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Proceedings 2001 IEEE International Conference on Cluster Computing (2001)
Las Vegas, Nevada
Oct. 14, 2001 to Oct. 17, 2001
ISBN: 0-7695-1390-5
pp: 442
<p>Given access to independent samples of a distribution A over [n] × [m], we show how to test whether the distributions formed by projecting A to each coordinate are independent, i.e., whether A is \varepsilon-close in the L<sub>1</sub> norm to the product distribution A<sub>1</sub> × A<sub>2</sub> for some distributions A<sub>1</sub> over [n] and A<sub>2</sub> over [m]. The sample complexity of our test is \widetilde0(n^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} m^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} poly(\varepsilon ^{ - 1} )), assuming without loss of generality that m \leqslant n. We also give a matching lower bound, up to poly(\log n,\varepsilon ^{ - 1} ) factors.</p> <p>Furthermore, given access to samples of a distribution X over [n], we show how to test if X is \varepsilon-close in L<sub>1</sub> norm to an explicitly specified distribution Y . Our test uses \widetilde0(n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} poly(\varepsilon ^{ - 1} )) samples, which nearly matches the known tight bounds for the case when Y is uniform.</p>

E. Fischer, T. Batu, R. Kumar, R. Rubinfeld, P. White and L. Fortnow, "Testing Random Variables for Independence and Identity," Proceedings 2001 IEEE International Conference on Cluster Computing(FOCS), Las Vegas, Nevada, 2001, pp. 442.
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