RIPless compressed sensing from anisotropic measurements

TL;DR

This paper extends compressed sensing theory to anisotropic measurement ensembles, establishing measurement bounds proportional to the covariance matrix's condition number without relying on RIP, using convex geometry techniques.

Contribution

It provides new bounds for compressed sensing with anisotropic measurements, avoiding RIP and leveraging convex geometry methods for improved simplicity and bounds.

Findings

Measurement bounds grow with the covariance matrix's condition number.

The approach does not rely on restricted isometry properties (RIP).

Results are derived using convex geometry concepts from low-rank matrix recovery.

Abstract

Compressed sensing is the art of reconstructing a sparse vector from its inner products with respect to a small set of randomly chosen measurement vectors. It is usually assumed that the ensemble of measurement vectors is in isotropic position in the sense that the associated covariance matrix is proportional to the identity matrix. In this paper, we establish bounds on the number of required measurements in the anisotropic case, where the ensemble of measurement vectors possesses a non-trivial covariance matrix. Essentially, we find that the required sampling rate grows proportionally to the condition number of the covariance matrix. In contrast to other recent contributions to this problem, our arguments do not rely on any restricted isometry properties (RIP's), but rather on ideas from convex geometry which have been systematically studied in the theory of low-rank matrix recovery. This allows for a simple argument and slightly improved bounds, but may lead to a worse dependency on noise (which we do not consider in the present paper).