Minimum--Entropy Couplings and their Applications

Given two discrete random variables X and Y, with probability distributions bf p=(p1, ldots , pn) and bf q=(q1, ldots , qm), respectively, denote by cal C(bf p, bf q) the set of all couplings of bf p and bf q, that is, the set of all bivariate probability distributions that have bf p and bf q as marginals. In this paper, we study the problem of finding a joint probability distribution in cal C(bf p, bf 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 cal C(bf p, bf 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 X1, ldots , Xk, consistent with the known k marginal distributions of the individual random variables X1, ldots , Xk. 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.