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

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 has thus led many to use neural networks to learn to estimate SPD matrices in a data-driven fashion. For learning structured outputs, one promising strategy involves architectures designed by unrolling iterative algorithms, which potentially benefit from inductive bias properties. However, designing correct unrolled architectures for SPD learning is difficult: they either do not guarantee that their output has all the desired properties, rely on heavy computations, or are overly restrained to specific matrices which hinders their expressivity. In this paper, we propose a novel and generic learning module with guaranteed SPD outputs called SpodNet, that also enables learning a larger class of functions than existing approaches. Notably, it solves the challenging task of learning jointly SPD and sparse matrices. Our experiments demonstrate the versatility of SpodNet layers.