Low solution rank of the matrix LASSO under RIP with consequences for rank-constrained algorithms

TL;DR

This paper demonstrates that solutions to the matrix LASSO problem are low rank under RIP conditions, with implications for rank-constrained algorithms and convergence guarantees.

Contribution

It provides the first proof that matrix LASSO solutions are low rank under RIP, impacting nonconvex optimization approaches.

Findings

LASSO solutions have low rank under RIP assumptions.

Low-rank projected gradient descent converges linearly to the LASSO solution.

All second-order critical points in the low-rank formulation are globally optimal.

Abstract

We show that solutions to the popular convex matrix LASSO problem (nuclear-norm--penalized linear least-squares) have low rank under similar assumptions as required by classical low-rank matrix sensing error bounds. Although the purpose of the nuclear norm penalty is to promote low solution rank, a proof has not yet (to our knowledge) been provided outside very specific circumstances. Furthermore, we show that this result has significant theoretical consequences for nonconvex rank-constrained optimization approaches. Specifically, we show that if (a) the ground truth matrix has low rank, (b) the (linear) measurement operator has the matrix restricted isometry property (RIP), and (c) the measurement error is small enough relative to the nuclear norm penalty, then the (unique) LASSO solution has rank (approximately) bounded by that of the ground truth. From this, we show (a) that a low-rank--projected proximal gradient descent algorithm will converge linearly to the LASSO solution from any initialization, and (b) that the nonconvex landscape of the low-rank Burer-Monteiro--factored problem formulation is benign in the sense that all second-order critical points are globally optimal and yield the LASSO solution.