One-Bit Compressed Sensing by Linear Programming

TLDR: In this paper, the authors give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x ∈ R n from the signs of O(s log 2 (n/s) random linear measurements of x.

Abstract: ONE-BIT COMPRESSED SENSING BY LINEAR PROGRAMMING arXiv:1109.4299v5 [cs.IT] 16 Mar 2012 YANIV PLAN AND ROMAN VERSHYNIN Abstract. We give the first computationally tractable and almost optimal solution to the problem of one-bit compressed sensing, showing how to accurately recover an s-sparse vector x ∈ R n from the signs of O(s log 2 (n/s)) random linear measurements of x. The recovery is achieved by a simple linear program. This result extends to approximately sparse vectors x. Our result is universal in the sense that with high probability, one measurement scheme will successfully recover all sparse vectors simultaneously. The argument is based on solving an equivalent geometric problem on random hyperplane tessellations. 1. Introduction Compressed sensing is a modern paradigm of data acquisition, which is having an impact on several disciplines, see [21]. The scientist has access to a measurement vector v ∈ R m obtained as v = Ax, where A is a given m × n measurement matrix and x ∈ R n is an unknown signal that one needs to recover from v. One would like to take m n, rendering A non-invertible; the key ingredient to successful recovery of x is take into account its assumed structure – sparsity. Thus one assumes that x has at most s nonzero entries, although the support pattern is unknown. The strongest known results are for random measurement matrices A. In particular, if A has Gaussian i.i.d. entries, then we may take m = O(s log(n/s)) and still recover x exactly with high probability [10, 7]; see [26] for an overview. Furthermore, this recovery may be achieved in polynomial time by solving the convex minimization program min x 0 1 subject to Ax 0 = v. Stability results are also available when noise is added to the problem [9, 8, 3, 27]. However, while the focus of compressed sensing is signal recovery with minimal information, the classical set-up (1.1), (1.2) assumes infinite bit precision of the measurements. This disaccord raises an important question: how many bits per measurement (i.e. per coordinate of v) are sufficient for tractable and accurate sparse recovery? This paper shows that one bit per measurement is enough. There are many applications where such severe quantization may be inherent or preferred — analog-to-digital conversion [20, 18], binomial regression in statistical modeling and threshold group testing [12], to name a few. 1.1. Main results. This paper demonstrates that a simple modification of the convex program (1.2) is able to accurately estimate x from extremely quantized measurement vector y = sign(Ax). Date: September 19, 2011. 2000 Mathematics Subject Classification. 94A12; 60D05; 90C25. Y.P. is supported by an NSF Postdoctoral Research Fellowship under award No. 1103909. R.V. is supported by NSF grants DMS 0918623 and 1001829.