Surrounding the solution of a Linear System of Equations from all sides

09/03/2020

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by   Stefan Steinerberger, et al.

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Suppose A ∈ℝ^n × n is invertible and we are looking for the solution of Ax = b. Given an initial guess x_1 ∈ℝ, we show that by reflecting through hyperplanes generated by the rows of A, we can generate an infinite sequence (x_k)_k=1^∞ such that all elements have the same distance to the solution, i.e. x_k - x = x_1 - x. If the hyperplanes are chosen at random, averages over the sequence converge and 𝔼 x - 1/m∑_k=1^m x_k≤1 + A_F A^-1/√(m)·x-x_1. The bound does not depend on the dimension of the matrix. This introduces a purely geometric way of attacking the problem: are there fast ways of estimating the location of the center of a sphere from knowing many points on the sphere? Our convergence rate (coinciding with that of the Random Kaczmarz method) comes from averaging, can one do better?