Related papers: Hodgelets: Localized Spectral Representations of F…
Obtaining sparse, interpretable representations of observable data is crucial in many machine learning and signal processing tasks. For data representing flows along the edges of a graph, an intuitively interpretable way to obtain such…
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…
Simplicial complexes are generalizations of graphs that describe higher-order network interactions among nodes in the graph. Network dynamics described by graph Laplacian flows have been widely studied in network science and control theory,…
Our previous multiscale graph basis dictionaries/graph signal transforms -- Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives -- were…
We propose principled Gaussian processes (GPs) for modeling functions defined over the edge set of a simplicial 2-complex, a structure similar to a graph in which edges may form triangular faces. This approach is intended for learning…
Networks and network processes have emerged as powerful tools for modeling social interactions, disease propagation, and a variety of additional dynamics driven by relational structures. Recently, neural networks have been generalized to…
We consider the following inference problem: Given a set of edge-flow signals observed on a graph, lift the graph to a cell complex, such that the observed edge-flow signals can be represented as a sparse combination of gradient and curl…
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…
In this paper, we study linear filters to process signals defined on simplicial complexes, i.e., signals defined on nodes, edges, triangles, etc. of a simplicial complex, thereby generalizing filtering operations for graph signals. We…
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher…
This paper focuses on devising graph signal processing tools for the treatment of data defined on the edges of a graph. We first show that conventional tools from graph signal processing may not be suitable for the analysis of such signals.…
We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the…
A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimises the choice of wavelet for a dataset of graphs, such that their…
Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. Here we introduce a Hodge-theoretic approach to spectral theory and dimensionality reduction for…
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…
Recently, graph neural networks have been widely used for network embedding because of their prominent performance in pairwise relationship learning. In the real world, a more natural and common situation is the coexistence of pairwise…
Focusing on coupling between edges, we generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian -- the generalization of the graph…
We establish explicit operator norm bounds and essential self-adjointness criteria for discrete Hodge Laplacians on weighted graphs and simplicial complexes. For unweighted $d$-regular graphs we prove the universal estimate…
In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on…
We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian…