17 accepted papers
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…
Optimal transport (OT) provides a robust framework for comparing probability distributions. Its effectiveness is significantly influenced by the choice of the underlying ground metric. Traditionally, the ground metric has either been (i) predefined, e.g., as a Euclidean metric, or (ii) learned in a…
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 give a single complex description of the global structure of the point cloud. However, common machine learning appli…
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 perspec…
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 distributi…
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 ord…
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 features of all nodes exponentially converge to the same vector over layers, prohibiting the design of deep GNNs. In t…
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 incr…
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 th…
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 incorporat…
Similar to community detection, partitioning the nodes of a complex network according to their structural roles aims to identify fundamental building blocks of a network, which can be used, e.g., to find simplified descriptions of the network connectivity, to derive reduced order models for dynamica…
We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our frame…
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 spec…
Finding equitable partitions is closely related to the extraction of graph symmetries and of interest in a variety of applications context such as node role detection, cluster synchronization, consensus dynamics, and network control problems. In this work we study a blind identification problem in w…
We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to construct a family of wavelets for higher-order signals on sim…
The processing of signals supported on non-Euclidean domains has attracted large interest recently. Thus far, such non-Euclidean domains have been abstracted primarily as graphs with signals supported on the nodes, though the processing of signals on more general structures such as simplicial comple…
We discuss a variant of `blind' community detection, in which we aim to partition an unobserved network from the observation of a (dynamical) graph signal defined on the network. We consider a scenario where our observed graph signals are obtained by filtering white noise input, and the underlying n…