QRP Variation of Cross--Approximation Iterations for Low Rank Approximation
Victor Y. Pan, John Svadlenka
TL;DR
The paper addresses efficient Low Rank Approximation for extremely large matrices encountered in Big Data. It advocates superfast LRA via Cross--Approximation iterations to compress M into AB+E with relative error up to ε, using only $(m+n)r$ data instead of $mn$. It then applies this idea to the Fast Multipole Method by obtaining LRA of neutered off-diagonal blocks of HSS matrices as products $N=FH$, turning FMM into a Superfast Multipole Method and achieving near-optimal mat-vec and, under mild assumptions, $O(n r \log^2 n)$ solves. Numerical experiments on 1024×1024 HSS matrices from Toeplitz-derived Cauchy-like families demonstrate accurate LRA after a few CA iterations, with ranks chosen by a binary search and limited improvement from random preprocessing.
Abstract
We call matrix algorithms superfast if they use much fewer flops and memory cells than the input matrix has entries. Using such algorithms is indispensable for Big Data Mining and Analysis, where the input matrices are so immense that one can only access a small fraction of all their entries. A natural remedy is Low Rank Approximation (LRA) of these matrices, which is routinely computed by means of Cross-Approximation iterations for more than a decade of worldwide application in computational practice. We point out and extensively test an important application of superfast LRA to significant acceleration of the celebrated Fast Multipole Method, which turns it into Superfast Multipole Method.