Structural Convergence Results for Approximation of Dominant Subspaces from Block Krylov Spaces

TL;DR

This paper provides theoretical bounds on approximating dominant singular subspaces of matrices using block Krylov methods, combining classical convergence analysis with optimal approximation techniques.

Contribution

It introduces new structural bounds for Krylov subspace approximations of dominant singular vectors, integrating Lanczos analysis with least squares optimization.

Findings

Bounds on subspace distance in terms of principal angles

Approximation quality bounds relative to best Frobenius norm approximation

Dependence of bounds on tangent of principal angles

Abstract

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.