Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure

TL;DR

This paper introduces SpodNet, a neural network module that guarantees outputs are symmetric positive-definite matrices with optional structural constraints like sparsity, enabling more expressive and constrained matrix estimation in various applications.

Contribution

The paper presents SpodNet, a novel neural network component that ensures SPD outputs with support for structural constraints, addressing limitations of previous methods.

Findings

SpodNet successfully learns SPD matrices with structural constraints.

It outperforms existing methods in accuracy and expressivity.

The approach is versatile across different applications.

Abstract

Estimating matrices in the symmetric positive-definite (SPD) cone is of interest for many applications ranging from computer vision to graph learning. While there exist various convex optimization-based estimators, they remain limited in expressivity due to their model-based approach. The success of deep learning motivates the use of learning-based approaches to estimate SPD matrices with neural networks in a data-driven fashion. However, designing effective neural architectures for SPD learning is challenging, particularly when the task requires additional structural constraints, such as element-wise sparsity. Current approaches either do not ensure that the output meets all desired properties or lack expressivity. In this paper, we introduce SpodNet, a novel and generic learning module that guarantees SPD outputs and supports additional structural constraints. Notably, it solves the challenging task of learning jointly SPD and sparse matrices. Our experiments illustrate the versatility and relevance of SpodNet layers for such applications.