Evolution of K¸ 3 æ {pı} the invariant probability distribution of K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 ( ¸ ) def. ( ¸ ) ( ¸ ) p 3 K 3 K 2 3 {K¸ 3} Figure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distribution of K œ K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has K 3 K 2 3 {K¸ 3}¸ tion of K¸ 3 æ {pı} the invariant probability distribution of K œ R3 ◊ 3 + ,ú with One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). (4.22) d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 (4.23) ed P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has 3 K 3 K 2 3 Figure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distr K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄ One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one Î log(P ( ¸ ) ) ≠ log(Pı)ÎŒ Æ d H (u ( ¸ ) , uı) + d H (v where Pı is the unique solution of (4.2). igure 4.8: Evolution of K¸ 3 æ {pı} the invariant probability distribution of K œ R K€13 = 13. Theorem 4.2. One has (u ( ¸ ) , v ( ¸ ) ) æ (uı, vı) and d H (u ( ¸ ) , uı) = O(⁄(K)2 ¸), d H (v ( ¸ ) , vı) = O(⁄(K)2 ¸). One also has d H (u ( ¸ ) , uı) Æ d H (P ( ¸ ) 1 m , a) 1 ≠ ⁄(K)2 d H (v ( ¸ ) , vı) Æ d H (P ( ¸ ) ,€1 n , b) 1 ≠ ⁄(K)2 where we denoted P ( ¸ ) def. = diag(u ( ¸ ) )K diag(v ( ¸ ) ). Lastly, one has Î log(P ( ¸ ) ) ≠ log(Pı)ÎŒ Æ d H (u ( ¸ ) , uı) + d H (v ( ¸ ) , vı) where Pı is the unique solution of (4.2). Proof. One notices that for any (v, vÕ) œ (Rm + ,ú )2 , one has Sinkhorn under Hilbert’s Metric Theorem: <latexit 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<latexit 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Rothblum, 1999] and references therein). An intuitive way to handle these is to solve them iteratively, by modifying first u so that it satisfies the left of Equation (4.14) and then v to satisfy its right-hand side. These two upd Sinkhorn’s algorithm: u ( ¸ +1) def. = a Kv ( ¸ ) and v ( ¸ +1) def. = b K T u ( ¸ +1) , initialized with an arbitrary positive vector v (0) = 1 m . The division operator between two vectors is to be understood entry-wise. Note that a di erent in will likely lead to a di erent solution for u, v, since u, v are only define multiplicative constant (if u, v satisfy (4.13) then so do ⁄u, v/⁄ for any turns out however that these iterations converge (see Remark 4.7 for a justifica iterative projections, and Remark 4.13 for a strict contraction result) and a the same optimal coupling diag(u)K diag(v). Figure 4.5, top row, shows the of the coupling diag(u ( ¸ ) )K diag(v ( ¸ ) ) computed by Sinkhorn iterations. It ev the Gibbs kernel K towards the optimal coupling solving (4.2) by progressive the mass away from the diagonal. Remark 4.4 (Historical Perspective). This algorithm was originally introdu proof of convergence by Sinkhorn [1964] with later contributions [Sinkhorn a 1967, Sinkhorn, 1967]. It was used earlier as a heuristic to scale a matrix fits desired marginals (typically uniform) and known as the iterative propo ting procedure (IPFP) [Deming and Stephan, 1940] and RAS [Bacharach, 1 ods [Idel, 2016], and later extended in infinite dimensions by Ruschendorf [19 adopted very early as well in the field of economics, precisely to obtain ap [Franklin and Lorenz, 1989] <latexit 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<latexit 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