The Kronecker Product in Approximation and Fast Transform Geration

Two aspects of the Kronecker product are studied. First, we start with the Kronecker product approximation problem. Given a matrix A and a factorization of its dimensions, we find matrices B and C of the corresponding dimensions whose Kronecker product is as close as possible to A in Frobenius norm. Various constrained Kronecker product approximation problems are also considered along with examples. Second, we develop a source-to-source compiler that processes matrix factorization formulae to generate efficient implementations of fast transformation algorithms. Our goal is to automate the implementation of efficient FFT and other fast transforms that can be characterized using Kronecker product factorizations. The Kronecker product notation plays a crucial role, allowing the simplification of expression of fast transform algorithms (Walsh-Hadamard, Haar, Slant, Hartley) and capitalizing on the experience from the FFT field. The compiler is based on a set of term rewriting rules that translate high level matrix descriptions into loops and assignment statements in any imperative programming language (parallel or sequential). We study the Haar and Daubechies wavelet algorithms with an effort to express them using matrix language. We provide back-end translators for FORTRAN, C and MATLAB.