2025/05/18 by Ma, Ya-Jing, Wang, Feng, Wu, Xian-Yuan +1
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2505.12227
Given probability distributions \bf p=(p1,p2,…,pm) and \bf q=(q1,q2,…, qn) with m,n≥ 2, denote by \cal C(\bf p,q) the set of all couplings of \bf p,q, a convex subset of \Rmn. Denote by \cal Ce(\bf p,\bf q) the finite set of all extreme points of \cal C(\bf p,q). It is well known that, as a strictly concave function, the Shannan entropy H on \cal C(\bf p,q) takes its minimal value in \cal Ce(\bf p,\bf q). In this paper, first, the detailed structure of \cal Ce(\bf p,\bf q) is well specified and all extreme points are enumerated by a special algorithm. As an application, the exact solution of the minimum-entropy coupling problem is obtained. Second, it is proved that for any strict Schur-concave function Ψ on \cal C(\bf p,q), Ψ also takes its minimal value on \cal Ce(\bf p,\bf q). As an application, the exact solution of the minimum-entropy coupling problem is obtained for (Φ,ℏ)-entropy, a large class of entropy including Shannon entropy, Rényi entropy and Tsallis entropy etc. Finally, all the above are generalized to multi-marginal case.