Streaming Quantiles Algorithms with Small Space and Update Time.

Approximating quantiles and distributions over streaming data has been studied for roughly two decades now. Recently, Karnin, Lang, and Liberty proposed the first asymptotically optimal algorithm for doing so. This manuscript complements their theoretical result by providing a practical variants of their algorithm with improved constants. For a given sketch size, our techniques provably reduce the upper bound on the sketch error by a factor of two. These improvements are verified experimentally. Our modified quantile sketch improves the latency as well by reducing the worst-case update time from O(1ε) down to O(log1ε).

Randomized algorithms for the low-rank approximation of matrices.

We describe two recently proposed randomized algorithms for the construction of low-rank approximations to matrices, and demonstrate their application (inter alia) to the evaluation of the singular value decompositions of numerically low-rank matrices. Being probabilistic, the schemes described here have a finite probability of failure; in most cases, this probability is rather negligible (10(-17) is a typical value). In many situations, the new procedures are considerably more efficient and reliable than the classical (deterministic) ones; they also parallelize naturally. We present several numerical examples to illustrate the performance of the schemes.