Learning Sheaf Laplacian Optimizing Restriction Maps

TL;DR

This paper introduces a new, efficient method to learn the sheaf Laplacian, including graph topology and restriction maps, from data, enhancing graph signal processing with a numerically efficient approach based on sheaf theory.

Contribution

It presents a novel framework for inferring sheaf Laplacians from data, improving computational efficiency over existing semidefinite programming methods.

Findings

The method accurately infers graph topology and restriction maps from observed data.

The approach is significantly more numerically efficient than semidefinite programming methods.

Data characteristics like cross-correlation and dimensionality differences influence the inferred graph.

Abstract

The aim of this paper is to propose a novel framework to infer the sheaf Laplacian, including the topology of a graph and the restriction maps, from a set of data observed over the nodes of a graph. The proposed method is based on sheaf theory, which represents an important generalization of graph signal processing. The learning problem aims to find the sheaf Laplacian that minimizes the total variation of the observed data, where the variation over each edge is also locally minimized by optimizing the associated restriction maps. Compared to alternative methods based on semidefinite programming, our solution is significantly more numerically efficient, as all its fundamental steps are resolved in closed form. The method is numerically tested on data consisting of vectors defined over subspaces of varying dimensions at each node. We demonstrate how the resulting graph is influenced by two key factors: the cross-correlation and the dimensionality difference of the data residing on the graph's nodes.