The Kaczmarz method is a way to iteratively solve a linear system of equations $Ax = b$. One interprets the solution $x$ as the point where hyperplanes intersect and then iteratively projects an approximate solution onto these hyperplanes to get better and better approximations. We note a somewhat related idea: one could take two random hyperplanes and project one into the orthogonal complement of the other. This leads to a sequence of linear systems $A^{(k)} x = b^{(k)}$ which is fast to compute, preserves the original solution and whose small singular values grow like $\sigma_{\ell}(A^{(k)}) \sim \exp(k/n^2) \cdot \sigma_{\ell}(A)$.

翻译：Kaczmarz 方法是一种迭代求解线性方程组 $Ax = b$ 的方法。该方法将解 $x$ 解释为超平面的交点，然后迭代地将近似解投影到这些超平面上，从而得到越来越精确的近似解。我们注意到一个与之有一定关联的想法：可以选取两个随机超平面，并将其中一个投影到另一个的正交补空间中。这导致产生一系列线性系统 $A^{(k)} x = b^{(k)}$，这些系统计算速度快，能保持原始解，并且其小奇异值以 $\sigma_{\ell}(A^{(k)}) \sim \exp(k/n^2) \cdot \sigma_{\ell}(A)$ 的方式增长。