arXiv subjects

Bastian Rieck

Publications and source records attributed to Bastian Rieck.

At least 55 records · Page 3Linked to original sources

arXiv2024

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

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.

cs.LG↗

arXiv2024

Mapping the Multiverse of Latent Representations

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.

cs.LG↗

arXiv2024

Differentiable Euler Characteristic Transforms for Shape Classification

The Euler Characteristic Transform (ECT) has proven to be a powerful representation, 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.

cs.LG↗

arXiv2024

Simplicial Representation Learning with Neural $k$-Forms

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.

cs.LG↗

arXiv2024

The Manifold Density Function: An Intrinsic Method for the Validation of Manifold Learning

We introduce the manifold density function, which is an intrinsic method to validate manifold learning techniques. Our approach adapts and extends Ripley's $K$-function, and categorizes in an unsupervised setting the extent to which an output of a manifold learning algorithm captures the structure of a latent manifold. Our manifold density function generalizes to broad classes of Riemannian manifolds. In particular, we extend the manifold density function to general two-manifolds using the Gauss-Bonnet theorem, and demonstrate that the manifold density function for hypersurfaces is well approximated using the first Laplacian eigenvalue. We prove desirable convergence and robustness properties.

cs.LG↗

arXiv2023

All the World's a (Hyper)Graph: A Data Drama

We introduce Hyperbard, a dataset of diverse relational data representations derived from Shakespeare's plays. Our representations range from simple graphs capturing character co-occurrence in single scenes to hypergraphs encoding complex communication settings and character contributions as hyperedges with edge-specific node weights. By making multiple intuitive representations readily available for experimentation, we facilitate rigorous representation robustness checks in graph learning, graph mining, and network analysis, highlighting the advantages and drawbacks of specific representations. Leveraging the data released in Hyperbard, we demonstrate that many solutions to popular graph mining problems are highly dependent on the representation choice, thus calling current graph curation practices into question. As an homage to our data source, and asserting that science can also be art, we present all our points in the form of a play.

cs.LG↗

arXiv2023

MAGNet: Motif-Agnostic Generation of Molecules from Shapes

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 struggle to represent substructures beyond their known motif set. To alleviate this issue and increase flexibility across datasets, we propose MAGNet, a graph-based model that generates abstract shapes before allocating atom and bond types. To this end, we introduce a novel factorisation of the molecules' data distribution that accounts for the molecules' global context and facilitates learning adequate assignments of atoms and bonds onto shapes. Despite the added complexity of shape abstractions, MAGNet outperforms most other graph-based approaches on standard benchmarks. Importantly, we demonstrate that MAGNet's improved expressivity leads to molecules with more topologically distinct structures and, at the same time, diverse atom and bond assignments.

physics.chem-ph↗

arXiv2023

Curvature Filtrations for Graph Generative Model Evaluation

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 that has recently proved its utility 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.

cs.LG↗

arXiv2023

Filtration Surfaces for Dynamic Graph Classification

Existing approaches for classifying dynamic graphs either lift graph kernels to the temporal domain, or use graph neural networks (GNNs). However, current baselines have scalability issues, cannot handle a changing node set, or do not take edge weight information into account. We propose filtration surfaces, a novel method that is scalable and flexible, to alleviate said restrictions. We experimentally validate the efficacy of our model and show that filtration surfaces outperform previous state-of-the-art baselines on datasets that rely on edge weight information. Our method does so while being either completely parameter-free or having at most one parameter, and yielding the lowest overall standard deviation among similarly scalable methods.

cs.LG↗

arXiv2023

Weisfeiler and Leman go Machine Learning: The Story so far

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.

cs.LG↗

arXiv2023

Topological Singularity Detection at Multiple Scales

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.

cs.LG↗

arXiv2023

Evaluating the "Learning on Graphs" Conference Experience

With machine learning conferences growing ever larger, and reviewing processes becoming increasingly elaborate, more data-driven insights into their workings are required. In this report, we present the results of a survey accompanying the first "Learning on Graphs" (LoG) Conference. The survey was directed to evaluate the submission and review process from different perspectives, including authors, reviewers, and area chairs alike.

cs.LG↗

arXiv2023

Metric Space Magnitude and Generalisation in Neural Networks

Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

cs.LG↗

arXiv2023

DONUT -- Creation, Development, and Opportunities of a Database

DONUT is a database of papers about practical, real-world uses of Topological Data Analysis (TDA). Its original seed was planted in a group chat formed during the HIM Spring School on Applied and Computational Algebraic Topology in April 2017. This document describes the creation, curation, and maintenance process of the database.

cs.DL↗

arXiv2023

Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework

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.

cs.LG↗

arXiv2023

A Diffusion Model Predicts 3D Shapes from 2D Microscopy Images

Diffusion models are a special type of generative model, capable of synthesising new data from a learnt distribution. We introduce DISPR, a diffusion-based model for solving the inverse problem of three-dimensional (3D) cell shape prediction from two-dimensional (2D) single cell microscopy images. Using the 2D microscopy image as a prior, DISPR is conditioned to predict realistic 3D shape reconstructions. To showcase the applicability of DISPR as a data augmentation tool in a feature-based single cell classification task, we extract morphological features from the red blood cells grouped into six highly imbalanced classes. Adding features from the DISPR predictions to the three minority classes improved the macro F1 score from $F1_\text{macro} = 55.2 \pm 4.6\%$ to $F1_\text{macro} = 72.2 \pm 4.9\%$. We thus demonstrate that diffusion models can be successfully applied to inverse biomedical problems, and that they learn to reconstruct 3D shapes with realistic morphological features from 2D microscopy images.

cs.CV↗

arXiv2023

Euler Characteristic Transform Based Topological Loss for Reconstructing 3D Images from Single 2D Slices

The computer vision task of reconstructing 3D images, i.e., shapes, from their single 2D image slices is extremely challenging, more so in the regime of limited data. Deep learning models typically optimize geometric loss functions, which may lead to poor reconstructions as they ignore the structural properties of the shape. To tackle this, we propose a novel topological loss function based on the Euler Characteristic Transform. This loss can be used as an inductive bias to aid the optimization of any neural network toward better reconstructions in the regime of limited data. We show the effectiveness of the proposed loss function by incorporating it into SHAPR, a state-of-the-art shape reconstruction model, and test it on two benchmark datasets, viz., Red Blood Cells and Nuclei datasets. We also show a favourable property, namely injectivity and discuss the stability of the topological loss function based on the Euler Characteristic Transform.

cs.LG↗

arXiv2023

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.

cs.LG↗