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Ziv Goldfeld

Ziv Goldfeld contributes to research discovery and scholarly infrastructure.

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Machine Learning math.ST Statistics Theory Information Theory math.IT math.PR Artificial Intelligence math.OC

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preprint2026arXiv

PLOT: Progressive Localization via Optimal Transport in Neural Causal Abstraction

Causal abstraction offers a principled framework for mechanistic interpretability, aligning a high-level causal model with the low-level computation realized by a neural network through counterfactual intervention analysis. Existing methods such as distributed alignment search (DAS) learn expressive subspace interventions, but the relevant neural site is unknown a priori, so finding a handle requires a computationally burdensome search over candidate sites. We introduce PLOT (Progressive Localization via Optimal Transport), a transport-based framework that localizes causal variables from the output effect geometry of abstract and neural interventions. PLOT fits an optimal transport coupling between abstract variables and candidate neural sites, yielding a global soft correspondence that can be calibrated into intervention handles. In simple settings, a single coupling over individual neurons suffices. In larger models, PLOT is applied progressively, moving from coarse sites such as tokens, timesteps, or layers to finer supports such as coordinate groups or PCA spans, and optionally guiding DAS based on the localized signal. Across experiments of increasing complexity, transport-only PLOT handles are exceedingly fast and competitive on accuracy, while PLOT-guided DAS reaches DAS-level accuracy at a fraction of full DAS runtime, providing an efficient localization engine for causal abstraction research at scale.

preprint2026arXiv

Sliced Inner Product Gromov-Wasserstein Distances

The Gromov-Wasserstein (GW) problem provides a framework for aligning heterogeneous datasets by matching their intrinsic geometry, but its statistical and computational scaling remains an issue for high-dimensional problems. Slicing techniques offer an appealing route to scalability, but, unlike Wasserstein distances, GW problems do not generally admit closed-form solutions in one-dimension. We resolve this problem for the GW problem with inner product cost (IGW), propose a sliced IGW distance that enjoys a natural rotational invariance property, and comprehensively study its structural and computational properties. Numerical experiments validating our theory are presented, followed by applications to heterogeneous clustering of text data and language model representation comparison.

preprint2023arXiv

Data-Driven Optimization of Directed Information over Discrete Alphabets

Directed information (DI) is a fundamental measure for the study and analysis of sequential stochastic models. In particular, when optimized over input distributions it characterizes the capacity of general communication channels. However, analytic computation of DI is typically intractable and existing optimization techniques over discrete input alphabets require knowledge of the channel model, which renders them inapplicable when only samples are available. To overcome these limitations, we propose a novel estimation-optimization framework for DI over discrete input spaces. We formulate DI optimization as a Markov decision process and leverage reinforcement learning techniques to optimize a deep generative model of the input process probability mass function (PMF). Combining this optimizer with the recently developed DI neural estimator, we obtain an end-to-end estimation-optimization algorithm which is applied to estimating the (feedforward and feedback) capacity of various discrete channels with memory. Furthermore, we demonstrate how to use the optimized PMF model to (i) obtain theoretical bounds on the feedback capacity of unifilar finite-state channels; and (ii) perform probabilistic shaping of constellations in the peak power-constrained additive white Gaussian noise channel.

preprint2022arXiv

Limit distribution theory for smooth $p$-Wasserstein distances

The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked by the curse of dimensionality, whereby the number of data points needed to accurately estimate them grows exponentially with dimension. Gaussian smoothing was recently introduced as a means to alleviate the curse of dimensionality, giving rise to a parametric convergence rate in any dimension, while preserving the Wasserstein metric and topological structure. To facilitate valid statistical inference, in this work, we develop a comprehensive limit distribution theory for the empirical smooth Wasserstein distance. The limit distribution results leverage the functional delta method after embedding the domain of the Wasserstein distance into a certain dual Sobolev space, characterizing its Hadamard directional derivative for the dual Sobolev norm, and establishing weak convergence of the smooth empirical process in the dual space. To estimate the distributional limits, we also establish consistency of the nonparametric bootstrap. Finally, we use the limit distribution theory to study applications to generative modeling via minimum distance estimation with the smooth Wasserstein distance, showing asymptotic normality of optimal solutions for the quadratic cost.

preprint2022arXiv

Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

The smooth 1-Wasserstein distance (SWD) $W_1^σ$ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure. Indeed, SWD exhibits parametric convergence rates and inherits the metric and topological structure of the classic Wasserstein distance. Motivated by the above, this work conducts a thorough statistical study of the SWD, including a high-dimensional limit distribution result for empirical $W_1^σ$, bootstrap consistency, concentration inequalities, and Berry-Esseen type bounds. The derived nondegenerate limit stands in sharp contrast with the classic empirical $W_1$, for which a similar result is known only in the one-dimensional case. We also explore asymptotics and characterize the limit distribution when the smoothing parameter $σ$ is scaled with $n$, converging to $0$ at a sufficiently slow rate. The dimensionality of the sampled distribution enters empirical SWD convergence bounds only through the prefactor (i.e., the constant). We provide a sharp characterization of this prefactor's dependence on the smoothing parameter and the intrinsic dimension. This result is then used to derive new empirical convergence rates for classic $W_1$ in terms of the intrinsic dimension. As applications of the limit distribution theory, we study two-sample testing and minimum distance estimation (MDE) under $W_1^σ$. We establish asymptotic validity of SWD testing, while for MDE, we prove measurability, almost sure convergence, and limit distributions for optimal estimators and their corresponding $W_1^σ$ error. Our results suggest that the SWD is well suited for high-dimensional statistical learning and inference.

preprint2022arXiv

Neural Estimation and Optimization of Directed Information over Continuous Spaces

This work develops a new method for estimating and optimizing the directed information rate between two jointly stationary and ergodic stochastic processes. Building upon recent advances in machine learning, we propose a recurrent neural network (RNN)-based estimator which is optimized via gradient ascent over the RNN parameters. The estimator does not require prior knowledge of the underlying joint and marginal distributions. The estimator is also readily optimized over continuous input processes realized by a deep generative model. We prove consistency of the proposed estimation and optimization methods and combine them to obtain end-to-end performance guarantees. Applications for channel capacity estimation of continuous channels with memory are explored, and empirical results demonstrating the scalability and accuracy of our method are provided. When the channel is memoryless, we investigate the mapping learned by the optimized input generator.

preprint2022arXiv

Neural Estimation of Statistical Divergences

Statistical divergences (SDs), which quantify the dissimilarity between probability distributions, are a basic constituent of statistical inference and machine learning. A modern method for estimating those divergences relies on parametrizing an empirical variational form by a neural network (NN) and optimizing over parameter space. Such neural estimators are abundantly used in practice, but corresponding performance guarantees are partial and call for further exploration. We establish non-asymptotic absolute error bounds for a neural estimator realized by a shallow NN, focusing on four popular $\mathsf{f}$-divergences -- Kullback-Leibler, chi-squared, squared Hellinger, and total variation. Our analysis relies on non-asymptotic function approximation theorems and tools from empirical process theory to bound the two sources of error involved: function approximation and empirical estimation. The bounds characterize the effective error in terms of NN size and the number of samples, and reveal scaling rates that ensure consistency. For compactly supported distributions, we further show that neural estimators of the first three divergences above with appropriate NN growth-rate are minimax rate-optimal, achieving the parametric convergence rate.

preprint2022arXiv

Statistical inference with regularized optimal transport

Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing, and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator, and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of: (i) average- and max-sliced $p$-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

preprint2020arXiv

Capacity of Continuous Channels with Memory via Directed Information Neural Estimator

Calculating the capacity (with or without feedback) of channels with memory and continuous alphabets is a challenging task. It requires optimizing the directed information (DI) rate over all channel input distributions. The objective is a multi-letter expression, whose analytic solution is only known for a few specific cases. When no analytic solution is present or the channel model is unknown, there is no unified framework for calculating or even approximating capacity. This work proposes a novel capacity estimation algorithm that treats the channel as a `black-box', both when feedback is or is not present. The algorithm has two main ingredients: (i) a neural distribution transformer (NDT) model that shapes a noise variable into the channel input distribution, which we are able to sample, and (ii) the DI neural estimator (DINE) that estimates the communication rate of the current NDT model. These models are trained by an alternating maximization procedure to both estimate the channel capacity and obtain an NDT for the optimal input distribution. The method is demonstrated on the moving average additive Gaussian noise channel, where it is shown that both the capacity and feedback capacity are estimated without knowledge of the channel transition kernel. The proposed estimation framework opens the door to a myriad of capacity approximation results for continuous alphabet channels that were inaccessible until now.

preprint2020arXiv

Convergence of Smoothed Empirical Measures with Applications to Entropy Estimation

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast\mathcal{N}_σ$, for $\mathcal{N}_σ\triangleq\mathcal{N}(0,σ^2 \mathrm{I}_d)$, by $\hat{P}_n\ast\mathcal{N}_σ$, where $\hat{P}_n$ is the empirical measure, under different statistical distances. The convergence is examined in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $χ^2$-divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf{W}_1$) converges at rate $e^{O(d)}n^{-\frac{1}{2}}$ in remarkable contrast to a typical $n^{-\frac{1}{d}}$ rate for unsmoothed $\mathsf{W}_1$ (and $d\ge 3$). For the KL divergence, squared 2-Wasserstein distance ($\mathsf{W}_2^2$), and $χ^2$-divergence, the convergence rate is $e^{O(d)}n^{-1}$, but only if $P$ achieves finite input-output $χ^2$ mutual information across the additive white Gaussian noise channel. If the latter condition is not met, the rate changes to $ω(n^{-1})$ for the KL divergence and $\mathsf{W}_2^2$, while the $χ^2$-divergence becomes infinite - a curious dichotomy. As a main application we consider estimating the differential entropy $h(P\ast\mathcal{N}_σ)$ in the high-dimensional regime. The distribution $P$ is unknown but $n$ i.i.d samples from it are available. We first show that any good estimator of $h(P\ast\mathcal{N}_σ)$ must have sample complexity that is exponential in $d$. Using the empirical approximation results we then show that the absolute-error risk of the plug-in estimator converges at the parametric rate $e^{O(d)}n^{-\frac{1}{2}}$, thus establishing the minimax rate-optimality of the plug-in. Numerical results that demonstrate a significant empirical superiority of the plug-in approach to general-purpose differential entropy estimators are provided.

preprint2020arXiv

Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency

Optimal transport (OT), and in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances suffers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. However, while it enjoys fast algorithms and better statistical properties, it looses the metric structure that Wasserstein distances enjoy. This work proposes a novel Gaussian-smoothed OT (GOT) framework, that achieves the best of both worlds: preserving the 1-Wasserstein metric structure while alleviating the empirical approximation curse of dimensionality. Furthermore, as the Gaussian-smoothing parameter shrinks to zero, GOT $Γ$-converges towards classic OT (with convergence of optimizers), thus serving as a natural extension. An empirical study that supports the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.

preprint2020arXiv

Limit Distribution for Smooth Total Variation and $χ^2$-Divergence in High Dimensions

Statistical divergences are ubiquitous in machine learning as tools for measuring discrepancy between probability distributions. As these applications inherently rely on approximating distributions from samples, we consider empirical approximation under two popular $f$-divergences: the total variation (TV) distance and the $χ^2$-divergence. To circumvent the sensitivity of these divergences to support mismatch, the framework of Gaussian smoothing is adopted. We study the limit distributions of $\sqrt{n}δ_{\mathsf{TV}}(P_n\ast\mathcal{N},P\ast\mathcal{N})$ and $nχ^2(P_n\ast\mathcal{N}\|P\ast\mathcal{N})$, where $P_n$ is the empirical measure based on $n$ independently and identically distributed (i.i.d.) observations from $P$, $\mathcal{N}_σ:=\mathcal{N}(0,σ^2\mathrm{I}_d)$, and $\ast$ stands for convolution. In arbitrary dimension, the limit distributions are characterized in terms of Gaussian process on $\mathbb{R}^d$ with covariance operator that depends on $P$ and the isotropic Gaussian density of parameter $σ$. This, in turn, implies optimality of the $n^{-1/2}$ expected value convergence rates recently derived for $δ_{\mathsf{TV}}(P_n\ast\mathcal{N},P\ast\mathcal{N})$ and $χ^2(P_n\ast\mathcal{N}\|P\ast\mathcal{N})$. These strong statistical guarantees promote empirical approximation under Gaussian smoothing as a potent framework for learning and inference based on high-dimensional data.

preprint2020arXiv

The Information Bottleneck Problem and Its Applications in Machine Learning

Inference capabilities of machine learning (ML) systems skyrocketed in recent years, now playing a pivotal role in various aspect of society. The goal in statistical learning is to use data to obtain simple algorithms for predicting a random variable $Y$ from a correlated observation $X$. Since the dimension of $X$ is typically huge, computationally feasible solutions should summarize it into a lower-dimensional feature vector $T$, from which $Y$ is predicted. The algorithm will successfully make the prediction if $T$ is a good proxy of $Y$, despite the said dimensionality-reduction. A myriad of ML algorithms (mostly employing deep learning (DL)) for finding such representations $T$ based on real-world data are now available. While these methods are often effective in practice, their success is hindered by the lack of a comprehensive theory to explain it. The information bottleneck (IB) theory recently emerged as a bold information-theoretic paradigm for analyzing DL systems. Adopting mutual information as the figure of merit, it suggests that the best representation $T$ should be maximally informative about $Y$ while minimizing the mutual information with $X$. In this tutorial we survey the information-theoretic origins of this abstract principle, and its recent impact on DL. For the latter, we cover implications of the IB problem on DL theory, as well as practical algorithms inspired by it. Our goal is to provide a unified and cohesive description. A clear view of current knowledge is particularly important for further leveraging IB and other information-theoretic ideas to study DL models.

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