Principles of Modern Radar - Volume 2 1891121537

176 CHAPTER 5 Radar Applications of Sparse Reconstructionapproaches discussed in Section 5.3.1.2. An analysis and discussion of one of these iterativereweighted least squares (IRLS) approaches in terms of RIP is provided in [85].One class of IRLS algorithms solves a sequence of l 2 -regularized problems with aweighting matrix derived from the previous iterate. An example is the focal underdeterminedsystem solver (FOCUSS) algorithm [86], which predates the work on CS. Here,we provide an example of an algorithm that solves a sequence of reweighted l 1 problemsas proposed in [87]. In particular, the noisy-data version of the algorithm is given asˆx k+1 = argminx1W i,i k =| ˆx i k |+ε∥ W k x ∥ ∥1subject to ‖Ax − y‖ 2 ≤ σIn other words, W k is a diagonal matrix with elements equal to the inverses of the amplitudesof the elements of the previous iterate 31 . The next estimate of x is obtained bysolving a new BP σ problem with this weighting matrix applied to x. As a coefficientbecomes small, the weight applied to it will become very large, driving the coefficienttoward zero. In this way, the reweighting scheme promotes sparse solutions. Indeed, thisapproach can reconstruct sparse solutions with fewer measurements than a straightforwardBP σ approach. Unfortunately, the price for this performance improvement is the need tosolve a sequence of l 1 optimization problems. These subproblems can be solved with oneof the techniques we have already discussed.This algorithm can be derived from a MM framework where the cost function tobe minimized is the sum of the logarithms of the absolute values of the coefficientsof x [87]. This function’s unit ball is more similar to the l 0 norm than that of the l 1norm, intuitively explaining the improved performance. This approach can be extendedto nonconvex minimization with p < 1; see [88].5.3.4 Greedy MethodsWe now turn our attention to a somewhat different class of algorithms for sparse reconstruction.Greedy algorithms are principally motivated by computational efficiency, althoughthey often exhibit somewhat inferior performance to the approaches already considered.They do not arise from optimizing an l p -penalized cost function but instead rely on iterativeattempts to identify the support of the unknown vector x true . Notice that if we knowa priori which elements of x true are nonzero, indexed by the set Ɣ, then we can solve themuch easier problemy = A Ɣ x trueƔ + e (5.35)where subscripting by Ɣ indicates that we have thrown away the entries or columns notincluded in Ɣ. Since x true is sparse, the matrix A Ɣ now has many more rows than columns.Put simply, this problem is overdetermined and can be solved easily using least squares.In particular, we can estimate x true as 32ˆx Ɣ = ( A H Ɣ A Ɣ) −1AHƔ y (5.36)31 The ε factor is included to ensure finite weights.32 Naturally we set entries outside the set Ɣ to zero.