Non-Adaptive Adaptive Sampling in Turnstile Streams
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Abstract: 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 n by d matrix A, 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 on turnstile streams that either improve state-of-the-art or have only been previously studied in the more relaxed row-arrival model. This includes column subset selection, subspace approximation, projective clustering, and volume maximization. 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.
This is a joint work with Ilya Razenshteyn, David Woodruff, and Samson Zhou.