Isotropically Random Orthogonal Matrices: Performance of LASSO and Minimum Conic Singular Values

TL;DR

This paper analyzes the performance of the Generalized LASSO and minimum conic singular values for isotropically random orthogonal matrices, showing their superiority over Gaussian matrices through a modified GMT framework.

Contribution

It extends the GMT analysis to i.r.o. matrices and provides sharp performance characterizations and bounds, highlighting their advantages over Gaussian ensembles.

Findings

i.r.o. matrices outperform Gaussian matrices in LASSO recovery

Derived sharp normalized squared error for i.r.o. ensemble

Established larger lower bounds on conic singular values for i.r.o. matrices

Abstract

Recently, the precise performance of the Generalized LASSO algorithm for recovering structured signals from compressed noisy measurements, obtained via i.i.d. Gaussian matrices, has been characterized. The analysis is based on a framework introduced by Stojnic and heavily relies on the use of Gordon's Gaussian min-max theorem (GMT), a comparison principle on Gaussian processes. As a result, corresponding characterizations for other ensembles of measurement matrices have not been developed. In this work, we analyze the corresponding performance of the ensemble of isotropically random orthogonal (i.r.o.) measurements. We consider the constrained version of the Generalized LASSO and derive a sharp characterization of its normalized squared error in the large-system limit. When compared to its Gaussian counterpart, our result analytically confirms the superiority in performance of the i.r.o. ensemble. Our second result, derives an asymptotic lower bound on the minimum conic singular values of i.r.o. matrices. This bound is larger than the corresponding bound on Gaussian matrices. To prove our results we express i.r.o. matrices in terms of Gaussians and show that, with some modifications, the GMT framework is still applicable.