Kaczmarz Algorithm and Frames

TL;DR

This paper characterizes sequences of vectors in Hilbert spaces where the Kaczmarz algorithm converges, linking them to frame theory, and identifies conditions under which the algorithm reliably finds solutions to linear systems.

Contribution

It generalizes the characterization of convergence sequences to frames and bases, and identifies matrices for which the Kaczmarz algorithm always converges.

Findings

Sequences with the Kaczmarz algorithm convergence are characterized by tight frames with constant 1.

Only orthonormal bases are effective Riesz bases for the Kaczmarz algorithm.

Characterization of matrices A for which the Kaczmarz algorithm converges to solutions of Ax=b.

Abstract

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations $A x = b$ and characterize the (bounded) matrices $A$ for which the Kaczmarz algorithm always converges to a solution.