Learning Sheaf Laplacian Optimizing Restriction Maps
Leonardo Di Nino, Sergio Barbarossa, Paolo Di Lorenzo
TL;DR
The paper addresses learning the sheaf Laplacian and the underlying cellular graph structure from node-observed data, generalizing traditional graph Laplacian learning to cellular sheaves. It decouples the problem into local Procrustes-based estimation of edge restriction maps, preceded by a denoising step to obtain node subspace representations, and selects a fixed number of edges to minimize total variation. The approach yields closed-form solutions for local problems, enabling efficient learning and capturing cross-node correlations and subspace-dimension effects; it outperforms conventional graph methods in smoothing data and reveals meaningful clusters by subspace dimension. These results suggest substantial practical impact for topology-aware signal processing, enabling scalable learning of higher-order network structures with interpretable alignment of local data spaces.
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.