Learning multiplication-free linear transformations

Constructing fast approximate eigenspaces with application to the fast graph Fourier transforms

“…In this paper, we assume we do not have information about the eigenspace of L, but have access to L itself. Previous work [Rusu andRosasco, 2019, Frerix andBruna, 2019] considered the possibility of performing first the eigendecomposition and then working directly with the eigenspace U. For the same graphs, we show in Figure 3 the evolution of the accuracy of the overall Laplacian L as a function of the number of basic transformations in their factorization.…”

“…we have the extended orthonormal Givens transformations [Rusu andThompson, 2017, Rusu andRosasco, 2019], which for simplicity we call a G-transform:…”

“…The general, unsymmetric, case [Golub and van Loan, 1996][Chapter 7] is much more challenging as the given matrix might not even be diagonalizable and furthermore, even when it is, the factorization has to be done over the complex-valued field in general. As the eigenvector matrix is generally unstructured, in this case, we rely on a given number of scaling and shear transformations to approximate it [Rusu, 2018]. We formulate optimization problems for each of these basic components and show how to locally, optimally solve them with an iterative algorithm and closed-form solutions.…”