Summary Based Structures with Improved Sublinear Recovery for Compressed Sensing

TL;DR

This paper introduces a new measurement matrix for compressed sensing that enables sublinear time recovery of sparse signals with fewer measurements, outperforming existing algorithms in empirical oversampling efficiency.

Contribution

It proposes a novel class of measurement matrices based on binary sequence summaries and demonstrates recovery guarantees for multiple algorithms, including a sublinear time method.

Findings

Sublinear time recovery of sparse vectors with fewer measurements.

Empirical oversampling constants better than existing algorithms.

Potential applications in market analysis and real-time sensing.

Abstract

We introduce a new class of measurement matrices for compressed sensing, using low order summaries over binary sequences of a given length. We prove recovery guarantees for three reconstruction algorithms using the proposed measurements, including $\ell_1$ minimization and two combinatorial methods. In particular, one of the algorithms recovers $k$-sparse vectors of length $N$ in sublinear time $\text{poly}(k\log{N})$, and requires at most $\Omega(k\log{N}\log\log{N})$ measurements. The empirical oversampling constant of the algorithm is significantly better than existing sublinear recovery algorithms such as Chaining Pursuit and Sudocodes. In particular, for $10^3\leq N\leq 10^8$ and $k=100$, the oversampling factor is between 3 to 8. We provide preliminary insight into how the proposed constructions, and the fast recovery scheme can be used in a number of practical applications such as market basket analysis, and real time compressed sensing implementation.