Near-Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time

Currently, in the numerical linear algebra community, it is thought that to obtain nearly-optimal bounds for various problems such as rank computation, finding a maximal linearly independent subset of columns, regression, low rank approximation, maximum matching on general graphs and linear matroid union, one would need to resolve the main open question of Nelson and Nguyen (FOCS, 2013) regarding the logarithmic factors in the sketching dimension for existing constant factor approximation oblivious subspace embeddings. We show how to bypass this question using a refined sketching technique, and obtain optimal or nearly optimal bounds for these problems. A key technique we use is an explicit mapping of Indyk based on uncertainty principles and extractors, which after first applying known oblivious subspace embeddings, allows us to quickly spread out the mass of the vector so that sampling is now effective, and we avoid a logarithmic factor that is standard in the sketching dimension resulting from matrix Chernoff bounds. For the fundamental problems of rank computation and finding a linearly independent subset of columns, our algorithms improve Cheung, Kwok, and Lau (JACM, 2013) and are optimal to within a constant factor and a $\log\log(n)$-factor, respectively. Further, for constant factor regression and low rank approximation we give the first optimal algorithms, for the current matrix multiplication exponent.