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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

An important challenge in the streaming model is to maintain small-space approximations of entrywise functions performed on a matrix that is generated by the outer product of two vectors given as a stream. In other works, streams typically define matrices in a standard way via a sequence of updates, as in the
work of Woodruff [22] and others. We describe the matrix formed by the outer product, and other matrices that do not fall into this category, as implicit matrices. As such, we consider the general problem of computing over such implicit matrices with Hadamard functions, which are functions applied entrywise on a matrix. In this paper, we apply this generalization to provide new techniques for identifying independence between two data streams. The previous state of the art algorithm of Braverman and Ostrovsky [9] gave a (1 +- epsilon)-approximation for the L_1 distance between the joint and product of the marginal distributions, using space O(log^{1024}(nm) epsilon^{-1024}), where m is the length of the stream and n denotes the size of the universe from which stream elements are drawn. Our general techniques include the L_1 distance as a special case, and we give an improved space bound of O(log^{12}(n) log^{2}({nm}/epsilon) epsilon^{-7}).

Vladimir Braverman, Alan Roytman, and Gregory Vorsanger. Approximating Subadditive Hadamard Functions on Implicit Matrices. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2016.25, author = {Braverman, Vladimir and Roytman, Alan and Vorsanger, Gregory}, title = {{Approximating Subadditive Hadamard Functions on Implicit Matrices}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {25:1--25:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.25}, URN = {urn:nbn:de:0030-drops-66483}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.25}, annote = {Keywords: Streaming Algorithms, Measuring Independence, Hadamard Functions, Implicit Matrices} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Given a stream of data, a typical approach in streaming algorithms is to design a sophisticated algorithm with small memory that computes a specific statistic over the streaming data. Usually, if one wants to compute a different statistic after the stream is gone, it is impossible. But what if we want to compute a different statistic after the fact? In this paper, we consider the following fascinating possibility: can we collect some small amount of specific data during the stream that is "universal," i.e., where we do not know anything about the statistics we will want to later compute, other than the guarantee that had we known the statistic ahead of time, it would have been possible to do so with small memory? This is indeed what we introduce (and show) in this paper with matching upper and lower bounds: we show that it is possible to collect universal statistics of polylogarithmic size, and prove that these universal statistics allow us after the fact to compute all other statistics that are computable with similar amounts of memory. We show that this is indeed possible, both for the standard unbounded streaming model and the sliding window streaming model.

Vladimir Braverman, Rafail Ostrovsky, and Alan Roytman. Zero-One Laws for Sliding Windows and Universal Sketches. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 573-590, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2015.573, author = {Braverman, Vladimir and Ostrovsky, Rafail and Roytman, Alan}, title = {{Zero-One Laws for Sliding Windows and Universal Sketches}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {573--590}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.573}, URN = {urn:nbn:de:0030-drops-53248}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.573}, annote = {Keywords: Streaming Algorithms, Universality, Sliding Windows} }

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