Learning Sheaf Laplacian Optimizing Restriction Maps

• URL: http://arxiv.org/abs/2501.19207v1
• Date: Fri, 31 Jan 2025 15:15:08 GMT
• Title: Learning Sheaf Laplacian Optimizing Restriction Maps
• Authors: Leonardo Di Nino, Sergio Barbarossa, Paolo Di Lorenzo,
• Abstract summary: We propose a novel framework to infer the sheaf Laplacian from a set of data observed over the nodes of a graph. 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.
• Score: 19.477754758501707
• License:
• 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.

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