arXiv subjects

Michael T. Schaub

Publications and source records attributed to Michael T. Schaub.

At least 37 records · Page 2Linked to original sources

arXiv2025

Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs

Residual connections and normalization layers have become standard design choices for graph neural networks (GNNs), and were proposed as solutions to the mitigate the oversmoothing problem in GNNs. However, how exactly these methods help alleviate the oversmoothing problem from a theoretical perspective is not well understood. In this work, we provide a formal and precise characterization of (linearized) GNNs with residual connections and normalization layers. We establish that (a) for residual connections, the incorporation of the initial features at each layer can prevent the signal from becoming too smooth, and determines the subspace of possible node representations; (b) batch normalization prevents a complete collapse of the output embedding space to a one-dimensional subspace through the individual rescaling of each column of the feature matrix. This results in the convergence of node representations to the top-$k$ eigenspace of the message-passing operator; (c) moreover, we show that the centering step of a normalization layer -- which can be understood as a projection -- alters the graph signal in message-passing in such a way that relevant information can become harder to extract. We therefore introduce a novel, principled normalization layer called GraphNormv2 in which the centering step is learned such that it does not distort the original graph signal in an undesirable way. Experimental results confirm the effectiveness of our method.

cs.LG↗

arXiv2025

Topological Point Cloud Clustering

We present Topological Point Cloud Clustering (TPCC), a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features. TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud. As it is based on considering sparse eigenvector computations, TPCC is similarly easy to interpret and implement as spectral clustering. However, by focusing not just on a single matrix associated to a graph created from the point cloud data, but on a whole set of Hodge-Laplacians associated to an appropriately constructed simplicial complex, we can leverage a far richer set of topological features to characterize the data points within the point cloud and benefit from the relative robustness of topological techniques against noise. We test the performance of TPCC on both synthetic and real-world data and compare it with classical spectral clustering.

math.AT↗

arXiv2025

A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals

We present a Bayesian perspective on quantifying the uncertainty of graph signals estimated or reconstructed from imperfect observations. We show that many conventional methods of graph signal estimation, reconstruction and imputation, can be reinterpreted as finding the mean of a posterior Gaussian distribution, with a covariance matrix shaped by the graph structure. In this perspective, assumptions of signal smoothness as well as bandlimitedness are naturally expressible as the choice of certain prior distributions; noisy, noise-free or partial observations are expressible in terms of certain likelihood models. In addition to providing a point estimate, as most standard estimation strategies do, our probabilistic framework enables us to characterize the shape of the estimated signal distribution around the estimate in terms of the posterior covariance matrix.

eess.SP↗

arXiv2025

Point-Level Topological Representation Learning on Point Clouds

Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise and heterogeneous sampling.

math.AT↗

arXiv2025

Convergence of gradient based training for linear Graph Neural Networks

Graph Neural Networks (GNNs) are powerful tools for addressing learning problems on graph structures, with a wide range of applications in molecular biology and social networks. However, the theoretical foundations underlying their empirical performance are not well understood. In this article, we examine the convergence of gradient dynamics in the training of linear GNNs. Specifically, we prove that the gradient flow training of a linear GNN with mean squared loss converges to the global minimum at an exponential rate. The convergence rate depends explicitly on the initial weights and the graph shift operator, which we validate on synthetic datasets from well-known graph models and real-world datasets. Furthermore, we discuss the gradient flow that minimizes the total weights at the global minimum. In addition to the gradient flow, we study the convergence of linear GNNs under gradient descent training, an iterative scheme viewed as a discretization of gradient flow.

cs.LG↗

arXiv2025

Efficient Sampling of Temporal Networks with Preserved Causality Structure

In this paper, we extend the classical Color Refinement algorithm for static networks to temporal (undirected and directed) networks. This enables us to design an algorithm to sample synthetic networks that preserves the $d$-hop neighborhood structure of a given temporal network. The higher $d$ is chosen, the better the temporal neighborhood structure of the original network is preserved. Specifically, we provide efficient algorithms that preserve time-respecting ("causal") paths in the networks up to length $d$, and scale to real-world network sizes. We validate our approach theoretically (for Degree and Katz centrality) and experimentally (for edge persistence, causal triangles, and burstiness). An experimental comparison shows that our method retains these key temporal characteristics more effectively than existing randomization methods.

cs.SI↗

arXiv2024

TopoX: A Suite of Python Packages for Machine Learning on Topological Domains

We introduce TopoX, a Python software suite that provides reliable and user-friendly building blocks for computing and machine learning on topological domains that extend graphs: hypergraphs, simplicial, cellular, path and combinatorial complexes. TopoX consists of three packages: TopoNetX facilitates constructing and computing on these domains, including working with nodes, edges and higher-order cells; TopoEmbedX provides methods to embed topological domains into vector spaces, akin to popular graph-based embedding algorithms such as node2vec; TopoModelX is built on top of PyTorch and offers a comprehensive toolbox of higher-order message passing functions for neural networks on topological domains. The extensively documented and unit-tested source code of TopoX is available under MIT license at https://pyt-team.github.io/}{https://pyt-team.github.io/.

cs.LG↗

arXiv2024

Topological Trajectory Classification and Landmark Inference on Simplicial Complexes

We consider the problem of classifying trajectories on a discrete or discretised 2-dimensional manifold modelled by a simplicial complex. Previous works have proposed to project the trajectories into the harmonic eigenspace of the Hodge Laplacian, and then cluster the resulting embeddings. However, if the considered space has vanishing homology (i.e., no "holes"), then the harmonic space of the 1-Hodge Laplacian is trivial and thus the approach fails. Here we propose to view this issue akin to a sensor placement problem and present an algorithm that aims to learn "optimal holes" to distinguish a set of given trajectory classes. Specifically, given a set of labelled trajectories, which we interpret as edge-flows on the underlying simplicial complex, we search for 2-simplices whose deletion results in an optimal separation of the trajectory labels according to the corresponding spectral embedding of the trajectories into the harmonic space. Finally, we generalise this approach to the unsupervised setting.

cs.SI↗

arXiv2024

Graph Neural Networks Do Not Always Oversmooth

Graph neural networks (GNNs) have emerged as powerful tools for processing relational data in applications. However, GNNs suffer from the problem of oversmoothing, the property that the features of all nodes exponentially converge to the same vector over layers, prohibiting the design of deep GNNs. In this work we study oversmoothing in graph convolutional networks (GCNs) by using their Gaussian process (GP) equivalence in the limit of infinitely many hidden features. By generalizing methods from conventional deep neural networks (DNNs), we can describe the distribution of features at the output layer of deep GCNs in terms of a GP: as expected, we find that typical parameter choices from the literature lead to oversmoothing. The theory, however, allows us to identify a new, non-oversmoothing phase: if the initial weights of the network have sufficiently large variance, GCNs do not oversmooth, and node features remain informative even at large depth. We demonstrate the validity of this prediction in finite-size GCNs by training a linear classifier on their output. Moreover, using the linearization of the GCN GP, we generalize the concept of propagation depth of information from DNNs to GCNs. This propagation depth diverges at the transition between the oversmoothing and non-oversmoothing phase. We test the predictions of our approach and find good agreement with finite-size GCNs. Initializing GCNs near the transition to the non-oversmoothing phase, we obtain networks which are both deep and expressive.

stat.ML↗

arXiv2024

Detectability of hierarchical communities in networks

We study the problem of recovering a planted hierarchy of partitions in a network. The detectability of a single planted partition has previously been analysed in detail and a phase transition has been identified below which the partition cannot be detected. Here we show that, in the hierarchical setting, there exist additional phases in which the presence of multiple consistent partitions can either help or hinder detection. Accordingly, the detectability limit for non-hierarchical partitions typically provides insufficient information about the detectability of the complete hierarchical structure, as we highlight with several constructive examples.

cs.SI↗

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

Learning From Simplicial Data Based on Random Walks and 1D Convolutions

Triggered by limitations of graph-based deep learning methods in terms of computational expressivity and model flexibility, recent years have seen a surge of interest in computational models that operate on higher-order topological domains such as hypergraphs and simplicial complexes. While the increased expressivity of these models can indeed lead to a better classification performance and a more faithful representation of the underlying system, the computational cost of these higher-order models can increase dramatically. To this end, we here explore a simplicial complex neural network learning architecture based on random walks and fast 1D convolutions (SCRaWl), in which we can adjust the increase in computational cost by varying the length and number of random walks considered while accounting for higher-order relationships. Importantly, due to the random walk-based design, the expressivity of the proposed architecture is provably incomparable to that of existing message-passing simplicial neural networks. We empirically evaluate SCRaWl on real-world datasets and show that it outperforms other simplicial neural networks.

cs.LG↗

arXiv2024

Ellipsoidal embeddings of graphs

Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis tools are based on the algebraic representation of graphs via matrices such as the graph Laplacian, or on associated graph embeddings. Such embeddings associate to each node a set of coordinates in a vector space, a representation which can then be employed for learning tasks such as the classification or alignment of the nodes of the graph. As the geometric picture provided by embedding methods enables the use of a multitude of methods developed for vector space data, embeddings have thus gained interest both from a theoretical as well as a practical perspective. Inspired by trace-optimization problems, often encountered in the analysis of graph-based data, here we present a method to derive ellipsoidal embeddings of the nodes of a graph, in which each node is assigned a set of coordinates on the surface of a hyperellipsoid. Our method may be seen as an alternative to popular spectral embedding techniques, to which it shares certain similarities we discuss. To illustrate the utility of the embedding we conduct a case study in which we analyse synthetic and real world networks with modular structure, and compare the results obtained with known methods in the literature.

cs.SI↗

arXiv2024

Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal

The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for applications such as graph classification, clustering, or eigenmode analysis. Recently, the Hodge Laplacian has come into focus as a generalisation of the ordinary Laplacian for higher-order graph models such as simplicial and cellular complexes. Akin to the traditional analysis of graph Laplacians, many authors analyse the smallest eigenvalues of the Hodge Laplacian, which are connected to important topological properties such as homology. However, small eigenvalues of the Hodge Laplacian can carry different information depending on whether they are related to curl or gradient eigenmodes, and thus may not be comparable. We therefore introduce the notion of persistent eigenvector similarity and provide a method to track individual harmonic, curl, and gradient eigenvectors/-values through the so-called persistence filtration, leveraging the full information contained in the Hodge-Laplacian spectrum across all possible scales of a point cloud. Finally, we use our insights (a) to introduce a novel form of Hodge spectral clustering and (b) to classify edges and higher-order simplices based on their relationship to the smallest harmonic, curl, and gradient eigenvectors.

math.AT↗

arXiv2024

Optimal transport distances for directed, weighted graphs: a case study with cell-cell communication networks

Comparing graphs by means of optimal transport has recently gained significant attention, as the distances induced by optimal transport provide both a principled metric between graphs as well as an interpretable description of the associated changes between graphs in terms of a transport plan. As the lack of symmetry introduces challenges in the typically considered formulations, optimal transport distances for graphs have mostly been developed for undirected graphs. Here, we propose two distance measures to compare directed graphs based on variants of optimal transport: (i) an earth movers distance (Wasserstein) and (ii) a Gromov-Wasserstein (GW) distance. We evaluate these two distances and discuss their relative performance for both simulated graph data and real-world directed cell-cell communication graphs, inferred from single-cell RNA-seq data.

cs.LG↗

arXiv2024

Simplicial Convolutional Filters

We study linear filters for processing signals supported on abstract topological spaces modeled as simplicial complexes, which may be interpreted as generalizations of graphs that account for nodes, edges, triangular faces etc. To process such signals, we develop simplicial convolutional filters defined as matrix polynomials of the lower and upper Hodge Laplacians. First, we study the properties of these filters and show that they are linear and shift-invariant, as well as permutation and orientation equivariant. These filters can also be implemented in a distributed fashion with a low computational complexity, as they involve only (multiple rounds of) simplicial shifting between upper and lower adjacent simplices. Second, focusing on edge-flows, we study the frequency responses of these filters and examine how we can use the Hodge-decomposition to delineate gradient, curl and harmonic frequencies. We discuss how these frequencies correspond to the lower- and the upper-adjacent couplings and the kernel of the Hodge Laplacian, respectively, and can be tuned independently by our filter designs. Third, we study different procedures for designing simplicial convolutional filters and discuss their relative advantages. Finally, we corroborate our simplicial filters in several applications: to extract different frequency components of a simplicial signal, to denoise edge flows, and to analyze financial markets and traffic networks.

eess.SP↗

arXiv2024

A Wasserstein Graph Distance Based on Distributions of Probabilistic Node Embeddings

Distance measures between graphs are important primitives for a variety of learning tasks. In this work, we describe an unsupervised, optimal transport based approach to define a distance between graphs. Our idea is to derive representations of graphs as Gaussian mixture models, fitted to distributions of sampled node embeddings over the same space. The Wasserstein distance between these Gaussian mixture distributions then yields an interpretable and easily computable distance measure, which can further be tailored for the comparison at hand by choosing appropriate embeddings. We propose two embeddings for this framework and show that under certain assumptions about the shape of the resulting Gaussian mixture components, further computational improvements of this Wasserstein distance can be achieved. An empirical validation of our findings on synthetic data and real-world Functional Brain Connectivity networks shows promising performance compared to existing embedding methods.

cs.CE↗

arXiv2024

Faster optimal univariate microgaggregation

Microaggregation is a method to coarsen a dataset, by optimally clustering data points in groups of at least $k$ points, thereby providing a $k$-anonymity type disclosure guarantee for each point in the dataset. Previous algorithms for univariate microaggregation had a $O(k n)$ time complexity. By rephrasing microaggregation as an instance of the concave least weight subsequence problem, in this work we provide improved algorithms that provide an optimal univariate microaggregation on sorted data in $O(n)$ time and space. We further show that our algorithms work not only for sum of squares cost functions, as typically considered, but seamlessly extend to many other cost functions used for univariate microaggregation tasks. In experiments we show that the presented algorithms lead to real world performance improvements.

cs.DS↗