Block transform adaptation by stochastic gradient descent

The problem of computing the eigendecomposition of anN N symmetric matrix is cast as an unconstrained minimization of ei-ther of two performance measures. TheK = N(N 1)=2 inde-pendent 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 of-ten desirable for the transform to approximate a local Karhunen-Loève 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. 1