulk conditions is required. We view the object detection problem as a multiple hypothesis testing for discrete statistical inverse problems. We present an algorithm that allows to detect greyscale objects of various shapes in noisy images. We prove results on consistency and algorithmic complexity of our procedures. Applications to cryo-electron microscopy are presented.

preprint2013arXiv

The Randomized Dependence Coefficient

We introduce the Randomized Dependence Coefficient (RDC), a measure of non-linear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper.

preprint2012arXiv

Causal Inference on Time Series using Structural Equation Models

Causal inference uses observations to infer the causal structure of the data generating system. We study a class of functional models that we call Time Series Models with Independent Noise (TiMINo). These models require independent residual time series, whereas traditional methods like Granger causality exploit the variance of residuals. There are two main contributions: (1) Theoretical: By restricting the model class (e.g. to additive noise) we can provide a more general identifiability result than existing ones. This result incorporates lagged and instantaneous effects that can be nonlinear and do not need to be faithful, and non-instantaneous feedbacks between the time series. (2) Practical: If there are no feedback loops between time series, we propose an algorithm based on non-linear independence tests of time series. When the data are causally insufficient, or the data generating process does not satisfy the model assumptions, this algorithm may still give partial results, but mostly avoids incorrect answers. An extension to (non-instantaneous) feedbacks is possible, but not discussed. It outperforms existing methods on artificial and real data. Code can be provided upon request.

preprint2012arXiv

easyGWAS: An integrated interspecies platform for performing genome-wide association studies

Motivation: The rapid growth in genome-wide association studies (GWAS) in plants and animals has brought about the need for a central resource that facilitates i) performing GWAS, ii) accessing data and results of other GWAS, and iii) enabling all users regardless of their background to exploit the latest statistical techniques without having to manage complex software and computing resources. Results: We present easyGWAS, a web platform that provides methods, tools and dynamic visualizations to perform and analyze GWAS. In addition, easyGWAS makes it simple to reproduce results of others, validate findings, and access larger sample sizes through merging of public datasets. Availability: Detailed method and data descriptions as well as tutorials are available in the supplementary materials. easyGWAS is available at http://easygwas.tuebingen.mpg.de/. Contact: dominik.grimm@tuebingen.mpg.de

preprint2012arXiv

Influence Maximization in Continuous Time Diffusion Networks

The problem of finding the optimal set of source nodes in a diffusion network that maximizes the spread of information, influence, and diseases in a limited amount of time depends dramatically on the underlying temporal dynamics of the network. However, this still remains largely unexplored to date. To this end, given a network and its temporal dynamics, we first describe how continuous time Markov chains allow us to analytically compute the average total number of nodes reached by a diffusion process starting in a set of source nodes. We then show that selecting the set of most influential source nodes in the continuous time influence maximization problem is NP-hard and develop an efficient approximation algorithm with provable near-optimal performance. Experiments on synthetic and real diffusion networks show that our algorithm outperforms other state of the art algorithms by at least ~20% and is robust across different network topologies.

preprint2012arXiv

Structure and Dynamics of Information Pathways in Online Media

Diffusion of information, spread of rumors and infectious diseases are all instances of stochastic processes that occur over the edges of an underlying network. Many times networks over which contagions spread are unobserved, and such networks are often dynamic and change over time. In this paper, we investigate the problem of inferring dynamic networks based on information diffusion data. We assume there is an unobserved dynamic network that changes over time, while we observe the results of a dynamic process spreading over the edges of the network. The task then is to infer the edges and the dynamics of the underlying network. We develop an on-line algorithm that relies on stochastic convex optimization to efficiently solve the dynamic network inference problem. We apply our algorithm to information diffusion among 3.3 million mainstream media and blog sites and experiment with more than 179 million different pieces of information spreading over the network in a one year period. We study the evolution of information pathways in the online media space and find interesting insights. Information pathways for general recurrent topics are more stable across time than for on-going news events. Clusters of news media sites and blogs often emerge and vanish in matter of days for on-going news events. Major social movements and events involving civil population, such as the Libyan's civil war or Syria's uprise, lead to an increased amount of information pathways among blogs as well as in the overall increase in the network centrality of blogs and social media sites.

preprint2012arXiv

Submodular Inference of Diffusion Networks from Multiple Trees

Diffusion and propagation of information, influence and diseases take place over increasingly larger networks. We observe when a node copies information, makes a decision or becomes infected but networks are often hidden or unobserved. Since networks are highly dynamic, changing and growing rapidly, we only observe a relatively small set of cascades before a network changes significantly. Scalable network inference based on a small cascade set is then necessary for understanding the rapidly evolving dynamics that govern diffusion. In this article, we develop a scalable approximation algorithm with provable near-optimal performance based on submodular maximization which achieves a high accuracy in such scenario, solving an open problem first introduced by Gomez-Rodriguez et al (2010). Experiments on synthetic and real diffusion data show that our algorithm in practice achieves an optimal trade-off between accuracy and running time.

preprint2012arXiv

The representer theorem for Hilbert spaces: a necessary and sufficient condition

A family of regularization functionals is said to admit a linear representer theorem if every member of the family admits minimizers that lie in a fixed finite dimensional subspace. A recent characterization states that a general class of regularization functionals with differentiable regularizer admits a linear representer theorem if and only if the regularization term is a non-decreasing function of the norm. In this report, we improve over such result by replacing the differentiability assumption with lower semi-continuity and deriving a proof that is independent of the dimensionality of the space.

preprint2011arXiv

Gravitational Lensing Accuracy Testing 2010 (GREAT10) Challenge Handbook

GRavitational lEnsing Accuracy Testing 2010 (GREAT10) is a public image analysis challenge aimed at the development of algorithms to analyze astronomical images. Specifically, the challenge is to measure varying image distortions in the presence of a variable convolution kernel, pixelization and noise. This is the second in a series of challenges set to the astronomy, computer science and statistics communities, providing a structured environment in which methods can be improved and tested in preparation for planned astronomical surveys. GREAT10 extends upon previous work by introducing variable fields into the challenge. The "Galaxy Challenge" involves the precise measurement of galaxy shape distortions, quantified locally by two parameters called shear, in the presence of a known convolution kernel. Crucially, the convolution kernel and the simulated gravitational lensing shape distortion both now vary as a function of position within the images, as is the case for real data. In addition, we introduce the "Star Challenge" that concerns the reconstruction of a variable convolution kernel, similar to that in a typical astronomical observation. This document details the GREAT10 Challenge for potential participants. Continually updated information is also available from http://www.greatchallenges.info.

preprint2011arXiv

Robust Learning via Cause-Effect Models

We consider the problem of function estimation in the case where the data distribution may shift between training and test time, and additional information about it may be available at test time. This relates to popular scenarios such as covariate shift, concept drift, transfer learning and semi-supervised learning. This working paper discusses how these tasks could be tackled depending on the kind of changes of the distributions. It argues that knowledge of an underlying causal direction can facilitate several of these tasks.

preprint2011arXiv

Uncovering the Temporal Dynamics of Diffusion Networks

Time plays an essential role in the diffusion of information, influence and disease over networks. In many cases we only observe when a node copies information, makes a decision or becomes infected -- but the connectivity, transmission rates between nodes and transmission sources are unknown. Inferring the underlying dynamics is of outstanding interest since it enables forecasting, influencing and retarding infections, broadly construed. To this end, we model diffusion processes as discrete networks of continuous temporal processes occurring at different rates. Given cascade data -- observed infection times of nodes -- we infer the edges of the global diffusion network and estimate the transmission rates of each edge that best explain the observed data. The optimization problem is convex. The model naturally (without heuristics) imposes sparse solutions and requires no parameter tuning. The problem decouples into a collection of independent smaller problems, thus scaling easily to networks on the order of hundreds of thousands of nodes. Experiments on real and synthetic data show that our algorithm both recovers the edges of diffusion networks and accurately estimates their transmission rates from cascade data.

preprint2010arXiv

Hilbert space embeddings and metrics on probability measures

A Hilbert space embedding for probability measures has recently been proposed, with applications including dimensionality reduction, homogeneity testing, and independence testing. This embedding represents any probability measure as a mean element in a reproducing kernel Hilbert space (RKHS). A pseudometric on the space of probability measures can be defined as the distance between distribution embeddings: we denote this as $γ_k$, indexed by the kernel function $k$ that defines the inner product in the RKHS. We present three theoretical properties of $γ_k$. First, we consider the question of determining the conditions on the kernel $k$ for which $γ_k$ is a metric: such $k$ are denoted {\em characteristic kernels}. Unlike pseudometrics, a metric is zero only when two distributions coincide, thus ensuring the RKHS embedding maps all distributions uniquely (i.e., the embedding is injective). While previously published conditions may apply only in restricted circumstances (e.g. on compact domains), and are difficult to check, our conditions are straightforward and intuitive: bounded continuous strictly positive definite kernels are characteristic. Alternatively, if a bounded continuous kernel is translation-invariant on $\bb{R}^d$, then it is characteristic if and only if the support of its Fourier transform is the entire $\bb{R}^d$. Second, we show that there exist distinct distributions that are arbitrarily close in $γ_k$. Third, to understand the nature of the topology induced by $γ_k$, we relate $γ_k$ to other popular metrics on probability measures, and present conditions on the kernel $k$ under which $γ_k$ metrizes the weak topology.

preprint2009arXiv

Causal Inference on Discrete Data using Additive Noise Models

Inferring the causal structure of a set of random variables from a finite sample of the joint distribution is an important problem in science. Recently, methods using additive noise models have been suggested to approach the case of continuous variables. In many situations, however, the variables of interest are discrete or even have only finitely many states. In this work we extend the notion of additive noise models to these cases. We prove that whenever the joint distribution $\prob^{(X,Y)}$ admits such a model in one direction, e.g. $Y=f(X)+N, N \independent X$, it does not admit the reversed model $X=g(Y)+\tilde N, \tilde N \independent Y$ as long as the model is chosen in a generic way. Based on these deliberations we propose an efficient new algorithm that is able to distinguish between cause and effect for a finite sample of discrete variables. In an extensive experimental study we show that this algorithm works both on synthetic and real data sets.

preprint2006arXiv

Comment on "Support Vector Machines with Applications"

Comment on ``Support Vector Machines with Applications'' [math.ST/0612817]

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