Block Transform Adaptation by Stochastic Gradient Descent

The problem of computing the eigendecomposition of an N N symmetric matrix is cast as an unconstrained minimization of either of two performance measures. The K = N(N 1)=2 independent parameters represent angles of distinct Givens rotations. Gradient descent is applied to the minimization problem, step size bounds for local convergence are given, and similarities to LMS adaptive filtering are noted. In adaptive transform coding it is often desirable for the transform to approximate a local KarhunenLoève Transform for the source. Determining such a transform is equivalent to finding the eigenvectors of the correlation matrix of the source; thus, the eigendecomposition methods developed here are applicable to adaptive transform coding.