Compressive distilled sensing: Sparse recovery using adaptivity in compressive measurements

Abstract

Abstract-The recently-proposed theory of distilled sensing establishes that adaptivity in sampling can dramatically improve the performance of sparse recovery in noisy settings. In particular, it is now known that adaptive point sampling enables the detection and/or support recovery of sparse signals that are otherwise too weak to be recovered using any method based on non-adaptive point sampling. In this paper the theory of distilled sensing is extended to highly-undersampled regimes, as in compressive sensing. A simple adaptive sampling-and-refinement procedure called compressive distilled sensing is proposed, where each step of the procedure utilizes information from previous observations to focus subsequent measurements into the proper signal subspace, resulting in a significant improvement in effective measurement SNR on the signal subspace. As a result, for the same budget of sensing resources, compressive distilled sensing can result in significantly improved error bounds compared to those for traditional compressive sensing.

Sequentially designed compressed sensing

“…This SCS procedure proposed here (depicted in Algorithm 1) also improves on our earlier compressive distilled sensing (CDS) work [7], which employed a different measurement and estimation process (based on random Gaussian sensing matrices). The recovery results we obtained in [7] were valid only for sparse signals with dynamic range (the ratio of the largest to smallest nonzero component amplitudes) on the order of a constant.…”

Adaptive sensing for sparse recovery

“…The following theorem describes the error performance of this support estimator obtained using the CDS adaptive compressive sampling procedure. The result follows from iterated application of Lemmas 1 and 2 in [25], which are analogous to Lemmas 1.1 and 1.2 here, as well as the results in [26] which describe the model selection performance of the LASSO. Theorem 1.3.…”

Improved bounds for sparse recovery from adaptive measurements

“…If the linear combinations are non-adaptive, this leads to a regression model commonly studied in the Lasso and Compressed Sensing literature [18], [19]. However, sequentially tuning the linear combinations leads to an adaptive version of the regression model which can be shown to provide significant improvements, as well [20].…”

“…ISIT 2010, Austin, Texas, U.S.A., June 13 -18, 2010 1563 978-1-4244-7892-7/10/$26.00 ©2010 IEEE ISIT 2010 this paper is a theoretical analysis that reveals the dramatic gains that can be attained using such sequential procedures. Our focus here is on a sequential adaptive sampling procedure called distilled sensing (DS).…”