arXiv subjects

Bastian Rieck

Publications and source records attributed to Bastian Rieck.

At least 37 records · Page 2Linked to original sources

arXiv2026

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.

cs.LG↗

arXiv2026

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.

cs.CV↗

arXiv2025

Less is More: Local Intrinsic Dimensions of Contextual Language Models

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.

cs.CL↗

arXiv2025

EmbedOR: Provable Cluster-Preserving Visualizations with Curvature-Based Stochastic Neighbor Embeddings

Stochastic Neighbor Embedding (SNE) algorithms like UMAP and tSNE often produce visualizations that do not preserve the geometry of noisy and high dimensional data. In particular, they can spuriously separate connected components of the underlying data submanifold and can fail to find clusters in well-clusterable data. To address these limitations, we propose EmbedOR, a SNE algorithm that incorporates discrete graph curvature. Our algorithm stochastically embeds the data using a curvature-enhanced distance metric that emphasizes underlying cluster structure. Critically, we prove that the EmbedOR distance metric extends consistency results for tSNE to a much broader class of datasets. We also describe extensive experiments on synthetic and real data that demonstrate the visualization and geometry-preservation capabilities of EmbedOR. We find that, unlike other SNE algorithms and UMAP, EmbedOR is much less likely to fragment continuous, high-density regions of the data. Finally, we demonstrate that the EmbedOR distance metric can be used as a tool to annotate existing visualizations to identify fragmentation and provide deeper insight into the underlying geometry of the data.

cs.LG↗

arXiv2025

Low-dimensional embeddings of high-dimensional data

Large collections of high-dimensional data have become nearly ubiquitous across many academic fields and application domains, ranging from biology to the humanities. Since working directly with high-dimensional data poses challenges, the demand for algorithms that create low-dimensional representations, or embeddings, for data visualization, exploration, and analysis is now greater than ever. In recent years, numerous embedding algorithms have been developed, and their usage has become widespread in research and industry. This surge of interest has resulted in a large and fragmented research field that faces technical challenges alongside fundamental debates, and it has left practitioners without clear guidance on how to effectively employ existing methods. Aiming to increase coherence and facilitate future work, in this review we provide a detailed and critical overview of recent developments, derive a list of best practices for creating and using low-dimensional embeddings, evaluate popular approaches on a variety of datasets, and discuss the remaining challenges and open problems in the field.

cs.LG↗

arXiv2025

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

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.

cs.LG↗

arXiv2025

Molecular Machine Learning Using Euler Characteristic Transforms

The shape of a molecule determines its physicochemical and biological properties. However, it is often underrepresented in standard molecular representation learning approaches. Here, we propose using the Euler Characteristic Transform (ECT) as a geometrical-topological descriptor. Computed directly on a molecular graph derived from handcrafted atomic features, the ECT enables the extraction of multiscale structural features, offering a novel way to represent and encode molecular shape in the feature space. We assess the predictive performance of this representation across nine benchmark regression datasets, all centered around predicting the inhibition constant $K_i$. In addition, we compare our proposed ECT-based representation against traditional molecular representations and methods, such as molecular fingerprints/descriptors and graph neural networks (GNNs). Our results show that our ECT-based representation achieves competitive performance, ranking among the best-performing methods on several datasets. More importantly, its combination with traditional representations, particularly with the AVALON fingerprint, significantly \emph{enhances predictive performance}, outperforming other methods on most datasets. These findings highlight the complementary value of multiscale topological information and its potential for being combined with established techniques. Our study suggests that hybrid approaches incorporating explicit shape information can lead to more informative and robust molecular representations, enhancing and opening new avenues in molecular machine learning tasks. To support reproducibility and foster open biomedical research, we provide open access to all experiments and code used in this work.

cs.LG↗

arXiv2025

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

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 particularly 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 exhibits superior performance compared to standard GNNs on a variety of node-classification tasks, while also offering theoretical guarantees that demonstrate its effectiveness.

cs.LG↗

arXiv2025

Detecting Spatial Dependence in Transcriptomics Data using Vectorised Persistence Diagrams

Evaluating spatial patterns in data is an integral task across various domains, including geostatistics, astronomy, and spatial tissue biology. The analysis of transcriptomics data in particular relies on methods for detecting spatially-dependent features that exhibit significant spatial patterns for both explanatory analysis and feature selection. However, given the complex and high-dimensional nature of these data, there is a need for robust, stable, and reliable descriptors of spatial dependence. We leverage the stability and multiscale properties of persistent homology to address this task. To this end, we introduce a novel framework using functional topological summaries, such as Betti curves and persistence landscapes, for identifying and describing non-random patterns in spatial data. In particular, we propose a non-parametric one-sample permutation test for spatial dependence and investigate its utility across both simulated and real spatial omics data. Our vectorised approach outperforms baseline methods at accurately detecting spatial dependence. Further, we find that our method is more robust to outliers than alternative tests using Moran's I.

stat.ME↗

arXiv2025

Graph and Simplicial Complex Prediction Gaussian Process via the Hodgelet Representations

Predicting the labels of graph-structured data is crucial in scientific applications and is often achieved using graph neural networks (GNNs). However, when data is scarce, GNNs suffer from overfitting, leading to poor performance. Recently, Gaussian processes (GPs) with graph-level inputs have been proposed as an alternative. In this work, we extend the Gaussian process framework to simplicial complexes (SCs), enabling the handling of edge-level attributes and attributes supported on higher-order simplices. We further augment the resulting SC representations by considering their Hodge decompositions, allowing us to account for homological information, such as the number of holes, in the SC. We demonstrate that our framework enhances the predictions across various applications, paving the way for GPs to be more widely used for graph and SC-level predictions.

cs.LG↗

arXiv2025

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.

cs.LG↗

arXiv2025

MANTRA: The Manifold Triangulations Assemblage

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, leading the way for more effective higher-order models.

cs.LG↗

arXiv2025

Principal Curvatures Estimation with Applications to Single Cell Data

The rapidly growing field of single-cell transcriptomic sequencing (scRNAseq) presents challenges for data analysis due to its massive datasets. A common method in manifold learning consists in hypothesizing that datasets lie on a lower dimensional manifold. This allows to study the geometry of point clouds by extracting meaningful descriptors like curvature. In this work, we will present Adaptive Local PCA (AdaL-PCA), a data-driven method for accurately estimating various notions of intrinsic curvature on data manifolds, in particular principal curvatures for surfaces. The model relies on local PCA to estimate the tangent spaces. The evaluation of AdaL-PCA on sampled surfaces shows state-of-the-art results. Combined with a PHATE embedding, the model applied to single-cell RNA sequencing data allows us to identify key variations in the cellular differentiation.

cs.LG↗

arXiv2025

Graph Classification Gaussian Processes via Hodgelet Spectral Features

The problem of classifying graphs is ubiquitous in machine learning. While it is standard to apply graph neural networks or graph kernel methods, Gaussian processes can be employed by transforming spatial features from the graph domain into spectral features in the Euclidean domain, and using them as the input points of classical kernels. However, this approach currently only takes into account features on vertices, whereas some graph datasets also support features on edges. In this work, we present a Gaussian process-based classification algorithm that can leverage one or both vertex and edges features. Furthermore, we take advantage of the Hodge decomposition to better capture the intricate richness of vertex and edge features, which can be beneficial on diverse tasks.

cs.LG↗

arXiv2025

Metric Space Magnitude for Evaluating the Diversity of Latent Representations

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

cs.LG↗

arXiv2024

On the Expressivity of Persistent Homology in Graph Learning

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

cs.LG↗

arXiv2024

CliquePH: Higher-Order Information for Graph Neural Networks through Persistent Homology on Clique Graphs

Graph neural networks have become the default choice by practitioners for graph learning tasks such as graph classification and node classification. Nevertheless, popular graph neural network models still struggle to capture higher-order information, i.e., information that goes \emph{beyond} pairwise interactions. Recent work has shown that persistent homology, a tool from topological data analysis, can enrich graph neural networks with topological information that they otherwise could not capture. Calculating such features is efficient for dimension 0 (connected components) and dimension 1 (cycles). However, when it comes to higher-order structures, it does not scale well, with a complexity of $O(n^d)$, where $n$ is the number of nodes and $d$ is the order of the structures. In this work, we introduce a novel method that extracts information about higher-order structures in the graph while still using the efficient low-dimensional persistent homology algorithm. On standard benchmark datasets, we show that our method can lead to up to $31\%$ improvements in test accuracy.

cs.LG↗

arXiv2024

Characterizing Physician Referral Networks with Ricci Curvature

Identifying (a) systemic barriers to quality healthcare access and (b) key indicators of care efficacy in the United States remains a significant challenge. To improve our understanding of regional disparities in care delivery, we introduce a novel application of curvature, a geometrical-topological property of networks, to Physician Referral Networks. Our initial findings reveal that Forman-Ricci and Ollivier-Ricci curvature measures, which are known for their expressive power in characterizing network structure, offer promising indicators for detecting variations in healthcare efficacy while capturing a range of significant regional demographic features. We also present APPARENT, an open-source tool that leverages Ricci curvature and other network features to examine correlations between regional Physician Referral Networks structure, local census data, healthcare effectiveness, and patient outcomes.

cs.SI↗