Abstract
Abstract An accurate and efficient algorithm for solving the constrained ℓ 1-norm minimization problem is highly needed and is crucial for the success of sparse signal recovery in compressive sampling. We tackle the constrained ℓ 1-norm minimization problem by reformulating it via an indicator function which describes the constraints. The resulting model is solved efficiently and accurately by using an elegant proximity operator-based algorithm. Numerical experiments show that the proposed algorithm performs well for sparse signals with magnitudes over a high dynamic range. Furthermore, it performs significantly better than the well-known algorithm NESTA (a shorthand for Nesterov’s algorithm) and DADM (dual alternating direction method) in terms of the quality of restored signals and the computational complexity measured in the CPU-time consumed.
Highlights
We study the recovery of an unknown vector u0 ∈ Rn from the observed data b ∈ Rm and the model b = Au0 + z, (1)
We briefly review an alternating direction method for model (2) that is equivalently written as the following constrained optimization problem min u 1 + ιB (w − b) : Au = w, u ∈ Rn, w ∈ Rm . (12)
We focus on sparse signals with various dynamic ranges and various measurement matrices including randomly partial discrete cosine transforms (DCTs), randomly partial discrete Walsh-Hadamard transforms (DWHTs), and random Gaussian matrices and evaluate performance of algorithms in terms of various error metrics, speed, and robustness-to-noise
Summary
Non-differentiability of both the 1-norm and the indicator of the set imposes challenges for solving the associated optimization problem. Enough, their proximity operators have explicit expressions. The main advantage of this approach is that solving (QPλ) or smoothing the 1-norm are no longer necessary This makes the proposed algorithm attractive for solving (BP) and (BP ). The rest of the paper is organized as follows: in Section 2, we reformulate the 1-norm minimization problems (BP) and (BP ) via an indicator function and characterize solutions of the proposed model in terms of fixed-point equations.
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