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• W2963484654 abstract "Given two discrete random variables X and Y, with probability distributions p = (p <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> , ..., p <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n</sub> ) and q = (q <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> , ..., q <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>m</sub> ), respectively, let us denote by C(p, q) the set of all couplings of p and q, that is, the set of all bivariate probability distributions that have p and q as marginals. In this paper, we study the problem of finding a joint probability distribution in C(p, q) of minimum entropy (equivalently, a coupling that maximizes the mutual information between X and Y), and we discuss several situations where the need for this kind of optimization naturally arises. Since the optimization problem is known to be NP-hard, we give an efficient algorithm to find a joint probability distribution in C(p, q) with entropy exceeding the minimum possible at most by 1 bit, thus providing an approximation algorithm with an additive gap of at most 1 bit. Leveraging on this algorithm, we extend our result to the problem of finding a minimum-entropy joint distribution of arbitrary k ≥ 2 discrete random variables X <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> , ..., X <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> , consistent with the known k marginal distributions of the individual random variables X <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>1</sub> , ..., X <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>k</sub> . In this case, our algorithm has an additive gap of at most log k from optimum. We also discuss several related applications of our findings and extensions of our results to entropies different from the Shannon entropy." @default.
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• W2963484654 date "2019-06-01" @default.
• W2963484654 modified "2023-09-26" @default.
• W2963484654 title "Minimum-Entropy Couplings and Their Applications" @default.
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• W2963484654 doi "https://doi.org/10.1109/tit.2019.2894519" @default.
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