Читать книгу Graph Spectral Image Processing - Gene Cheung - Страница 40

The Krylov subspace of an arbitrary square matrix and a vector x is defined as follows:

[1.64]

The Krylov subspace method, in terms of GSP, refers to filtering, i.e. evaluating an arbitrary filtered response h(W)x, realized in a Krylov subspace . Many methods to evaluate h(W)x in a Krylov subspace have been proposed, mainly in computational linear algebra and numerical computation (Golub and Van Loan 1996). A famous approximation method is the Arnoldi approximation, which is given by

[1.65]

where h(H_K) is evaluating h(·) for the upper Heisenberg matrix H_K, which is obtained by using the Arnoldi process. Furthermore, H_K is expected to be much smaller than the original matrix; therefore, evaluating h(H_K) using full eigendecomposition will be feasible and light-weighted.