Non-Convex Compressed Sensing from Noisy Measurements

ABSTRACT

This paper proposes solution to the following non-convex optimization problem:

min || x ||_p subject to || y – Ax ||_q≤ ε

Such an optimization problem arises in a rapidly advancing branch of signal processing called ‘Compressed Sensing’ (CS). The problem of CS is to reconstruct a k-sparse vector x_nX1, from noisy measurements y = Ax +η , where A_mXn (m < n) is the measurement matrix and η_nX1 is additive noise.

In general the optimization methods developed for CS minimizes a sparsity promoting l_1-norm (p=1) for Gaussian noise (q=2). This is restrictive for two reasons: i) theoretically it has been shown that, with positive fractional norms (0 < p < 1), the sparse vector x can be reconstructed by fewer measurements than required by l_1-norm; and ii) Noises other than Gaussian require the norm of the misfit (q) to be something other than 2. To address these two issues an Iterative Reweighted Least Squares based algorithm is proposed here to solve the aforesaid optimization problem.

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