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

Suppose $A \in \mathbb{R}^{n \times n}$ is invertible and we are looking for the solution of $Ax = b$. Given an initial guess $x_1 \in \mathbb{R}$, we show that by reflecting through hyperplanes generated by the rows of $A$, we can generate an infinite sequence $(x_k)_{k=1}^{\infty}$ 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 $$\mathbb{E} \left\| x - \frac{1}{m} \sum_{k=1}^{m}{ x_k} \right\| \leq \frac{1 + \|A\|_F \|A^{-1}\|}{\sqrt{m}} \cdot\|x-x_1\|.$$ 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

A Weighted Randomized Kaczmarz Method for Solving Linear Systems

The Kaczmarz method for solving a linear system $Ax = b$ interprets such a system as a collection of equations $\left\langle a_i, x\right\rangle = b_i$, where $a_i$ is the $i-$th row of $A$, then picks such an equation and corrects $x_{k+1} = x_k + \lambda a_i$ where $\lambda$ is chosen so that the $i-$th equation is satisfied. Convergence rates are difficult to establish. Assuming the rows to be normalized, $\|a_i\|_{\ell^2}=1$, Strohmer \& Vershynin established that if the order of equations is chosen at random, $\mathbb{E}~ \|x_k - x\|_{\ell^2}$ converges exponentially. We prove that if the $i-$th row is selected with likelihood proportional to $\left|\left\langle a_i, x_k \right\rangle - b_i\right|^{p}$, where $0<p<\infty$, then $\mathbb{E}~\|x_k - x\|_{\ell^2}$ converges faster than the purely random method. As $p \rightarrow \infty$, the method de-randomizes and explains, among other things, why the maximal correction method works well. We empirically observe that the method computes approximations of small singular vectors of $A$ as a byproduct