Modified Basis Pursuit Denoising(MODIFIED-BPDN) for Noisy Compressive Sensing with Partially Known Support

TL;DR

This paper introduces a modified basis pursuit denoising method for reconstructing sparse signals from noisy measurements when part of the support is known, providing computable error bounds and improved performance for small unknown supports.

Contribution

The work proposes a new modified-BPDN algorithm with computable error bounds, demonstrating tighter bounds and fewer measurement requirements when the unknown support is small.

Findings

Modified-BPDN bounds are tighter than BPDN bounds.

Bounds are computable and suitable for Monte Carlo analysis.

Fewer measurements are needed for accurate reconstruction with small unknown support.

Abstract

In this work, we study the problem of reconstructing a sparse signal from a limited number of linear 'incoherent' noisy measurements, when a part of its support is known. The known part of the support may be available from prior knowledge or from the previous time instant (in applications requiring recursive reconstruction of a time sequence of sparse signals, e.g. dynamic MRI). We study a modification of Basis Pursuit Denoising (BPDN) and bound its reconstruction error. A key feature of our work is that the bounds that we obtain are computable. Hence, we are able to use Monte Carlo to study their average behavior as the size of the unknown support increases. We also demonstrate that when the unknown support size is small, modified-BPDN bounds are much tighter than those for BPDN, and hold under much weaker sufficient conditions (require fewer measurements).