Kavli Affiliate: Feng Wang
| First 5 Authors: Ya-Jing Ma, Feng Wang, Xian-Yuan Wu, Kai-Yuan Cai,
| Summary:
Given probability distributions ${bf p}=(p_1,p_2,ldots,p_m)$ and ${bf
q}=(q_1,q_2,ldots, q_n)$ with $m,ngeq 2$, denote by ${cal C}(bf p,q)$ the
set of all couplings of $bf p,q$, a convex subset of $R^{mn}$. Denote by
${cal C}_e({bf p},{bf q})$ 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 ${cal
C}_e({bf p},{bf q})$. In this paper, first, the detailed structure of ${cal
C}_e({bf p},{bf q})$ 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 $Psi$ on ${cal C}(bf p,q)$, $Psi$ also takes
its minimal value on ${cal C}_e({bf p},{bf q})$. As an application, the
exact solution of the minimum-entropy coupling problem is obtained for
$(Phi,hbar)$-entropy, a large class of entropy including Shannon entropy,
R’enyi entropy and Tsallis entropy etc. Finally, all the above are generalized
to multi-marginal case.
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