Principles of Modern Radar - Volume 2 1891121537

5.2 CS Theory 163While the notion of Kruskal rank is important in sparse regularization, it has verylimited utility for the problems of interest to radar practitioners. The regular rank of amatrix has limited utility, because an arbitrarily small change in the matrix can alterthe rank. Put another way, a matrix can be “almost” rank deficient. In practice, we usemeasures like the condition number [20] to assess the sensitivity of matrix operations tosmall errors. The problem with Kruskal rank is analogous. If there exists a sparse vectorsuch that Ax ≈ 0, this will not violate the Kruskal rank condition. However, when even asmall amount of noise is added to the measurements, distinctions based on arbitrarily smalldifferences in the product Ax will not be robust. What we need is a condition on A thatguarantees sparse solutions will be unique, but also provides robustness in the presence ofnoise. As we shall see, this condition will also guarantee successful sparse reconstructionwhen solving the convex relaxation of our l 0 problem.5.2.5.2 The Restricted Isometry PropertyAn isometry is a continuous, one-to-one invertible mapping between metric spaces that preservesdistances [34]. As discussed in the previous section, we want to establish a conditionon A that provides robustness for sparse reconstruction. While several conditions are possible,we shall focus on the restricted isometry property (RIP). In particular, we will definethe restricted isometry constant (RIC) R n (A) as the smallest positive constant such that(1 − R n (A)) ‖x‖ 2 2 ≤ ‖Ax‖2 2 ≤ (1 + R n(A)) ‖x‖ 2 2 (5.14)for all x such that ‖x‖ 0 ≤ n. In other words, the mapping A preserves the energy in sparsesignals with n or fewer nonzero coefficients up to a small distortion. We refer to this conditionas RIP, since the required approximate isometry is restricted to the set of sparse signals.As we can see, this condition avoids the problem of arbitrarily small Ax values forsparse x that can occur when only the Kruskal rank condition is required. This propertyis analogous to a full rank matrix having a small condition number. Indeed, the RIP canbe interpreted as a requirement on the condition number of all submatrices of A withn or fewer columns. Furthermore, notice that A has Kruskal rank of at least n providedthat R n (A)