On sparsity, redundancy and quality of frame representations

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(N²) operations, as long as the number of non-zero coefficients in the sparse representation of r is ∈N for some 0 ≤ ∈ ≤ 0.5, thus improving on a result of Candès and Tao [3]. We also show that ∈ ≤ 0.5 cannot be relaxed without violating uniqueness. 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.