Uniform Sampling for Matrix Approximation
This work addresses the computational bottleneck of leverage score computation in matrix approximation for large-scale data processing, offering a simpler and faster iterative method, though it is incremental in improving understanding of uniform sampling.
The paper tackles the problem of approximating large matrices for efficient processing by showing that uniform random sampling, though insufficient for direct linear regression, can produce a matrix that approximates a large fraction of the original, enabling iterative algorithms that run in input-sparsity time and preserve sparsity.
Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time significantly. For theoretical performance guarantees, each row must be sampled with probability proportional to its statistical leverage score. Unfortunately, leverage scores are difficult to compute. A simple alternative is to sample rows uniformly at random. While this often works, uniform sampling will eliminate critical row information for many natural instances. We take a fresh look at uniform sampling by examining what information it does preserve. Specifically, we show that uniform sampling yields a matrix that, in some sense, well approximates a large fraction of the original. While this weak form of approximation is not enough for solving linear regression directly, it is enough to compute a better approximation. This observation leads to simple iterative row sampling algorithms for matrix approximation that run in input-sparsity time and preserve row structure and sparsity at all intermediate steps. In addition to an improved understanding of uniform sampling, our main proof introduces a structural result of independent interest: we show that every matrix can be made to have low coherence by reweighting a small subset of its rows.