Minimum-Entropy Couplings and Their Applications

Given two discrete random variables X and Y, with probability distributions p = (p
₁
, ..., p
_n
) and q = (q
₁
, ..., q
_m
), 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
₁
, ..., X
_k
, consistent with the known k marginal distributions of the individual random variables X
₁
, ..., X
_k
. 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.