On Sparsity, Redundancy and Quality of Frame Representations

Abstract — We consider approximations of signals by the elements of a frame in a complex vector space of dimension N and formulate both the noiseless and the noisy sparse representation problems. The noiseless representation problem is to find sparse representations of a signal r given that such representations exist. In this case, we explicitly construct a frame, referred to as the Vandermonde frame, for which the noiseless sparse representation problem can be solved uniquely using O(N 2) operations, as long as the number of non-zero coefficients in the sparse representation of r is ɛN for some 0 ≤ ɛ ≤ 0.5, thus improving on a result of Candes and Tao [4]. The noisy sparse representation problem is to find sparse representations of a signal r satisfying a distortion criterion. In this case, we establish a lower bound on the trade-off between the sparsity of the representation, the underlying distortion and the redundancy of any given frame. I