Consistency of trace norm minimization

TL;DR

This paper investigates the conditions under which trace norm minimization reliably estimates low-rank matrices, extending Lasso consistency results and proposing an adaptive method that ensures rank consistency.

Contribution

It provides necessary and sufficient conditions for rank consistency of trace norm minimization and introduces an adaptive approach that guarantees consistency beyond standard conditions.

Findings

Derived necessary and sufficient conditions for rank consistency.

Proposed an adaptive method that ensures rank consistency.

Extended Lasso consistency results to trace norm minimization.

Abstract

Regularization by the sum of singular values, also referred to as the trace norm, is a popular technique for estimating low rank rectangular matrices. In this paper, we extend some of the consistency results of the Lasso to provide necessary and sufficient conditions for rank consistency of trace norm minimization with the square loss. We also provide an adaptive version that is rank consistent even when the necessary condition for the non adaptive version is not fulfilled.