Daniel Spielman: “Approximate Elimination”
We provide context for and explain the recent Approximate Gaussian Elimination algorithm of Kyng and Sachdeva.
Gaussian Elimination is the first algorithm most of us learn for solving systems of linear equations. While it is simple and elegant, it can also be impractically slow. Kyng and Sachdeva show that, after carefully modifying elimination to randomly drop and rescale entries, it can provide very fast approximate solutions to systems of equations in Laplacian matrices. Our implementation of a refinement of this algorithm is now among the best Laplacian solvers in practice.
We will explain what Laplacian matrices are, what it means to approximately solve a system of linear equations over the reals, and how one analyzes this algorithm using recent results in Random Matrix Theory. We will also discuss what is means for an algorithm to be the “best in practice.”