Principles of Modern Radar - Volume 2 1891121537

168 CHAPTER 5 Radar Applications of Sparse Reconstruction5.3.1 Penalized Least SquaresThe convex relaxation of the l 0 reconstruction problem given in (5.12) can be viewedas a penalized least squares problem. We have already seen that this problem arises ina Bayesian framework by assuming a Gaussian noise prior and a Laplacian signal prior.This approach has a long history, for example, [25], and the use of the l 1 norm for a radarproblem was specifically proposed at least as early as [62].The good news is that (5.12) is a linear program for real data and a second-order coneprogram for complex data [1]. As a result, accurate and fairly efficient methods such asinterior point algorithms exist to solve (5.12) [24]. Unfortunately, these solvers are notwell suited to the extremely large A matrices in many problems of interest and do notcapitalize on the precise structure of (5.12). As a result, a host of specialized algorithmsfor solving these penalized least squares problems has been created.This section explores several of these algorithms. It should be emphasized that solversguaranteeing a solution to (5.12) or one of the equivalent formulations discussed hereininherit the RIP-based performance guarantee given in (5.15), provided that A meets therequired RIP requirement.5.3.1.1 Equivalent Optimization Problems and the Pareto FrontierFirst, we will discuss several equivalent formulations of (5.12). We will adopt the nomenclatureand terminology used in [63]. In this framework, the optimization problem solvedin (5.12) is referred to as basis pursuit de-noising (BPDN), or BP σ . This problem solvedin the noise-free setting with σ = 0 is called simply basis pursuit (BP), and its solutionis denoted ˆx BP . The theory of Lagrange multipliers indicates that we can solve an unconstrainedproblem that will yield the same solution, provided that the Lagrange multipler isselected correctly. We will refer to this unconstrained problem as l 1 penalized quadraticprogram and denote it as QP λ . Similarly, we can solve a constrained optimization problem,but with the constraint placed on the l 1 norm of the unknown vector instead of the l 2 normof the reconstruction error, to obtain yet a third equivalent problem. We will use the nameLASSO [64], popular in the statistics community, interchangeably with the notation LS τfor this problem. The three equivalent optimization problems can be written as(BP σ )(QP λ )(LS τ )ˆx σ = argminxˆx λ = argminxˆx τ = argminx‖x‖ 1 subject to ‖Ax − y‖ 2 ≤ σ (5.18)λ ‖x‖ 1 + ‖Ax − y‖ 2 2 (5.19)‖Ax − y‖ 2 subject to ‖x‖ 1 ≤ τ (5.20)We note that a fourth problem formulation known as the Dantzig selector also appears inthe literature [65] and can be expressed as(DS ζ )ˆx ζ = argminx‖x‖ 1 subject to ∥∥ A H (Ax − y) ∥ ∥∞≤ ζ (5.21)but this problem does not yield the same set of solutions as the other three. For a treatmentof the relationship between DS ζ and the other problems, see [41].The first three problems are all different ways of arriving at the same set of solutions.To be explicit, the solution to any one of these problems is characterized by a triplet ofvalues (σ, λ, τ) which renders ˆx σ = ˆx λ = ˆx τ . Unfortunately, it is very difficult to map