Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces
For researchers in numerical linear algebra and machine learning, this work offers rigorous theoretical guarantees for Krylov subspace approximations, though it is incremental as it extends existing Lanczos analysis to block Krylov spaces.
This paper provides theoretical bounds on the distance between block Krylov spaces and the dominant left singular vector space of a matrix, expressed in terms of principal angles, and quality of approximation bounds relative to best Frobenius norm approximation. These results serve as a structural foundation for analyzing randomized Krylov space methods.
This paper is concerned with approximating the dominant left singular vector space of a real matrix $A$ of arbitrary dimension, from block Krylov spaces generated by the matrix $AA^T$ and the block vector $AX$. Two classes of results are presented. First are bounds on the distance, in the two and Frobenius norms, between the Krylov space and the target space. The distance is expressed in terms of principal angles. Second are quality of approximation bounds, relative to the best approximation in the Frobenius norm. For starting guesses $X$ of full column-rank, the bounds depend on the tangent of the principal angles between $X$ and the dominant right singular vector space of $A$. The results presented here form the structural foundation for the analysis of randomized Krylov space methods. The innovative feature is a combination of traditional Lanczos convergence analysis with optimal approximations via least squares problems.