Structured regularization for constrained optimization on the SPD manifold

Abstract Matrix-valued optimization tasks, including those involving symmetric positive definite (SPD) matrices, arise in a wide range of applications in machine learning, data science and statistics. Classically, such problems are solved via constrained Euclidean optimization, where the domain is viewed as a Euclidean space and the structure of the matrices (e.g., positive definiteness) enters as constraints. More recently, geometric approaches that leverage parametrizations of the problem as unconstrained tasks on the corresponding matrix manifold have been proposed. While they exhibit algorithmic benefits in many settings, they cannot directly handle additional constraints, such as inequality or sparsity constraints. A remedy comes in the form of constrained Riemannian optimization methods, notably, Riemannian Frank–Wolfe and Projected Gradient Descent. However, both algorithms require potentially expensive subroutines that can introduce computational bottlenecks in practice. To mitigate these shortcomings we introduce a class of structured regularizers, based on symmetric gauge functions, which allow for solving constrained optimization on the SPD manifold with faster unconstrained methods. We show that our structured regularizers can be chosen to preserve or induce desirable structure, in particular convexity and ‘difference of convex’ structure. We demonstrate the effectiveness of our approach in numerical experiments.