2020/04/23 by Sepideh Mahabadi, Ilya Razenshteyn, Mahabadi, Sepideh +5 · 3 citations
Computer Science · Mathematics · #Adaptive sampling #Algorithm #Artificial intelligence #Automatic summarization #Complexity and Algorithms in Graphs #Computation #Computational Geometry (cs.CG) #Computer science #Data Structures and Algorithms (cs.DS) #Data mining #Data stream mining #FOS: Computer and information sciences #Machine Learning (cs.LG) #Mathematical optimization #Mathematics #Maximization #Privacy-Preserving Technologies in Data #Row #Sampling (signal processing) #Statistics #Stochastic Gradient Optimization Techniques #Streaming algorithm #Subspace topology #Turnstile #Upper and lower bounds #cs.CG #cs.DS #cs.LG
paper · pdf · doi:10.48550/arxiv.2004.10969
published in arXiv (Cornell University) (Cornell University) · To appear at STOC 2020
arxiv created 2020/04/23 · openalex publication_date 2020/04/23 · arxiv updated 2020/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
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∈ℝn× 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 algorithms for column subset selection, subspace approximation, projective clustering, and volume maximization on turnstile streams that use space sublinear in n. 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. See paper for full abstract.