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Given data drawn from an unknown distribution, D, to what extent is it possible to amplify'' this dataset and faithfully output an even larger set of samples that appear to have been drawn from D? We formalize this question as follows: an (n,m) amplification procedure takes as input n independent draws from an unknown distribution D, and outputs a set of m > n
samples'' which must be indistinguishable from m samples drawn iid from D. We consider this sample amplification problem in two fundamental settings: the case where D is an arbitrary discrete distribution supported on k elements, and the case where D is a ddimensional Gaussian with unknown mean, and fixed covariance matrix. Perhaps surprisingly, we show a valid amplification procedure exists for both of these settings, even in the regime where the size of the input dataset, n, is significantly less than what would be necessary to learn distribution D to nontrivial accuracy. We also show that our procedures are optimal up to constant factors. Beyond these results, we describe potential applications of such data amplification, and formalize a number of curious directions for future research along this vein.
Author Information
Brian Axelrod (Stanford)
Shivam Garg (Stanford University)
Vatsal Sharan (Stanford University)
Gregory Valiant (Stanford University)
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