Improving compressed sensing with the diamond norm

TL;DR

This paper introduces the diamond norm as a new regularizer for low-rank matrix recovery, showing it can outperform the nuclear norm in certain applications by leveraging tensorial structures.

Contribution

The work identifies the diamond norm as an improved regularizer for low-rank matrix recovery, characterizes the matrices where it outperforms nuclear norm, and demonstrates its advantages numerically.

Findings

Diamond norm outperforms nuclear norm in specific matrix recovery tasks.

The descent cone of the diamond norm is contained within that of the nuclear norm for certain matrices.

Numerical experiments show improved reconstruction in signal analysis and quantum information applications.

Abstract

In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a minimal number of linear measurements. Within the paradigm of compressed sensing, this is made computationally efficient by minimizing the nuclear norm as a convex surrogate for rank. In this work, we identify an improved regularizer based on the so-called diamond norm, a concept imported from quantum information theory. We show that -for a class of matrices saturating a certain norm inequality- the descent cone of the diamond norm is contained in that of the nuclear norm. This suggests superior reconstruction properties for these matrices. We explicitly characterize this set of matrices. Moreover, we demonstrate numerically that the diamond norm indeed outperforms the nuclear norm in a number of relevant applications: These include signal analysis tasks such as blind matrix deconvolution or the retrieval of certain unitary basis changes, as well as the quantum information problem of process tomography with random measurements. The diamond norm is defined for matrices that can be interpreted as order-4 tensors and it turns out that the above condition depends crucially on that tensorial structure. In this sense, this work touches on an aspect of the notoriously difficult tensor completion problem.