Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure
This addresses a challenge in applications like computer vision and graph learning by providing a data-driven, learning-based approach for SPD estimation with structural constraints, representing a novel method for a known bottleneck.
The paper tackles the problem of estimating symmetric positive-definite (SPD) matrices with structural constraints like sparsity, introducing SpodNet, a neural module that guarantees SPD outputs and supports such constraints, enabling joint learning of SPD and sparse matrices.
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.