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.

通过简单的迭代行抽样算法和加权少数行使每个矩阵的相干度低，我们能够使用随机抽取行的方式进行矩阵近似，以大大提高数据处理速度。