COMPRESSED SENSING AND BEST k-TERM APPROXIMATION

The typical paradigm for obtaining a compressed version of a discrete signal represented by a vector x ∈ R is to choose an appropriate basis, compute the coefficients of x in this basis, and then retain only the k largest of these with k < N . If we are interested in a bit stream representation, we also need in addition to quantize these k coefficients. Assuming, without loss of generality, that x already represents the coefficients of the signal in the appropriate basis, this means that we pick an approximation to x in the set Σk of k-sparse vectors (1.1) Σk := {x ∈ R : # supp(x) ≤ k}, where supp(x) is the support of x, i.e., the set of i for which xi = 0, and #A is the number of elements in the set A. The best performance that we can achieve by such an approximation process in some given norm ‖ · ‖X of interest is described by the best k-term approximation error