Fast Spectral Learning using Lanczos Eigenspace Projections

The core computational step in spectral learning – finding the projection of a function onto the eigenspace of a symmetric operator, such as a graph Laplacian – generally incurs a cubic computational complexity O(N). This paper describes the use of Lanczos eigenspace projections for accelerating spectral projections, which reduces the complexity to O(nTop + nN) operations, where n is the number of distinct eigenvalues, and Top is the complexity of multiplying T by a vector. This approach is based on diagonalizing the restriction of the operator to the Krylov space spanned by the operator and a projected function. Even further savings can be accrued by constructing an approximate Lanczos tridiagonal representation of the Krylov-space restricted operator. A key novelty of this paper is the use of Krylov-subspace modulated Lanczos acceleration for multi-resolution wavelet analysis. A challenging problem of learning to control a robot arm is used to test the proposed approach.