Quantile-Based Random Kaczmarz for corrupted linear systems of equations

07/12/2021

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by   Stefan Steinerberger, et al.

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We consider linear systems Ax = b where A ∈ℝ^m × n consists of normalized rows, a_i_ℓ^2 = 1, and where up to β m entries of b have been corrupted (possibly by arbitrarily large numbers). Haddock, Needell, Rebrova and Swartworth propose a quantile-based Random Kaczmarz method and show that for certain random matrices A it converges with high likelihood to the true solution. We prove a deterministic version by constructing, for any matrix A, a number β_A such that there is convergence for all perturbations with β < β_A. Assuming a random matrix heuristic, this proves convergence for tall Gaussian matrices with up to ∼ 0.5% corruption (a number that can likely be improved).