Learning Semidefinite Regularizers

TL;DR

This paper introduces a novel method for learning semidefinite regularizers from data, enabling structured inverse problem solutions without requiring explicit domain knowledge, using advanced matrix factorization and optimization techniques.

Contribution

It generalizes dictionary learning to semidefinite regularizers, providing an algorithm with local linear convergence based on Operator Sinkhorn scaling and geometric analysis.

Findings

Algorithm converges locally linearly under certain conditions.

Regularizers improve semidefinite programming for inverse problems.

Framework leverages structured matrix factorizations and stability analysis.

Abstract

Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function, which is specified based on prior domain-specific expertise to induce a desired structure in the solution. We consider the problem of learning suitable regularization functions from data in settings in which precise domain knowledge is not directly available. Previous work under the title of `dictionary learning' or `sparse coding' may be viewed as learning a regularization function that can be computed via linear programming. We describe generalizations of these methods to learn regularizers that can be computed and optimized via semidefinite programming. Our framework for learning such semidefinite regularizers is based on obtaining structured factorizations of data matrices, and our algorithmic approach for computing these factorizations combines recent techniques for rank minimization problems along with an operator analog of Sinkhorn scaling. Under suitable conditions on the input data, our algorithm provides a locally linearly convergent method for identifying the correct regularizer that promotes the type of structure contained in the data. Our analysis is based on the stability properties of Operator Sinkhorn scaling and their relation to geometric aspects of determinantal varieties (in particular tangent spaces with respect to these varieties). The regularizers obtained using our framework can be employed effectively in semidefinite programming relaxations for solving inverse problems.