Compressed Sensing with Very Sparse Gaussian Random Projections

TL;DR

This paper introduces a simple, efficient compressed sensing method using very sparse Gaussian random projections, achieving comparable accuracy to L1 decoding with fewer measurements by employing novel estimators.

Contribution

It proposes two new estimators for sparse signal recovery that require fewer measurements and are computationally efficient, improving upon existing sparse matrix methods.

Findings

Achieves $1.551 eK ext{ log } K/ ext{delta}$ measurements for signal recovery.

Supports support detection with $eK ext{ log } N/ ext{delta}$ measurements.

Comparable recovery accuracy to L1 decoding at the same measurement count.

Abstract

We study the use of very sparse random projections for compressed sensing (sparse signal recovery) when the signal entries can be either positive or negative. In our setting, the entries of a Gaussian design matrix are randomly sparsified so that only a very small fraction of the entries are nonzero. Our proposed decoding algorithm is simple and efficient in that the major cost is one linear scan of the coordinates. We have developed two estimators: (i) the {\em tie estimator}, and (ii) the {\em absolute minimum estimator}. Using only the tie estimator, we are able to recover a $K$-sparse signal of length $N$ using $1.551 eK \log K/\delta$ measurements (where $\delta\leq 0.05$ is the confidence). Using only the absolute minimum estimator, we can detect the support of the signal using $eK\log N/\delta$ measurements. For a particular coordinate, the absolute minimum estimator requires fewer measurements (i.e., with a constant $e$ instead of $1.551e$). Thus, the two estimators can be combined to form an even more practical decoding framework. Prior studies have shown that existing one-scan (or roughly one-scan) recovery algorithms using sparse matrices would require substantially more (e.g., one order of magnitude) measurements than L1 decoding by linear programming, when the nonzero entries of signals can be either negative or positive. In this paper, following a known experimental setup, we show that, at the same number of measurements, the recovery accuracies of our proposed method are (at least) similar to the standard L1 decoding.