Optimality of Operator-Like Wavelets for Representing Sparse AR(1) Processes

The discrete cosine transform (DCT) is known to be asymptotically equivalent to the Karhunen-Loève transform (KLT) of Gaussian first-order auto-regressive (AR(1)) processes. Since being uncorrelated under the Gaussian hypothesis is synonymous with independence, it also yields an independent-component analysis (ICA) of such signals. In this paper, we present a constructive non-Gaussian generalization of this result: the characterization of the optimal orthogonal transform (ICA) for the family of symmetric-stable AR(1) processes. The degree of sparsity of these processes is controlled by the stability parameter 0 <; α ≤ 2 with the only non-sparse member of the family being the classical Gaussian AR(1) process with α = 2. Specifically, we prove that, for α <; 2, a fixed family of operator-like wavelet bases systematically outperforms the DCT in terms of compression and denoising ability. The effect is quantified with the help of two performance criteria (one based on the Kullback-Leibler divergence, and the other on Stein´s formula for the minimum estimation error) that can also be viewed as statistical measures of independence. Finally, we observe that, for the sparser kind of processes with 0 <; α ≤ 1, the operator-like wavelet basis, as dictated by linear system theory, is undistinguishable from the ICA solution obtained through numerical optimization. Our framework offers a unified view that encompasses sinusoidal transforms such as the DCT and a family of orthogonal Haar-like wavelets that is linked analytically to the underlying signal model.