Approximate Solutions of Linear Systems at a Universal Rate

07/07/2022

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by   Stefan Steinerberger, et al.

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Let A ∈ℝ^n × n be invertible, x ∈ℝ^n unknown and b =Ax given. We are interested in approximate solutions: vectors y ∈ℝ^n such that Ay - b is small. We prove that for all 0< ε <1 there is a composition of k orthogonal projections onto the n hyperplanes generated by the rows of A, where k ≤ 2 log(1/ε) n/ε^2 which maps the origin to a vector y∈ℝ^n satisfying A y - Ax≤ε·A· x. We note that this upper bound on k is independent of the matrix A. This procedure is stable in the sense that y≤ 2x. The existence proof is based on a probabilistically refined analysis of the Random Kaczmarz method which seems to achieve this rate when solving for A x = b with high likelihood.