Researcher profile

Bastian Rieck

Bastian Rieck contributes to research discovery and scholarly infrastructure.

ResearcherAffiliation not importedOpen to collaborate

Machine Learning math.AT Computer Vision Computational Geometry Databases Digital Libraries eess.IV math.CO Social and Information Networks

Trust snapshot

Quick read

Trust 21 - EmergingVerification L1Unclaimed author

23works

0followers

9topics

4close collaborators

Actions

Decide how to stay connected

Follow researcher0

Claiming links this public author record to a researcher profile and unlocks direct collaboration workflows.

Direct collaboration

Open a focused conversation when the fit is right

Claim this author entity first to unlock direct invitations.

Inspect adjacent work, topics, institutions and collaborators without jumping out to a separate graph page.

Building this graph slice

BZPEER is loading the nearby papers, people, topics and institutions for this page.

preprint2026arXiv

Diversity Curves for Graph Representation Learning

Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging. Unsupervised tasks in particular require interpretable, scalable, and reliable size-aware graph representations. Our work addresses these issues by tracking the structural diversity of a graph across coarsening levels. The resulting graph embeddings, which we denote diversity curves, are interpretable by construction, efficient, and directly comparable across coarsening hierarchies. Specifically, we track the spread of graphs, a novel isometry invariant that is inherently well-suited for encoding the metric diversity and geometry of graphs. We utilise edge contraction coarsening and prove that this improves expressivity, thus leading to more powerful graph-level representations than structural descriptors alone. Demonstrating their utility over a range of baseline methods in practice, we use diversity curves to (i) cluster and visualise simulated graphs across varying sizes, (ii) distinguish the geometry of single-cell graphs, (iii) compare the structure of molecular graph datasets, and (iv) characterise geometric shapes.

preprint2026arXiv

Geometry-Aware Edge Pooling for Graph Neural Networks

Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-world applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation. However, existing pooling operations often optimise for the learning task at the expense of discarding fundamental graph structures, thus reducing interpretability. This leads to unreliable performance across dataset types, downstream tasks and pooling ratios. Addressing these concerns, we propose novel graph pooling layers for structure-aware pooling via edge collapses. Our methods leverage diffusion geometry and iteratively reduce a graph's size while preserving both its metric structure and its structural diversity. We guide pooling using magnitude, an isometry-invariant diversity measure, which permits us to control the fidelity of the pooling process. Further, we use the spread of a metric space as a faster and more stable alternative ensuring computational efficiency. Empirical results demonstrate that our methods (i) achieve top performance compared to alternative pooling layers across a range of diverse graph classification tasks, (ii) preserve key spectral properties of the input graphs, and (iii) retain high accuracy across varying pooling ratios.

preprint2026arXiv

Geometry-Aware Simplicial Message Passing

The Weisfeiler--Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry, i.e., meshes with identical connectivity but different embeddings are indistinguishable. We introduce the Geometric Simplicial Weisfeiler--Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. In addition, we show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), a complete invariant for geometric simplicial complexes, this yields a geometric expressivity characterization together with an approximation framework. Experiments on synthetic and mesh datasets serve to validate our theory, showing a clear hierarchy from combinatorial to geometry-aware models.

preprint2026arXiv

Have Graph -- Will Lift? The Case for Higher-Order Benchmarks

After a somewhat rocky start, geometry and topology have established a foothold in machine learning. Message passing, either on graphs or higher-order complexes, is one of the main drivers of geometric deep learning, and paradigms that were once considered to be firmly in the realm of the abstract-like sheaves-have been "tamed" to serve as novel inductive biases for model architectures in topological deep learning. The veritable diversity of models, however, is in stark contrast to the scarcity of suitable benchmark datasets. As a result, researchers often resort to lifting existing graph datasets to include higher-order information. In this opinion paper, I want to encourage the community to also source new datasets, which may be used to prop up the foundations of our research field.

preprint2026arXiv

Invariant-Based Diagnostics for Graph Benchmarks

Progress on graph foundation models is hindered by benchmark practices that conflate the contributions of node features and graph structure, making it hard to tell whether a model actually learns from connectivity, or whether it even needs to. We propose addressing this using graph invariants, i.e., permutation-invariant, task-agnostic structural descriptors that serve as a diagnostic framework for graph benchmarks. We show that (i) invariants are more expressive than standard GNNs, (ii) invariants characterize structural heterogeneity within and across benchmark datasets, (iii) invariants predict multi-task performance, and (iv) simple invariant-based models are competitive with, and sometimes exceed, transformer and message-passing baselines across 26 datasets. Our results suggest that expressivity is not the main driver of predictive performance, and that on tasks where structure matters, a non-trainable structural proxy often matches trained message-passing models. We thus posit that invariant baselines should become a standard for evaluating whether structure is required for a task and whether a model picks up on it, serving as a stepping stone towards graph foundation models.

preprint2026arXiv

No Triangulation Without Representation: Generalization in Topological Deep Learning

Despite an ever-increasing interest in topological deep learning models that target higher-order datasets, there is no consensus on how to evaluate such models. This is exacerbated by the fact that topological objects permit operations, such as structural refinements, that are not appropriate for graph data. In this work, we extend MANTRA, a benchmark dataset containing manifold triangulations, to a larger class of manifolds with more diverse homeomorphism types. We show that, unlike prior claims, both graph neural networks (GNNs) and higher-order message passing (HOMP) methods can saturate the benchmark. However, we find that this is contingent on the right representation and feature assignment, emphasizing their importance in baseline models. We thus provide a novel evaluation protocol based on representational diversity and triangulation refinement. Surprisingly, we find no indication that existing models are capable of generalizing beyond the combinatorial structure of the data. This points towards a research gap in developing models that understand topological structure independent of scale. Our work thus provides the necessary scaffolding to evaluate future models and enable the development of topology-aware inductive biases.

preprint2026arXiv

Persistent Homology via Ellipsoids

Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from, with Čech, Rips, alpha, and witness complexes being popular choices. In this manuscript, we build a novel type of geometrically informed simplicial complex, called a Rips-type ellipsoid complex. This complex is based on the idea that ellipsoids aligned with tangent directions better approximate the data compared to conventional (Euclidean) balls centered at sample points, as used in the construction of Rips and Alpha complexes. We use Principal Component Analysis to estimate tangent spaces directly from samples and present an algorithm for computing Rips-type ellipsoid barcodes, i.e., topological descriptors based on Rips-type ellipsoid complexes. Additionally, we show that the ellipsoid barcodes depend continuously on the input data so that small perturbations of a k-generic point cloud lead to proportionally small changes in the resulting ellipsoid barcodes. This provides a theoretical guarantee analogous, if somewhat weaker, to the classical stability results for Rips and Čech filtrations. We also conduct extensive experiments and compare Rips-type ellipsoid barcodes with standard Rips barcodes. Our findings indicate that Rips-type ellipsoid complexes are particularly effective for estimating the homology of manifolds and spaces with bottlenecks from samples. In particular, the persistence intervals corresponding to ground-truth topological features are longer compared to those obtained using the Rips complex of the data. Furthermore, Rips-type ellipsoid barcodes lead to better classification results in sparsely sampled point clouds. Finally, we demonstrate that Rips-type ellipsoid barcodes outperform Rips barcodes in classification tasks.

preprint2026arXiv

Point Cloud Synthesis Using Inner Product Transforms

Point cloud synthesis, i.e. the generation of novel point clouds from an input distribution, remains a challenging task, for which numerous complex machine learning models have been devised. We develop a novel method that encodes geometrical-topological characteristics of point clouds using inner products, leading to a highly-efficient point cloud representation with provable expressivity properties. Integrated into deep learning models, our encoding exhibits high quality in typical tasks like reconstruction, generation, and interpolation, with inference times orders of magnitude faster than existing methods.

preprint2025arXiv

Topology meets Machine Learning: An Introduction using the Euler Characteristic Transform

This overview article makes the case for how topological concepts can enrich research in machine learning. Using the Euler Characteristic Transform (ECT), a geometrical-topological invariant, as a running example, I present different use cases that result in more efficient models for analyzing point clouds, graphs, and meshes. Moreover, I outline a vision for how topological concepts could be used in the future, comprising (1) the learning of functions on topological spaces, (2) the building of hybrid models that imbue neural networks with knowledge about the topological information in data, and (3) the analysis of qualitative properties of neural networks. With current research already addressing some of these aspects, this article thus serves as an introduction and invitation to this nascent area of research.

preprint2023arXiv

On Measuring Excess Capacity in Neural Networks

We study the excess capacity of deep networks in the context of supervised classification. That is, given a capacity measure of the underlying hypothesis class - in our case, empirical Rademacher complexity - to what extent can we (a priori) constrain this class while retaining an empirical error on a par with the unconstrained regime? To assess excess capacity in modern architectures (such as residual networks), we extend and unify prior Rademacher complexity bounds to accommodate function composition and addition, as well as the structure of convolutions. The capacity-driving terms in our bounds are the Lipschitz constants of the layers and an (2, 1) group norm distance to the initializations of the convolution weights. Experiments on benchmark datasets of varying task difficulty indicate that (1) there is a substantial amount of excess capacity per task, and (2) capacity can be kept at a surprisingly similar level across tasks. Overall, this suggests a notion of compressibility with respect to weight norms, complementary to classic compression via weight pruning. Source code is available at https://github.com/rkwitt/excess_capacity.

preprint2023arXiv

Time-inhomogeneous diffusion geometry and topology

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of high-dimensional data. Diffusion condensation is constructed as a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. From a geometric perspective, we obtain convergence bounds based on the smallest transition probability and the radius of the data, whereas from a spectral perspective, our bounds are based on the eigenspectrum of the diffusion kernel. Our spectral results are of particular interest since most of the literature on data diffusion is focused on homogeneous processes. From a topological perspective, we show diffusion condensation generalizes centroid-based hierarchical clustering. We use this perspective to obtain a bound based on the number of data points, independent of their location. To understand the evolution of the data geometry beyond convergence, we use topological data analysis. We show that the condensation process itself defines an intrinsic condensation homology. We use this intrinsic topology as well as the ambient persistent homology of the condensation process to study how the data changes over diffusion time. We demonstrate both types of topological information in well-understood toy examples. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.

preprint2022arXiv

Evaluation Metrics for Graph Generative Models: Problems, Pitfalls, and Practical Solutions

Graph generative models are a highly active branch of machine learning. Given the steady development of new models of ever-increasing complexity, it is necessary to provide a principled way to evaluate and compare them. In this paper, we enumerate the desirable criteria for such a comparison metric and provide an overview of the status quo of graph generative model comparison in use today, which predominantly relies on the maximum mean discrepancy (MMD). We perform a systematic evaluation of MMD in the context of graph generative model comparison, highlighting some of the challenges and pitfalls researchers inadvertently may encounter. After conducting a thorough analysis of the behaviour of MMD on synthetically-generated perturbed graphs as well as on recently-proposed graph generative models, we are able to provide a suitable procedure to mitigate these challenges and pitfalls. We aggregate our findings into a list of practical recommendations for researchers to use when evaluating graph generative models.

preprint2022arXiv

Exploring the Geometry and Topology of Neural Network Loss Landscapes

Recent work has established clear links between the generalization performance of trained neural networks and the geometry of their loss landscape near the local minima to which they converge. This suggests that qualitative and quantitative examination of the loss landscape geometry could yield insights about neural network generalization performance during training. To this end, researchers have proposed visualizing the loss landscape through the use of simple dimensionality reduction techniques. However, such visualization methods have been limited by their linear nature and only capture features in one or two dimensions, thus restricting sampling of the loss landscape to lines or planes. Here, we expand and improve upon these in three ways. First, we present a novel "jump and retrain" procedure for sampling relevant portions of the loss landscape. We show that the resulting sampled data holds more meaningful information about the network's ability to generalize. Next, we show that non-linear dimensionality reduction of the jump and retrain trajectories via PHATE, a trajectory and manifold-preserving method, allows us to visualize differences between networks that are generalizing well vs poorly. Finally, we combine PHATE trajectories with a computational homology characterization to quantify trajectory differences.

preprint2022arXiv

On the Surprising Behaviour of node2vec

Graph embedding techniques are a staple of modern graph learning research. When using embeddings for downstream tasks such as classification, information about their stability and robustness, i.e., their susceptibility to sources of noise, stochastic effects, or specific parameter choices, becomes increasingly important. As one of the most prominent graph embedding schemes, we focus on node2vec and analyse its embedding quality from multiple perspectives. Our findings indicate that embedding quality is unstable with respect to parameter choices, and we propose strategies to remedy this in practice.

preprint2022arXiv

Topological Graph Neural Networks

Graph neural networks (GNNs) are a powerful architecture for tackling graph learning tasks, yet have been shown to be oblivious to eminent substructures such as cycles. We present TOGL, a novel layer that incorporates global topological information of a graph using persistent homology. TOGL can be easily integrated into any type of GNN and is strictly more expressive (in terms the Weisfeiler--Lehman graph isomorphism test) than message-passing GNNs. Augmenting GNNs with TOGL leads to improved predictive performance for graph and node classification tasks, both on synthetic data sets, which can be classified by humans using their topology but not by ordinary GNNs, and on real-world data.

preprint2020arXiv

Accelerating COVID-19 Differential Diagnosis with Explainable Ultrasound Image Analysis

Controlling the COVID-19 pandemic largely hinges upon the existence of fast, safe, and highly-available diagnostic tools. Ultrasound, in contrast to CT or X-Ray, has many practical advantages and can serve as a globally-applicable first-line examination technique. We provide the largest publicly available lung ultrasound (US) dataset for COVID-19 consisting of 106 videos from three classes (COVID-19, bacterial pneumonia, and healthy controls); curated and approved by medical experts. On this dataset, we perform an in-depth study of the value of deep learning methods for differential diagnosis of COVID-19. We propose a frame-based convolutional neural network that correctly classifies COVID-19 US videos with a sensitivity of 0.98+-0.04 and a specificity of 0.91+-08 (frame-based sensitivity 0.93+-0.05, specificity 0.87+-0.07). We further employ class activation maps for the spatio-temporal localization of pulmonary biomarkers, which we subsequently validate for human-in-the-loop scenarios in a blindfolded study with medical experts. Aiming for scalability and robustness, we perform ablation studies comparing mobile-friendly, frame- and video-based architectures and show reliability of the best model by aleatoric and epistemic uncertainty estimates. We hope to pave the road for a community effort toward an accessible, efficient and interpretable screening method and we have started to work on a clinical validation of the proposed method. Data and code are publicly available.

preprint2020arXiv

Graph Kernels: State-of-the-Art and Future Challenges

Graph-structured data are an integral part of many application domains, including chemoinformatics, computational biology, neuroimaging, and social network analysis. Over the last two decades, numerous graph kernels, i.e. kernel functions between graphs, have been proposed to solve the problem of assessing the similarity between graphs, thereby making it possible to perform predictions in both classification and regression settings. This manuscript provides a review of existing graph kernels, their applications, software plus data resources, and an empirical comparison of state-of-the-art graph kernels.

preprint2020arXiv

Path Imputation Strategies for Signature Models of Irregular Time Series

The signature transform is a 'universal nonlinearity' on the space of continuous vector-valued paths, and has received attention for use in machine learning on time series. However, real-world temporal data is typically observed at discrete points in time, and must first be transformed into a continuous path before signature techniques can be applied. We make this step explicit by characterising it as an imputation problem, and empirically assess the impact of various imputation strategies when applying signature-based neural nets to irregular time series data. For one of these strategies, Gaussian process (GP) adapters, we propose an extension~(GP-PoM) that makes uncertainty information directly available to the subsequent classifier while at the same time preventing costly Monte-Carlo (MC) sampling. In our experiments, we find that the choice of imputation drastically affects shallow signature models, whereas deeper architectures are more robust. Next, we observe that uncertainty-aware predictions (based on GP-PoM or indicator imputations) are beneficial for predictive performance, even compared to the uncertainty-aware training of conventional GP adapters. In conclusion, we have demonstrated that the path construction is indeed crucial for signature models and that our proposed strategy leads to competitive performance in general, while improving robustness of signature models in particular.

preprint2020arXiv

Set Functions for Time Series

Despite the eminent successes of deep neural networks, many architectures are often hard to transfer to irregularly-sampled and asynchronous time series that commonly occur in real-world datasets, especially in healthcare applications. This paper proposes a novel approach for classifying irregularly-sampled time series with unaligned measurements, focusing on high scalability and data efficiency. Our method SeFT (Set Functions for Time Series) is based on recent advances in differentiable set function learning, extremely parallelizable with a beneficial memory footprint, thus scaling well to large datasets of long time series and online monitoring scenarios. Furthermore, our approach permits quantifying per-observation contributions to the classification outcome. We extensively compare our method with existing algorithms on multiple healthcare time series datasets and demonstrate that it performs competitively whilst significantly reducing runtime.

preprint2019arXiv

Hierarchies and Ranks for Persistence Pairs

We develop a novel hierarchy for zero-dimensional persistence pairs, i.e., connected components, which is capable of capturing more fine-grained spatial relations between persistence pairs. Our work is motivated by a lack of spatial relationships between features in persistence diagrams, leading to a limited expressive power. We build upon a recently-introduced hierarchy of pairs in persistence diagrams that augments the pairing stored in persistence diagrams with information about which components merge. Our proposed hierarchy captures differences in branching structure. Moreover, we show how to use our hierarchy to measure the spatial stability of a pairing and we define a rank function for persistence pairs and demonstrate different applications.

preprint2019arXiv

Persistence Concepts for 2D Skeleton Evolution Analysis

In this work, we present concepts for the analysis of the evolution of two-dimensional skeletons. By introducing novel persistence concepts, we are able to reduce typical temporal incoherence, and provide insight in skeleton dynamics. We exemplify our approach by means of a simulation of viscous fingering---a highly dynamic process whose analysis is a hot topic in porous media research.

preprint2019arXiv

Persistent Intersection Homology for the Analysis of Discrete Data

Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional manifold. In such cases, persistent homology can be successfully employed to extract features, remove noise, and compare data sets. The underlying problems in some application domains, however, turn out to represent multiple manifolds with different dimensions. Algebraic topology typically analyzes such problems using intersection homology, an extension of homology that is capable of handling configurations with singularities. In this paper, we describe how the persistent variant of intersection homology can be used to assist data analysis in visualization. We point out potential pitfalls in approximating data sets with singularities and give strategies for resolving them.

preprint2019arXiv

Topological Machine Learning with Persistence Indicator Functions

Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.

Institution

Affiliation not imported yet

This author record came from a source that does not expose affiliation metadata. Once the author claims the profile or we enrich the record from another provider, this section will link to the concrete institution.

Topic footprint

Fields this researcher appears in

Machine Learning math.AT Computer Vision Computational Geometry Databases Digital Libraries eess.IV math.CO Social and Information Networks

Source provenance