Non-adaptive adaptive sampling on turnstile streams

Adaptive sampling is a useful algorithmic tool for data summarization problems in the classical centralized setting, where the entire dataset is available to the single processor performing the computation. Adaptive sampling repeatedly selects rows of an underlying matrix A ∈ R n×d , where n ≫ d, with probabilities proportional to their distances to the subspace of the previously selected rows. Intuitively, adaptive sampling seems to be limited to trivial multi-pass algorithms in the streaming model of computation due to its inherently sequential nature of assigning sampling probabilities to each row only after the previous iteration is completed. Surprisingly, we show this is not the case by giving the first one-pass algorithms for adaptive sampling on turnstile streams and using space poly(d, k, log n), where k is the number of adaptive sampling rounds to be performed. Our adaptive sampling procedure has a number of applications to various data summarization problems that either improve state-of-the-art or have only been previously studied in the more relaxed row-arrival model. We give the first relative-error algorithm for column subset selection on turnstile streams. We show our adaptive sampling algorithm also gives the first relative-error algorithm for subspace approximation on turnstile streams that returns k noisy rows of A. The quality of the output can be improved to a (1 + ǫ)-approximation at the tradeoff of a bicriteria algorithm that outputs a larger number of rows. We then give the first algorithm for projective clustering on turnstile streams that uses space sublinear in n. In fact, we use space poly d, k, s, 1 ǫ , log n to output a (1 + ǫ)-approximation, where s is the number of k-dimensional subspaces. Our adaptive sampling primitive also provides the first algorithm for volume maximization on turnstile streams. We complement our volume maximization algorithmic results with lower bounds that are tight up to lower order terms, even for multi-pass algorithms. By a similar construction, we also obtain lower bounds for volume maximization in the row-arrival model, which we match with competitive upper bounds.