Convex envelopes for fixed rank approximation

TL;DR

This paper introduces a convex envelope for fixed-rank matrix approximation, enabling the use of convex optimization methods with additional constraints, and analyzes its properties and proximity operator.

Contribution

A novel convex envelope for fixed-rank approximation is constructed, facilitating constrained optimization and providing insights into its properties and proximity operator.

Findings

Derived explicit expression for the convex envelope based on singular values.

Established global minimization properties of the convex envelope.

Studied the proximity operator associated with the convex envelope.

Abstract

A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added to the finite rank approximation problem. Expression for the dependence of the convex envelope on the singular values of the given matrix is derived and global minimization properties are derived. The corresponding proximity operator is also studied.