Principles of Modern Radar - Volume 2 1891121537

174 CHAPTER 5 Radar Applications of Sparse Reconstruction= argminxg(x) + P (2 ∥ x − b − 1 )∥ ∥∥∥2P ∇ f (b) 2λ ‖x‖ 1 + P 2 ∥(b x − − 2 )∥ ∥∥∥2P AH (Ab − y)2)= argminx(= η s b − 2 P AH (Ab − y), λ P(5.31)(5.32)Notice that the term in parenthesis in (5.31) is a constant for fixed b. Thus, the minimizationin (5.31) can be carried out component-wise, yielding a simple analytical solutionthat corresponds to the application of the soft threshold. Combining these ideas, weobtain ISTA(ˆx k+1 = η s ˆx k − 2 P AH (A ˆx k − y), λ )PUnfortunately, ISTA has been shown to enjoy only a sublinear, that is proportional to1/k, rate of convergence [79, Theorem 3.1]. Beck and Teboulle [79] propose a modifiedfast ISTA (FISTA) that uses a very simple modification to obtain a quadratic convergencerate. FISTA requires nearly identical computational cost, particularly for a large-scaleproblem, and is given byz 1 = ˆx 0 = 0t 1 = 1 (ˆx k = η s z k − 2 P AH (Az k − y), λ )Pt k+1 = 1 + √ 1 + 4(t k ) 22z k+1 = ˆx k + t k − 1t ( k+1 ˆxk − ˆx k−1 )Intuitively, this algorithm uses knowledge of the previous two iterates to take faster stepstoward the global minimum. No additional applications of A and A H are required comparedwith ISTA, and thus the computational cost of this modification is negligible for thelarge-scale problems of interest.ISTA and FISTA 29 converge to true solutions of QP λ at sublinear and quadratic convergencerates, respectively. Thus, these algorithms inherit the RIP-based performanceguarantees already proven for minimization of this cost function, for example (5.15). Indeed,these algorithms are close cousins of the Kragh algorithm described in the previoussection. The key difference is that the derivation is restricted to the p = 1 norm case totake advantage of a simple analytical result for the minimization step of the algorithm.As a final note on these algorithms, in [61] the authors leverage the same previous workthat inspired FISTA to derive the NESTA algorithm. This approach provides extremelyfast computation, particularly when the forward operator A enjoys certain properties. Inaddition, the provided algorithm can solve more general problems, including minimization29 We should mention that the FISTA algorithm given in [79] is more general than the result providedhere, which has been specialized for our problem of interest.