Efficient Approximate Minimum Entropy Coupling of Multiple Probability Distributions
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Given a collection of probability distributions p_1,…,p_m, the minimum entropy coupling is the coupling X_1,…,X_m (X_i∼ p_i) with the smallest entropy H(X_1,…,X_m). While this problem is known to be NP-hard, we present an efficient algorithm for computing a coupling with entropy within 2 bits from the optimal value. More precisely, we construct a coupling with entropy within 2 bits from the entropy of the greatest lower bound of p_1,…,p_m with respect to majorization. This construction is also valid when the collection of distributions is infinite, and when the supports of the distributions are infinite. Potential applications of our results include random number generation, entropic causal inference, and functional representation of random variables.
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