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Bastian Rieck

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29 papers
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29

TMLR Journal 2026 Journal Article

Topological Inductive Bias fosters Multiple Instance Learning in Data-Scarce Scenarios

  • Salome Kazeminia
  • Carsten Marr
  • Bastian Rieck

Multiple instance learning (MIL) is a framework for weakly supervised classification, where labels are assigned to sets of instances, i.e., bags, rather than to individual data points. This paradigm has proven effective in tasks where fine-grained annotations are unavailable or costly to obtain. However, the effectiveness of MIL drops sharply when training data are scarce, such as for rare disease classification. To address this challenge, we propose incorporating topological inductive biases into the data representation space within the MIL framework. This bias introduces a topology-preserving constraint that encourages the instance encoder to maintain the topological structure of the instance distribution within each bag when mapping them to MIL latent space. As a result, our Topology Guided MIL (TG-MIL) method enhances the performance and generalizability of MIL classifiers across different aggregation functions, especially under scarce-data regimes. Our evaluations show average performance improvements of 15.3% for synthetic MIL datasets, 2.8% for MIL benchmarks, and 5.5% for rare anemia classification compared to current state-of-the-art MIL models, where only 17–120 samples per class are available. We make our code publicly available at https://github.com/SalomeKaze/TGMIL.

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ICML Conference 2025 Conference Paper

Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms

  • Julius von Rohrscheidt
  • Bastian Rieck

The Euler Characteristic Transform (ECT) is an efficiently computable geometrical-topological invariant that characterizes the global shape of data. In this paper, we introduce the local Euler Characteristic Transform ($\ell$-ECT), a novel extension of the ECT designed to enhance expressivity and interpretability in graph representation learning. Unlike traditional Graph Neural Networks (GNNs), which may lose critical local details through aggregation, the $\ell$-ECT provides a lossless representation of local neighborhoods. This approach addresses key limitations in GNNs by preserving nuanced local structures while maintaining global interpretability. Moreover, we construct a rotation-invariant metric based on $\ell$-ECTs for spatial alignment of data spaces. Our method demonstrates superior performance compared to standard GNNs on various benchmarking node classification tasks, while also offering theoretical guarantees of its effectiveness.

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NeurIPS Conference 2025 Conference Paper

Geometry-Aware Edge Pooling for Graph Neural Networks

  • Katharina Limbeck
  • Lydia Mezrag
  • Guy Wolf
  • Bastian Rieck

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.

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NeurIPS Conference 2025 Conference Paper

Less is More: Local Intrinsic Dimensions of Contextual Language Models

  • Benjamin Matthias Ruppik
  • Julius von Rohrscheidt
  • Carel van Niekerk
  • Michael Heck
  • Renato Vukovic
  • Shutong Feng
  • Hsien-Chin Lin
  • Nurul Lubis

Understanding the internal mechanisms of large language models (LLMs) remains a challenging and complex endeavor. Even fundamental questions, such as how fine-tuning affects model behavior, often require extensive empirical evaluation. In this paper, we introduce a novel perspective based on the geometric properties of contextual latent embeddings to study the effects of training and fine-tuning. To that end, we measure the local dimensions of a contextual language model's latent space and analyze their shifts during training and fine-tuning. We show that the local dimensions provide insights into the model's training dynamics and generalization ability. Specifically, the mean of the local dimensions predicts when the model’s training capabilities are exhausted, as exemplified in a dialogue state tracking task, overfitting, as demonstrated in an emotion recognition task, and grokking, as illustrated with an arithmetic task. Furthermore, our experiments suggest a practical heuristic: reductions in the mean local dimension tend to accompany and predict subsequent performance gains. Through this exploration, we aim to provide practitioners with a deeper understanding of the implications of fine-tuning on embedding spaces, facilitating informed decisions when configuring models for specific applications. The results of this work contribute to the ongoing discourse on the interpretability, adaptability, and generalizability of LLMs by bridging the gap between intrinsic model mechanisms and geometric properties in the respective embeddings.

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ICLR Conference 2025 Conference Paper

MAGNet: Motif-Agnostic Generation of Molecules from Scaffolds

  • Leon Hetzel
  • Johanna Sommer
  • Bastian Rieck
  • Fabian J. Theis
  • Stephan Günnemann

Recent advances in machine learning for molecules exhibit great potential for facilitating drug discovery from in silico predictions. Most models for molecule generation rely on the decomposition of molecules into frequently occurring substructures (motifs), from which they generate novel compounds. While motif representations greatly aid in learning molecular distributions, such methods fail to represent substructures beyond their known motif set, posing a fundamental limitation for discovering novel compounds. To address this limitation and enhance structural expressivity, we propose to separate structure from features by abstracting motifs to scaffolds and, subsequently, allocating atom and bond types. To this end, we introduce a novel factorisation of the molecules' data distribution that considers the entire molecular context and facilitates learning adequate assignments of atoms and bonds to scaffolds. Complementary to this, we propose MAGNet, the first model to freely learn motifs. Importantly, we demonstrate that MAGNet's improved expressivity leads to molecules with more structural diversity and, at the same time, diverse atom and bond assignments.

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ICLR Conference 2025 Conference Paper

MANTRA: The Manifold Triangulations Assemblage

  • Rubén Ballester
  • Ernst Röell
  • Daniel Bin Schmid
  • Mathieu Alain
  • Sergio Escalera
  • Carles Casacuberta
  • Bastian Rieck

The rising interest in leveraging higher-order interactions present in complex systems has led to a surge in more expressive models exploiting higher-order structures in the data, especially in topological deep learning (TDL), which designs neural networks on higher-order domains such as simplicial complexes. However, progress in this field is hindered by the scarcity of datasets for benchmarking these architectures. To address this gap, we introduce MANTRA, the first large-scale, diverse, and intrinsically higher-order dataset for benchmarking higher-order models, comprising over 43,000 and 250,000 triangulations of surfaces and three-dimensional manifolds, respectively. With MANTRA, we assess several graph- and simplicial complex-based models on three topological classification tasks. We demonstrate that while simplicial complex-based neural networks generally outperform their graph-based counterparts in capturing simple topological invariants, they also struggle, suggesting a rethink of TDL. Thus, MANTRA serves as a benchmark for assessing and advancing topological methods, paving the way towards more effective higher-order models.

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ICML Conference 2025 Conference Paper

No Metric to Rule Them All: Toward Principled Evaluations of Graph-Learning Datasets

  • Corinna Coupette
  • Jeremy Wayland
  • Emily Simons
  • Bastian Rieck

Benchmark datasets have proved pivotal to the success of graph learning, and good benchmark datasets are crucial to guide the development of the field. Recent research has highlighted problems with graph-learning datasets and benchmarking practices—revealing, for example, that methods which ignore the graph structure can outperform graph-based approaches. Such findings raise two questions: (1) What makes a good graph-learning dataset, and (2) how can we evaluate dataset quality in graph learning? Our work addresses these questions. As the classic evaluation setup uses datasets to evaluate models, it does not apply to dataset evaluation. Hence, we start from first principles. Observing that graph-learning datasets uniquely combine two modes—graph structure and node features—, we introduce RINGS, a flexible and extensible mode-perturbation framework to assess the quality of graph-learning datasets based on dataset ablations —i. e. , quantifying differences between the original dataset and its perturbed representations. Within this framework, we propose two measures— performance separability and mode complementarity —as evaluation tools, each assessing the capacity of a graph dataset to benchmark the power and efficacy of graph-learning methods from a distinct angle. We demonstrate the utility of our framework for dataset evaluation via extensive experiments on graph-level tasks and derive actionable recommendations for improving the evaluation of graph-learning methods. Our work opens new research directions in data-centric graph learning, and it constitutes a step toward the systematic evaluation of evaluations.

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NeurIPS Conference 2025 Conference Paper

Point Cloud Synthesis Using Inner Product Transforms

  • Ernst Röell
  • Bastian Rieck

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.

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TMLR Journal 2024 Journal Article

Bayesian Computation Meets Topology

  • Julius von Rohrscheidt
  • Bastian Rieck
  • Sebastian M Schmon

Computational topology recently started to emerge as a novel paradigm for characterising the ‘shape’ of high-dimensional data, leading to powerful algorithms in (un)supervised representation learning. While capable of capturing prominent features at multiple scales, topological methods cannot readily be used for Bayesian inference. We develop a novel approach that bridges this gap, making it possible to perform parameter estimation in a Bayesian framework, using topology-based loss functions. Our method affords easy integration into topological machine learning algorithms. We demonstrate its efficacy for parameter estimation in different simulation settings.

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ICLR Conference 2024 Conference Paper

Differentiable Euler Characteristic Transforms for Shape Classification

  • Ernst Röell
  • Bastian Rieck

The _Euler Characteristic Transform_ (ECT) is a powerful invariant, combining geometrical and topological characteristics of shapes and graphs. However, the ECT was hitherto unable to learn task-specific representations. We overcome this issue and develop a novel computational layer that enables learning the ECT in an end-to-end fashion. Our method, the _Differentiable Euler Characteristic Transform_ (DECT) is fast and computationally efficient, while exhibiting performance on a par with more complex models in both graph and point cloud classification tasks. Moreover, we show that this seemingly simple statistic provides the same topological expressivity as more complex topological deep learning layers.

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ICML Conference 2024 Conference Paper

Mapping the Multiverse of Latent Representations

  • Jeremy Wayland
  • Corinna Coupette
  • Bastian Rieck

Echoing recent calls to counter reliability and robustness concerns in machine learning via multiverse analysis, we present PRESTO, a principled framework for mapping the multiverse of machine-learning models that rely on latent representations. Although such models enjoy widespread adoption, the variability in their embeddings remains poorly understood, resulting in unnecessary complexity and untrustworthy representations. Our framework uses persistent homology to characterize the latent spaces arising from different combinations of diverse machine-learning methods, (hyper)parameter configurations, and datasets, allowing us to measure their pairwise (dis)similarity and statistically reason about their distributions. As we demonstrate both theoretically and empirically, our pipeline preserves desirable properties of collections of latent representations, and it can be leveraged to perform sensitivity analysis, detect anomalous embeddings, or efficiently and effectively navigate hyperparameter search spaces.

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NeurIPS Conference 2024 Conference Paper

Metric Space Magnitude for Evaluating the Diversity of Latent Representations

  • Katharina Limbeck
  • Rayna Andreeva
  • Rik Sarkar
  • Bastian Rieck

The magnitude of a metric space is a novelinvariant that provides a measure of the 'effective size' of a space acrossmultiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy. We develop a family of magnitude-based measures of the intrinsicdiversity of latent representations, formalising a novel notion ofdissimilarity between magnitude functions of finite metric spaces. Our measures are provably stable under perturbations of the data, can beefficiently calculated, and enable a rigorous multi-scale characterisation and comparison oflatent representations. We show their utility and superior performance across different domains and tasks, includingthe automated estimation of diversity, the detection of mode collapse, andthe evaluation of generative models for text, image, and graph data.

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ICML Conference 2024 Conference Paper

Position: Topological Deep Learning is the New Frontier for Relational Learning

  • Theodore Papamarkou
  • Tolga Birdal
  • Michael M. Bronstein
  • Gunnar E. Carlsson
  • Justin Curry
  • Yue Gao
  • Mustafa Hajij
  • Roland Kwitt

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

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ICLR Conference 2024 Conference Paper

Simplicial Representation Learning with Neural k-Forms

  • Kelly Maggs
  • Celia Hacker
  • Bastian Rieck

Geometric deep learning extends deep learning to incorporate information about the geometry and topology data, especially in complex domains like graphs. Despite the popularity of message passing in this field, it has limitations such as the need for graph rewiring, ambiguity in interpreting data, and over-smoothing. In this paper, we take a different approach, focusing on leveraging geometric information from simplicial complexes embedded in $\mathbb{R}^n$ using node coordinates. We use differential $k$-forms in $\mathbb{R}^n$ to create representations of simplices, offering interpretability and geometric consistency without message passing. This approach also enables us to apply differential geometry tools and achieve universal approximation. Our method is efficient, versatile, and applicable to various input complexes, including graphs, simplicial complexes, and cell complexes. It outperforms existing message passing neural networks in harnessing information from geometrical graphs with node features serving as coordinates.

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NeurIPS Conference 2023 Conference Paper

Curvature Filtrations for Graph Generative Model Evaluation

  • Joshua Southern
  • Jeremy Wayland
  • Michael Bronstein
  • Bastian Rieck

Graph generative model evaluation necessitates understanding differences between graphs on the distributional level. This entails being able to harness salient attributes of graphs in an efficient manner. Curvature constitutes one such property of graphs, and has recently started to prove useful in characterising graphs. Its expressive properties, stability, and practical utility in model evaluation remain largely unexplored, however. We combine graph curvature descriptors with emerging methods from topological data analysis to obtain robust, expressive descriptors for evaluating graph generative models.

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ICLR Conference 2023 Conference Paper

Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework

  • Corinna Coupette
  • Sebastian Dalleiger
  • Bastian Rieck

Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging domain of hypergraphs has remained largely unexplored. On graphs, the Ollivier-Ricci curvature measures differences between random walks via Wasserstein distances, thus grounding a geometric concept in ideas from probability theory and optimal transport. We develop Orchid, a flexible framework generalizing Ollivier-Ricci curvature to hypergraphs, and prove that the resulting curvatures have favorable theoretical properties. Through extensive experiments on synthetic and real-world hypergraphs from different domains, we demonstrate that Orchid curvatures are both scalable and useful to perform a variety of hypergraph tasks in practice.

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ICML Conference 2023 Conference Paper

Topological Singularity Detection at Multiple Scales

  • Julius von Rohrscheidt
  • Bastian Rieck

The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct non-manifold structures, i. e. singularities, that can lead to erroneous findings. Detecting such singularities is therefore crucial as a precursor to interpolation and inference tasks. We address this issue by developing a topological framework that (i) quantifies the local intrinsic dimension, and (ii) yields a Euclidicity score for assessing the ’manifoldness’ of a point along multiple scales. Our approach identifies singularities of complex spaces, while also capturing singular structures and local geometric complexity in image data.

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JMLR Journal 2023 Journal Article

Weisfeiler and Leman go Machine Learning: The Story so far

  • Christopher Morris
  • Yaron Lipman
  • Haggai Maron
  • Bastian Rieck
  • Nils M. Kriege
  • Martin Grohe
  • Matthias Fey
  • Karsten Borgwardt

In recent years, algorithms and neural architectures based on the Weisfeiler–Leman algorithm, a well-known heuristic for the graph isomorphism problem, have emerged as a powerful tool for machine learning with graphs and relational data. Here, we give a comprehensive overview of the algorithm’s use in a machine-learning setting, focusing on the supervised regime. We discuss the theoretical background, show how to use it for supervised graph and node representation learning, discuss recent extensions, and outline the algorithm’s connection to (permutation-)equivariant neural architectures. Moreover, we give an overview of current applications and future directions to stimulate further research. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

  • Dhananjay Bhaskar
  • Kincaid MacDonald
  • Oluwadamilola Fasina
  • Dawson Thomas
  • Bastian Rieck
  • Ian Adelstein
  • Smita Krishnaswamy

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.

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ICLR Conference 2022 Conference Paper

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

  • Leslie O'Bray
  • Max Horn
  • Bastian Rieck
  • Karsten M. Borgwardt

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.

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NeurIPS Conference 2022 Conference Paper

On Measuring Excess Capacity in Neural Networks

  • Florian Graf
  • Sebastian Zeng
  • Bastian Rieck
  • Marc Niethammer
  • Roland Kwitt

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 a (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.

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ICLR Conference 2022 Conference Paper

Topological Graph Neural Networks

  • Max Horn
  • Edward De Brouwer
  • Michael Moor
  • Yves Moreau
  • Bastian Rieck
  • Karsten M. Borgwardt

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.

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NeurIPS Conference 2021 Conference Paper

A sandbox for prediction and integration of DNA, RNA, and proteins in single cells

  • Malte Luecken
  • Daniel Burkhardt
  • Robrecht Cannoodt
  • Christopher Lance
  • Aditi Agrawal
  • Hananeh Aliee
  • Ann Chen
  • Louise Deconinck

The last decade has witnessed a technological arms race to encode the molecular states of cells into DNA libraries, turning DNA sequencers into scalable single-cell microscopes. Single-cell measurement of chromatin accessibility (DNA), gene expression (RNA), and proteins has revealed rich cellular diversity across tissues, organisms, and disease states. However, single-cell data poses a unique set of challenges. A dataset may comprise millions of cells with tens of thousands of sparse features. Identifying biologically relevant signals from the background sources of technical noise requires innovation in predictive and representational learning. Furthermore, unlike in machine vision or natural language processing, biological ground truth is limited. Here we leverage recent advances in multi-modal single-cell technologies which, by simultaneously measuring two layers of cellular processing in each cell, provide ground truth analogous to language translation. We define three key tasks to predict one modality from another and learn integrated representations of cellular state. We also generate a novel dataset of the human bone marrow specifically designed for benchmarking studies. The dataset and tasks are accessible through an open-source framework that facilitates centralized evaluation of community-submitted methods.

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ICML Conference 2020 Conference Paper

Graph Filtration Learning

  • Christoph D. Hofer
  • Florian Graf
  • Bastian Rieck
  • Marc Niethammer
  • Roland Kwitt

We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem.

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ICML Conference 2020 Conference Paper

Set Functions for Time Series

  • Max Horn
  • Michael Moor
  • Christian Bock
  • Bastian Rieck
  • Karsten M. Borgwardt

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.

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ICML Conference 2020 Conference Paper

Topological Autoencoders

  • Michael Moor
  • Max Horn
  • Bastian Rieck
  • Karsten M. Borgwardt

We propose a novel approach for preserving topological structures of the input space in latent representations of autoencoders. Using persistent homology, a technique from topological data analysis, we calculate topological signatures of both the input and latent space to derive a topological loss term. Under weak theoretical assumptions, we construct this loss in a differentiable manner, such that the encoding learns to retain multi-scale connectivity information. We show that our approach is theoretically well-founded and that it exhibits favourable latent representations on a synthetic manifold as well as on real-world image data sets, while preserving low reconstruction errors.

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NeurIPS Conference 2020 Conference Paper

Uncovering the Topology of Time-Varying fMRI Data using Cubical Persistence

  • Bastian Rieck
  • Tristan Yates
  • Christian Bock
  • Karsten Borgwardt
  • Guy Wolf
  • Nicholas Turk-Browne
  • Smita Krishnaswamy

Functional magnetic resonance imaging (fMRI) is a crucial technology for gaining insights into cognitive processes in humans. Data amassed from fMRI measurements result in volumetric data sets that vary over time. However, analysing such data presents a challenge due to the large degree of noise and person-to-person variation in how information is represented in the brain. To address this challenge, we present a novel topological approach that encodes each time point in an fMRI data set as a persistence diagram of topological features, i. e. high-dimensional voids present in the data. This representation naturally does not rely on voxel-by-voxel correspondence and is robust towards noise. We show that these time-varying persistence diagrams can be clustered to find meaningful groupings between participants, and that they are also useful in studying within-subject brain state trajectories of subjects performing a particular task. Here, we apply both clustering and trajectory analysis techniques to a group of participants watching the movie 'Partly Cloudy'. We observe significant differences in both brain state trajectories and overall topological activity between adults and children watching the same movie.

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ICML Conference 2019 Conference Paper

A Persistent Weisfeiler-Lehman Procedure for Graph Classification

  • Bastian Rieck
  • Christian Bock
  • Karsten M. Borgwardt

The Weisfeiler–Lehman graph kernel exhibits competitive performance in many graph classification tasks. However, its subtree features are not able to capture connected components and cycles, topological features known for characterising graphs. To extract such features, we leverage propagated node label information and transform unweighted graphs into metric ones. This permits us to augment the subtree features with topological information obtained using persistent homology, a concept from topological data analysis. Our method, which we formalise as a generalisation of Weisfeiler–Lehman subtree features, exhibits favourable classification accuracy and its improvements in predictive performance are mainly driven by including cycle information.

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NeurIPS Conference 2019 Conference Paper

Wasserstein Weisfeiler-Lehman Graph Kernels

  • Matteo Togninalli
  • Elisabetta Ghisu
  • Felipe Llinares-López
  • Bastian Rieck
  • Karsten Borgwardt

Most graph kernels are an instance of the class of R-Convolution kernels, which measure the similarity of objects by comparing their substructures. Despite their empirical success, most graph kernels use a naive aggregation of the final set of substructures, usually a sum or average, thereby potentially discarding valuable information about the distribution of individual components. Furthermore, only a limited instance of these approaches can be extended to continuously attributed graphs. We propose a novel method that relies on the Wasserstein distance between the node feature vector distributions of two graphs, which allows to find subtler differences in data sets by considering graphs as high-dimensional objects, rather than simple means. We further propose a Weisfeiler--Lehman inspired embedding scheme for graphs with continuous node attributes and weighted edges, enhance it with the computed Wasserstein distance, and thus improve the state-of-the-art prediction performance on several graph classification tasks.

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