SHO-FA: Robust compressive sensing with order-optimal complexity, measurements, and bits

Abstract

Suppose x is any exactly k-sparse vector in n . We present a class of "sparse" matrices A, and a corresponding algorithm that we call SHO-FA (for Short and Fast 1 ) that, with high probability over A, can reconstruct x from Ax. The SHO-FA algorithm is related to the Invertible Bloom Lookup Tables (IBLTs) recently introduced by Goodrich et al., with two important distinctions -SHO-FA relies on linear measurements, and is robust to noise and approximate sparsity. The SHO-FA algorithm is the first to simultaneously have the following properties: (a) it requires only O(k) measurements, (b) the bitprecision of each measurement and each arithmetic operation is O (log(n) + P ) (here 2 −P corresponds to the desired relative error in the reconstruction of x), (c) the computational complexity of decoding is O(k) arithmetic operations, and (d) if the reconstruction goal is simply to recover a single component of x instead of all of x, with high probability over A this can be done in constant time. All constants above are independent of all problem parameters other than the desired probability of success. For a wide range of parameters these properties are information-theoretically order-optimal. In addition, our SHO-FA algorithm is robust to random noise, and (random) approximate sparsity for a large range of k. In particular, suppose the measured vector equals A(x + z) + e, where z and e correspond respectively to the source tail and measurement noise. Under reasonable statistical assumptions on z and e our decoding algorithm reconstructs x with an estimation error of O(||z||1 + (log k) 2 ||e||1). The SHO-FA algorithm works with high probability over A, z, and e, and still requires only O(k) steps and O(k) measurements over O(log(n))-bit numbers. This is in contrast to most existing algorithms which focus on the "worst-case" z model, where it is known Ω(k log(n/k)) measurements over O(log(n))-bit numbers are necessary.

A hybrid DFT-LDPC framework for fast, efficient and robust compressive sensing

“…1 As we were submitting this paper, we became aware of independently done contemporaneous work, by M. Bakshi, S. Jaggi, S. Cai and M. Chen [1], that appears to have similarities with the proposed algorithm in this work. From [2], there appear to be several differences between our two independent works as well. structure endows the resulting family of sparse measurement matrices with several useful properties that can be exploited to formulate fast and robust algorithms for compressive sensing, that are provably optimal (upto small constant multiple of k) in the number of measurements and the decoding complexity (k-steps) needed to recover the underlying highdimensional sparse signal, for certain important regimes of interest (noiseless and high signal-to-noise-ratio).…”

Robust sublinear complexity Walsh-Hadamard transform with arbitrary sparse support

“…The robust sublinear algorithm design can also be combined with the conventional peel-off structure for random support model [3]. Moreover, this idea has potential applications to the design of measurement matrix for compressed sensing [7] and practical code design for the so-called many-access channels [8], [9].…”