Abstract
AbstractWe 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 \input amssym $x \in {\Bbb R}^n$ 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.
Highlights
Compressed sensing is a modern paradigm of data acquisition, which is having an impact on several disciplines; see [21]
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 D sign.Ax/: Here y is the vector of signs of the coordinates of Ax
Note that y contains no information about the magnitude of x, and we can only hope to recover the normalized vector x=kxk2
Summary
Compressed sensing is a modern paradigm of data acquisition, which is having an impact on several disciplines; see [21]. V D Ax; where A is a given m n measurement matrix and x 2 Rn is an unknown signal that one needs to recover from v. One assumes that x has at most s nonzero entries, the support pattern is unknown. If A has Gaussian i.i.d. entries, we may take m D O.s log.n=s// and still recover x exactly with high probability [8, 9]; see [26] for an overview. This recovery may be achieved in polynomial time by solving the convex minimization program (1.2). There are many applications where such severe quantization may be inherent or preferred—analog-to-digital conversion [18, 20] and binomial regression in statistical modeling and threshold group testing [12], to name a few
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