SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. First, we have that (22) is a direct consequence of (Kober, 1940, Theorem 1) and A4. Let us now prove (a). Using (20), we have
$$
\begin{array}{l} \sum_ {m = 0} ^ {K n} \mathrm {K L} \left(\pi^ {m} \mid \pi^ {m - 1}\right) = \sum_ {\ell = 0} ^ {n - 1} \sum_ {k = 0} ^ {K - 1} \mathrm {K L} \left(\pi^ {\ell K + k + 1} \mid \pi^ {\ell K + k}\right) + \mathrm {K L} \left(\pi^ {0} \mid \pi^ {- 1}\right) \\ = \sum_ {\ell = 0} ^ {n - 1} \sum_ {i \in \mathbb {S}} \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i} ^ {\ell + 1} - \psi_ {i} ^ {\ell}\right) \mathrm {d} \mu_ {i} + \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i _ {K - 1}} ^ {0} - \psi_ {i _ {K - 1}} ^ {- 1}\right) \mathrm {d} \mu_ {i _ {K - 1}} \\ = \sum_ {i \in \mathbb {S}} \sum_ {\ell = 0} ^ {n - 1} \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i} ^ {\ell + 1} - \psi_ {i} ^ {\ell}\right) d \mu_ {i} + \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i _ {K - 1}} ^ {0} - \psi_ {i _ {K - 1}} ^ {- 1}\right) d \mu_ {i _ {K - 1}} \\ = \sum_ {i \in \mathbb {S}} \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i} ^ {n} - \psi_ {i} ^ {0}\right) \mathrm {d} \mu_ {i} + \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i _ {K - 1}} ^ {0} - \psi_ {i _ {K - 1}} ^ {- 1}\right) \mathrm {d} \mu_ {i _ {K - 1}} \\ = \sum_ {i \in \mathsf {S}} \int_ {\mathbb {R} ^ {d}} \psi_ {i} ^ {n} \mathrm {d} \mu_ {i} \leq \operatorname {K L} \left(\pi^ {\star} \mid \bar {\pi}\right). \\ \end{array}
$$
where the last inequality follows the proof of Proposition 5.
Since the first term in the inequality of (b) is equal to  $\mathrm{KL}(\pi^{\star}\mid \pi^{nK})$ , we obtain (b) using that  $\mathrm{KL}(\pi^{\star}\mid \pi^{nK})\leq \mathrm{KL}(\pi^{\star}\mid \bar{\pi})$  following the proof of Proposition 5.
Let us now prove (c). Since  $\mathrm{KL}(\pi^{\star}\mid \bar{\pi}) < \infty$ , using Proposition 17, we have that  $\bigoplus_{i\in S}\psi_i^\star \in L^1 (\pi^\star)$ . From (b) and Proposition 17, we also get that  $\bigoplus_{i\in S}(\psi_i^\star -\psi_i^n)\in L^1 (\pi^\star)$ , and thus  $\int_{(\mathbb{R}^d)^{\ell +1}}\left|\bigoplus_{i\in S}(\psi_i^\star -\psi_i^n)\right|\mathrm{d}\pi^\star \leq C_0$  with  $C_0 > 0$ . Therefore, we have
$$
\int_ {(\mathbb {R} ^ {d}) ^ {\ell + 1}} \left| \bigoplus_ {i \in \mathbf {S}} \psi_ {i} ^ {n} \right| d \pi^ {\star} \leq \int_ {(\mathbb {R} ^ {d}) ^ {\ell + 1}} \left| \bigoplus_ {i \in \mathbf {S}} \psi_ {i} ^ {\star} \right| d \pi^ {\star} + \int_ {(\mathbb {R} ^ {d}) ^ {\ell + 1}} \left| \bigoplus_ {i \in \mathbf {S}} \left(\psi_ {i} ^ {\star} - \psi_ {i} ^ {n}\right) \right| d \pi^ {\star} \leq 2 C _ {0}.
$$
Using (22), we conclude with A4 that for any  $i \in S$ , we have
$$
\int_ {\mathbb {R} ^ {d}} | \psi_ {i} ^ {n} | \mathrm {d} \mu_ {i} \leq 2 c _ {i} C _ {0},
$$
which concludes the proof of (c).
The next lemma gives an explicit expression for  $\mathrm{KL}(\pi^n\mid \bar{\pi})$ . It can be seen as the multi-marginal counterpart of (Ruschendorf, 1995, Lemma 4.2).
Lemma 19. For any  $n \in \mathbb{N}$ , with  $k_{n} = (n - 1) \bmod(K)$ , we have
$$
\begin{array}{l} \mathrm {K L} (\pi^ {n} \mid \bar {\pi}) = \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k _ {n}}} ^ {q _ {n} + 1} \mathrm {d} \mu_ {i _ {k _ {n}}} + \sum_ {\ell = 0} ^ {k _ {n} - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {\ell}} ^ {q _ {n} + 1} \exp [ \psi_ {i _ {\ell}} ^ {q _ {n} + 1} - \psi_ {i _ {\ell}} ^ {q _ {n} + 2} ] \mathrm {d} \mu_ {i _ {\ell}} \\ + \sum_ {m = k _ {n} + 1} ^ {K - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {m}} ^ {q _ {n}} \exp \left[ \psi_ {i _ {m}} ^ {q _ {n}} - \psi_ {i _ {m}} ^ {q _ {n} + 1} \right] \mathrm {d} \mu_ {i _ {m}}. \\ \end{array}
$$
Proof. Let  $n \in \mathbb{N}$ , with  $k_n = (n - 1) \bmod(K)$ . Using (18), we have
$$
\operatorname {K L} \left(\pi^ {n} \mid \bar {\pi}\right) = \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k _ {n}}} ^ {q _ {n} + 1} \mathrm {d} \mu_ {i _ {k _ {n}}} + \sum_ {\ell = 0} ^ {k _ {n} - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {\ell}} ^ {q _ {n} + 1} \mathrm {d} \pi_ {i _ {\ell}} ^ {n} + \sum_ {m = k _ {n} + 1} ^ {K - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {m}} ^ {q _ {n}} \mathrm {d} \pi_ {i _ {m}} ^ {n}. \tag {23}
$$
Consider  $m \in \{k_n + 1, \dots, K - 1\}$ . Let  $m_n$  be the closest integer to  $n$  such that  $m_n > n$  and  $m = (m_n - 1) \bmod(K)$ . By (19), we have
$$
\mathrm {d} \pi^ {n} = \exp \left[ \bigoplus_ {j = k _ {n} + 1} ^ {m} \psi_ {i _ {j}} ^ {q _ {n}} - \psi_ {i _ {j}} ^ {q _ {n} + 1} \right] \mathrm {d} \pi^ {m _ {n}}.
$$
Using (19) recursively, we obtain
$$
\mathrm {d} \pi_ {i _ {m}} ^ {n} = \exp \left[ \psi_ {i _ {m}} ^ {q _ {n}} - \psi_ {i _ {m}} ^ {q _ {n} + 1} \right] \mathrm {d} \pi_ {i _ {m}} ^ {m _ {n}}, \tag {24}
$$
where we recall that  $\pi_{i_m}^{m_n} = \mu_{i_m}$
Consider now  $\ell \in \{0, \dots, k_n - 1\}$ . Let  $\ell_n$  be the closest integer to  $n$  such that  $\ell_n > n$  and  $\ell = (\ell_n - 1) \bmod (K)$ . By (19), we have
$$
\mathrm {d} \pi^ {n} = \exp [ \bigoplus_ {j = k _ {n} + 1} ^ {K - 1} \{\psi_ {i _ {j}} ^ {q _ {n}} - \psi_ {i _ {j}} ^ {q _ {n} + 1} \} \bigoplus_ {j ^ {\prime} = 0} ^ {\ell} \{\psi_ {i _ {j ^ {\prime}}} ^ {q _ {n} + 1} - \psi_ {i _ {j ^ {\prime}}} ^ {q _ {n} + 2} \} ] \mathrm {d} \pi^ {\ell_ {n}},
$$
and using (19) recursively, we obtain
$$
\mathrm {d} \pi_ {i _ {\ell}} ^ {n} = \exp \left[ \psi_ {i _ {\ell}} ^ {q _ {n} + 1} - \psi_ {i _ {\ell}} ^ {q _ {n} + 2} \right] \mathrm {d} \pi_ {i _ {\ell}} ^ {\ell_ {n}}, \tag {25}
$$
where we recall that  $\pi_{i_{\ell}}^{\ell_{n}} = \mu_{i_{\ell}}$ . We conclude the proof upon combining (23), (24) and (25).
We are now ready to prove a uniform integrability result which is the multi-marginal counterpart of (Ruschendorf, 1995, Lemma 4.4). Before stating Lemma 21, we prove the following well-known lemma. We recall that a sequence  $(\Psi_{n})_{n\in \mathbb{N}}$  such that for any  $n\in \mathbb{N}$ ,  $\Psi_n\in \mathrm{L}^1 (\mu)$ , is uniformly integrable w.r.t.  $\mu$  if (i)  $\sup_{n\in \mathbb{N}}\int_{\mathbb{R}^d}|\Psi_n|\mathrm{d}\mu < + \infty$  and (ii) for any  $\varepsilon >0$ , there exists  $K > 0$  such that for any  $n\in \mathbb{N}$ ,  $\int_{\overline{\mathrm{B}} (0,K)^c}|\Psi_n|\mathrm{d}\mu \leq \varepsilon$ .
Lemma 20. Let  $f: \mathbb{R} \to \mathbb{R}$ , convex and non-decreasing on  $[A, +\infty)$  with  $A > 0$  and  $\lim_{x \to +\infty} f(x) / x = +\infty$ . Assume that  $\sup_{n \in \mathbb{N}} \int_{\mathbb{R}^d} f(|\Psi_n|) \, \mathrm{d}\mu < +\infty$ . Then,  $(\Psi_n)_{n \in \mathbb{N}}$  is uniformly integrable w.r.t.  $\mu$ .
Proof. Since  $f$  is convex, using Jensen's inequality, we get that  $\sup_{n\in \mathbb{N}}f(\int_{\mathbb{R}^d}|\Psi_n|\mathrm{d}\mu) < +\infty$  and since  $\lim_{x\to +\infty}f(x) / x = +\infty$  we have  $\sup_{n\in \mathbb{N}}\int_{\mathbb{R}^d}|\Psi_n|\mathrm{d}\mu < +\infty$ . Let  $\varepsilon >0$ , there exists  $K > 0$  such that for any  $x > K$ ,  $x\leq \varepsilon f(x) / B$  with  $B = \sup_{n\in \mathbb{N}}\int_{\mathbb{R}^d}f(|\Psi_n|)\mathrm{d}\mu < +\infty$ . Therefore, we have for any  $n\in \mathbb{N}$
$$
\int_ {\overline {{\mathrm {B}}} (0, K) ^ {\mathrm {c}}} | \Psi_ {n} | \mathrm {d} \mu \leq (\varepsilon / B) \int_ {\overline {{\mathrm {B}}} (0, K) ^ {\mathrm {c}}} f (| \Psi_ {n} |) \mathrm {d} \mu \leq \varepsilon ,
$$
Lemma 21. Assume A4 and A5. Then,  $\{\exp[\oplus_{i\in S}\psi_i^n]\}_{n\in \mathbb{N}}$  is uniformly integrable w.r.t.  $\bar{\pi}$ .
Proof. It is enough to show that the sequence  $\{f(\exp[\oplus_{i\in S}\psi_i^n])\}_{n\in \mathbb{N}}$  is bounded in  $\mathrm{L}^1(\bar{\pi})$ , where  $f: u \mapsto u\log(u)$  is continuous, convex and such that  $\lim_{u\to \infty}f(u)/u = +\infty$ , see Lemma 20. Let  $n \in \mathbb{N}$ . We have
$$
\begin{array}{l} \int_ {(\mathbb {R} ^ {d}) ^ {\ell + 1}} f \left(\exp \left[ \bigoplus_ {i \in \mathsf {S}} \psi_ {i} ^ {n} \right]\right) \mathrm {d} \bar {\pi} = \operatorname {K L} \left(\pi^ {n K} \mid \bar {\pi}\right) \\ = \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {K - 1}} ^ {n} \mathrm {d} \mu_ {i _ {K - 1}} + \sum_ {k = 0} ^ {K - 2} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} \exp \left[ \psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n + 1} \right] \mathrm {d} \mu_ {i _ {k}} (Lemma 19) \\ = \sum_ {k = 0} ^ {K - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} d \mu_ {i _ {k}} + \sum_ {k = 0} ^ {K - 2} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} \left\{\exp \left[ \psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n + 1} \right] - 1 \right\} d \mu_ {i _ {k}} \\ \leq \mathrm {K L} (\pi^ {\star} \mid \bar {\pi}) + (\bar {c} + 1) \sum_ {k = 0} ^ {K - 2} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} \mathrm {d} \mu_ {i _ {k}} (Lemma 18-(a), A5) \\ \leq \mathrm {K L} (\pi^ {\star} | \bar {\pi}) + (\bar {c} + 1) \sum_ {k = 0} ^ {K - 2} \sup  _ {n \in \mathbb {N}} \int_ {\mathbb {R} ^ {d}} \left| \psi_ {i _ {k}} ^ {n} \right| \mathrm {d} \mu_ {i _ {k}} <   \infty . (Lemma 18-(c)) \\ \end{array}
$$
With the preliminary results stated above, we are now ready to prove Proposition 6.
Proof of Proposition 6. Using A4 and A5, we have, by Lemma 21, uniform integrability of  $\{\exp[\bigoplus_{i\in\mathbb{S}}\psi_i^n]\}_{n\in\mathbb{N}}$  in  $\mathrm{L}^1(\bar{\pi})$ . Therefore, the sequence  $\{\pi^{nK}\}_{n\in\mathbb{N}}$  is relatively compact with respect to the weak topology of  $\sigma(\mathrm{L}^1(\bar{\pi}),\mathrm{L}^\infty(\bar{\pi}))$ , denoted as the  $\tau$ -topology. We recall that  $\lim_{n\to\infty}\mathrm{KL}(\pi^{nK+1} \mid \pi^{nK}) = 0$ . This implies that  $\{\pi^{nK+1}\}_{n\in\mathbb{N}}$  is also relatively  $\tau$ -compact. By trivial recursion, we obtain that the sequences  $\{\pi^{nK+k}\}_{n\in\mathbb{N}}$ , where  $k \in \{2,\dots,K-1\}$  are also relatively  $\tau$ -compact. Therefore,  $\{\pi^n\}_{n\in\mathbb{N}}$  is relatively  $\tau$ -compact and  $\tau$ -sequentially compact.
We consider an increasing function  $\Phi :\mathbb{N}\to \mathbb{N}$  such that  $\{\pi^m\}_{m\in \Phi (\mathbb{N})}$  is a  $\tau$  -convergent subsequence, and we denote by  $\tilde{\pi}$  its limit for this topology. In particular,  $\tilde{\pi}\in \mathcal{P}_S$  by Proposition 5. We assume without loss of generality that  $\Phi (\mathbb{N})\subset K\mathbb{N}$
Using the lower semi-continuity of the Kullback-Leibler divergence (Dupuis & Ellis, 2011, Lemma 1.4.3), we get
$$
\mathrm {K L} (\tilde {\pi} \mid \bar {\pi}) \leq \lim  \inf  \mathrm {K L} (\pi^ {m} \mid \bar {\pi}) \leq \lim  \sup  \mathrm {K L} (\pi^ {m} \mid \bar {\pi}).
$$
Consider  $k\in \{0,\ldots ,K - 2\}$  . By (19), we have
$$
\frac {\mathrm {d} \mu_ {i _ {k}}}{\mathrm {d} \pi_ {i _ {k}} ^ {n K + k}} = \frac {\mathrm {d} \pi_ {i _ {k}} ^ {n K + k + 1}}{\mathrm {d} \pi_ {i _ {k}} ^ {n K + k}} = \frac {\mathrm {d} \pi^ {n K + k + 1}}{\mathrm {d} \pi^ {n K + k}} = \exp [ \psi_ {i _ {k}} ^ {n + 1} - \psi_ {i _ {k}} ^ {n} ],
$$
$$
\| \mu_ {i _ {k}} - \pi_ {i _ {k}} ^ {n K + k} \| _ {\mathrm {T V}} = (1 / 2) \int_ {\mathbb {R} ^ {d}} \left| \mathrm {d} \pi_ {i _ {k}} ^ {n K + k} / \mathrm {d} \mu_ {i _ {k}} - 1 \right| \mathrm {d} \mu_ {i _ {k}} = (1 / 2) \int_ {\mathbb {R} ^ {d}} \left| \exp \left[ \psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n + 1} \right] - 1 \right| \mathrm {d} \mu_ {i _ {k}}.
$$
With Proposition 5, we obtain that  $\{\exp[\psi_{i_k}^n - \psi_{i_k}^{n+1}]\}_{n \in \mathbb{N}}$  converges to  $1$  in  $\mathrm{L}^1(\mu_{i_k})$ . In addition using the uniform integrability of  $\{\psi_{i_k}^n\}_{n \in \mathbb{N}}$  and A5, we get
$$
\lim  \sup  _ {n \rightarrow + \infty} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} \exp [ \psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n + 1} ] d \mu_ {i _ {k}} = \lim  \sup  _ {n \rightarrow + \infty} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {n} d \mu_ {i _ {k}}.
$$
We denote  $m = K\ell$ . Since  $\mathrm{KL}(\pi^m \mid \bar{\pi}) = \int_{\mathbb{R}^d} \psi_{i_{K-1}}^\ell \, \mathrm{d}\mu_{i_{K-1}} + \sum_{k=0}^{K-2} \int_{\mathbb{R}^d} \psi_{i_k}^\ell \, \exp[\psi_{i_k}^\ell - \psi_{i_k}^{\ell+1}] \, \mathrm{d}\mu_{i_k}$  by Lemma 19, we finally have
$$
\operatorname {K L} \left(\tilde {\pi} \mid \bar {\pi}\right) \leq \lim  \sup  \left\{\sum_ {k = 0} ^ {K - 1} \int_ {\mathbb {R} ^ {d}} \psi_ {i _ {k}} ^ {\ell} \mathrm {d} \mu_ {i _ {k}} \right\} \leq \operatorname {K L} \left(\pi^ {\star} \mid \bar {\pi}\right)
$$
where the last inequality comes from Lemma 18.
Since  $\tilde{\pi}_i = \mu_i$  for any  $i\in \mathsf{S}$ , using Proposition 5, we have  $\tilde{\pi} = \pi^{\star}$  by uniqueness of  $\pi^{\star}$ . Hence,  $\pi^{\star}$  is the only limit point of  $\{\pi^n\}_{n\in \mathbb{N}}$  in the  $\tau$ -topology. In particular,  $\mathrm{KL}(\pi^n\mid \bar{\pi})\to \mathrm{KL}(\pi^{\star}\mid \bar{\pi})$ . Since  $\mathcal{P}_{\mathsf{S}}$  is convex, this last result implies  $\| \pi^{\star} - \pi^{n}\|_{\mathrm{TV}}\to 0$ , see the proof of Theorem 2.1 in Csiszár (1975).
We finish this section by highlighting that A5 is stronger than (Ruschendorf, 1995, B1). A natural extension of the latter assumption would consist of having a guarantee on the  $(K - 1)$  first potentials given by (17), as presented below.
A6. There exist  $0 < \underline{c} < \bar{c}$  such that for any  $k\in \{0,\dots ,K - 2\}$ , we have  $\underline{c}\leq \exp (-\psi_{i_k}^1)\leq \bar{c}$ .
Under A6, (Ruschendorf, 1995, Lemma 4.3) can be adapted as written below.
Lemma 22. Assume A6. Then, for any  $n \in \mathbb{N}^*$
(a) for any  $k\in \{0,\ldots ,K - 2\}$ , there exists  $\alpha_{n,k}\in \mathbb{N}$  such that
$$
\underline {{c}} \cdot (\underline {{c}} / \bar {c}) ^ {\alpha_ {n, k} (K - 2)} \leq \exp [ \psi_ {i _ {k}} ^ {n - 1} - \psi_ {i _ {k}} ^ {n} ] \leq \bar {c} \cdot (\bar {c} / \underline {{c}}) ^ {\alpha_ {n, k} (K - 2)}
$$
(b) there exists  $\alpha_{n,K - 1}\in \mathbb{N}$  such that
$$
1 / \bar {c} ^ {K - 1} \cdot (\underline {{c}} / \bar {c}) ^ {\alpha_ {n, K - 1} (K - 2)} \leq \exp [ \psi_ {i _ {K - 1}} ^ {n - 1} - \psi_ {i _ {K - 1}} ^ {n} ] \leq 1 / \underline {{c}} ^ {K - 1} \cdot (\bar {c} / \underline {{c}}) ^ {\alpha_ {n, K - 1} (K - 2)}
$$
where  $\{\alpha_{n,k}\}_{n\in \mathbb{N}^*,k\in \{0,\dots ,K - 1\}}$  is a strictly increasing sequence that can be explicitly defined.
Proof. We prove the result by recursion on  $n \in \mathbb{N}^*$ .
Take  $n = 1$ . Let  $k \in \{0, \dots, K - 2\}$ . We define  $\alpha_{1,k} = 0$  and directly obtain (a) by A5 since  $\psi_{i_k}^0 = 0$ . Let us prove (b). We have by (17)
$$
\begin{array}{l} \exp \bigl [ \psi_ {i _ {K - 1}} ^ {0} - \psi_ {i _ {K - 1}} ^ {1} \bigr ] = \frac {\int_ {(\mathbb {R} ^ {d}) ^ {\ell}} \exp \bigl [ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {1} \bigr ] h d \nu_ {- i _ {K - 1}}}{\int_ {(\mathbb {R} ^ {d}) ^ {\ell}} \exp \bigl [ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {0} \bigr ] h d \nu_ {- i _ {K - 1}}} \\ = \frac {\int_ {(\mathbb {R} ^ {d}) ^ {\ell}} \exp [ \bigoplus_ {k = 0} ^ {K - 2} \{\psi_ {i _ {k}} ^ {1} - \psi_ {i _ {k}} ^ {0} \} + \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {0} ] h d \nu_ {- i _ {K - 1}}}{\int_ {(\mathbb {R} ^ {d}) ^ {\ell}} \exp [ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {0} ] h d \nu_ {- i _ {K - 1}}} . \\ \end{array}
$$
Using (a) at rank  $n = 1$ , we have
$$
1 / \bar {c} ^ {K - 1} \leq \exp \left[ \bigoplus_ {k = 0} ^ {K - 2} \left\{\psi_ {i _ {k}} ^ {1} - \psi_ {i _ {k}} ^ {0} \right\} \right] \leq 1 / \underline {{c}} ^ {K - 1},
$$
and therefore, we obtain (b) by taking  $\alpha_{1,K - 1} = 0$ . Let us assume that the result is verified for some  $n\in \mathbb{N}^*$ . We have
$$
\begin{array}{l} \exp [ \psi_ {i _ {0}} ^ {n} - \psi_ {i _ {0}} ^ {n + 1} ] = \frac {\int \exp [ \bigoplus_ {k = 1} ^ {K - 1} \psi_ {i _ {k}} ^ {n} ] h d \nu_ {- i _ {0}}}{\int \exp [ \bigoplus_ {k = 1} ^ {K - 1} \psi_ {i _ {k}} ^ {n - 1} ] h d \nu_ {- i _ {0}}} \\ = \frac {\int \exp [ \bigoplus_ {k = 1} ^ {K - 2} \{\psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n - 1} \} \oplus \{\psi_ {i _ {K - 1}} ^ {n} - \psi_ {i _ {K - 1}} ^ {n - 1} \} + \bigoplus_ {k = 1} ^ {K - 1} \psi_ {i _ {k}} ^ {n - 1} ] h d \nu_ {- i _ {0}}}{\int \exp [ \bigoplus_ {k = 1} ^ {K - 1} \psi_ {i _ {k}} ^ {n - 1} ] h d \nu_ {- i _ {0}}} \\ \end{array}
$$
Using (a) and (b) at rank  $n$ , we have
$$
\begin{array}{l} 1 / \bar {c} ^ {K - 2} \cdot (\underline {{c}} / \bar {c}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 2} \alpha_ {n, k}} \leq \exp \left[ \bigoplus_ {k = 1} ^ {K - 2} \left\{\psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n - 1} \right\} \right] \\ \leq 1 / \underline {{c}} ^ {K - 2} \cdot (\bar {c} / \underline {{c}}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 2} \alpha_ {n, k}}, \\ \end{array}
$$
$$
\underline {{c}} ^ {K - 1} \cdot (\underline {{c}} / \bar {c}) ^ {\alpha_ {n, K - 1} (K - 2)} \leq \exp [ \psi_ {i _ {K - 1}} ^ {n} - \psi_ {i _ {K - 1}} ^ {n - 1} ] \leq \bar {c} ^ {K - 1} \cdot (\bar {c} / \underline {{c}}) ^ {\alpha_ {n, K - 1} (K - 2)}.
$$
$$
\begin{array}{l} \underline {{c}} \cdot (\underline {{c}} / \bar {c}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 1} \alpha_ {n, k}} \leq \exp [ \bigoplus_ {k = 1} ^ {K - 2} \{\psi_ {i _ {k}} ^ {n} - \psi_ {i _ {k}} ^ {n - 1} \} \oplus \{\psi_ {i _ {K - 1}} ^ {n} - \psi_ {i _ {K - 1}} ^ {n - 1} \} ] \\ \leq \bar {c} \cdot (\bar {c} / \underline {{c}}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 1} \alpha_ {n, k}}, \\ \end{array}
$$
$$
\underline {{c}} \cdot (\underline {{c}} / \bar {c}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 1} \alpha_ {n, k}} \leq \exp [ \psi_ {i _ {0}} ^ {n} - \psi_ {i _ {0}} ^ {n + 1} ] \leq \bar {c} \cdot (\bar {c} / \underline {{c}}) ^ {(K - 2) \sum_ {k = 1} ^ {K - 1} \alpha_ {n, k}}.
$$
Now, we define  $\alpha_{n + 1,0} = \sum_{k = 1}^{K - 1}\alpha_{n,k}$  to obtain (a) for  $k = 0$ . Consider now  $k\in \{1,\dots ,K - 2\}$ . Following the same steps as above, we recursively define
$$
\alpha_ {n + 1, k} = \sum_ {j = 0} ^ {k - 1} \alpha_ {n + 1, j} + \sum_ {j ^ {\prime} = k + 1} ^ {K - 1} \alpha_ {n, j ^ {\prime}},
$$
which gives (a) at rank  $n + 1$ . Let us now prove (b) at rank  $n + 1$ . We have
$$
\begin{array}{l} \exp \left[ \psi_ {i _ {K - 1}} ^ {n} - \psi_ {i _ {K - 1}} ^ {n + 1} \right] = \frac {\int \exp \left[ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {n + 1} \right] h d \nu_ {- i _ {K - 1}}}{\int \exp \left[ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {n} \right] h d \nu_ {- i _ {K - 1}}} \\ = \frac {\int \exp [ \bigoplus_ {k = 0} ^ {K - 2} \{\psi_ {i _ {k}} ^ {n + 1} - \psi_ {i _ {k}} ^ {n} \} + \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {n} ] h d \nu_ {- i _ {k}}}{\int \exp [ \bigoplus_ {k = 0} ^ {K - 2} \psi_ {i _ {k}} ^ {n} ] h d \nu_ {- i _ {K - 1}}} . \\ \end{array}
$$
Using (a) at rank  $n + 1$ , we obtain
$$
\begin{array}{l} 1 / \bar {c} ^ {K - 1} \cdot (\underline {{c}} / \bar {c}) ^ {(K - 2) \sum_ {k = 0} ^ {K - 2} \alpha_ {n + 1, k}} \leq \exp \left[ \bigoplus_ {k = 0} ^ {K - 2} \left\{\psi_ {i _ {k}} ^ {n + 1} - \psi_ {i _ {k}} ^ {n} \right\} \right] \\ \leq 1 / \underline {{c}} ^ {K - 1} \cdot (\bar {c} / \underline {{c}}) ^ {(K - 2) \sum_ {k = 0} ^ {K - 2} \alpha_ {n + 1, k}}. \\ \end{array}
$$
Therefore, by taking  $\alpha_{n+1,K-1} = \sum_{k=0}^{K-2} \alpha_{n+1,k}$ , we obtain (b), which concludes the proof.
Unfortunately, Lemma 22 only yields non-vacuous bounds in the case  $K = 2$ . Indeed, when  $K > 2$ , the sequence  $\{\alpha_{n,k}\}_{n\in \mathbb{N}^*,k\in \{0,\ldots ,K - 1\}}$  leads to increase the bounds on the quantities  $\exp [\psi_{i_k}^{n - 1} - \psi_{i_k}^n]$ , which motivates the use of A5.
D.3 Proof of Section 5
For the rest of this section, we consider the multi-marginal Schrödinger bridge problem given by (TreeSB) and establish in Proposition 24 the correspondence with the regularized Wasserstein propagation problem presented in Solomon et al. (2014, 2015). We first state a technical result.
Lemma 23. Let  $\varepsilon >0$ . Assume that  $\pi^0$  is given by (2), where  $r\in \mathbb{V}$  is chosen arbitrarily. Then, for any  $\pi \in \mathcal{P}_{\Gamma_r}$ , we have
$$
\begin{array}{l} \varepsilon \mathrm {K L} (\pi \mid \pi^ {0}) = \sum_ {(v, v ^ {\prime}) \in \mathsf {E} _ {r}} \left\{w _ {v, v ^ {\prime}} \mathbb {E} _ {\pi_ {v, v ^ {\prime}}} \left[ \| X _ {v} - X _ {v ^ {\prime}} \| ^ {2} \right] - \varepsilon \mathrm {H} (\pi_ {v, v ^ {\prime}}) \right\} \\ + \varepsilon \sum_ {v \in \mathsf {V}} \mathrm {c a r d} (\mathrm {C} _ {v}) \mathrm {H} (\pi_ {v}) + \varepsilon \mathrm {K L} (\pi_ {r} \mid \pi_ {r} ^ {0}), \\ \end{array}
$$
where we recall that  $\mathbf{C}_v = \{v' \in \mathsf{V} : (v, v') \in \mathsf{E}_r\}$ .
Proof. Since  $\pi, \pi^0 \in \mathcal{P}_{\mathrm{T}_r}$ , we obtain the following decomposition
$$
\begin{array}{l} \operatorname {K L} (\pi \mid \pi^ {0}) \\ = \operatorname {K L} \left(\pi_ {r} \prod_ {\left(v, v ^ {\prime}\right) \in \mathsf {E} _ {r}} \pi_ {v ^ {\prime} | v} \mid \pi_ {r} ^ {0} \prod_ {\left(v, v ^ {\prime}\right) \in \mathsf {E} _ {r}} \pi_ {v ^ {\prime} | v} ^ {0}\right) \\ = \mathrm {K L} \left(\pi_ {r} \mid \pi_ {r} ^ {0}\right) + \sum_ {\left(v, v ^ {\prime}\right) \in E _ {r}} \int_ {\mathbb {R} ^ {d}} \mathrm {K L} \left(\pi_ {v ^ {\prime} \mid v} (\cdot | x _ {v}) \mid \pi_ {v ^ {\prime} \mid v} ^ {0} (\cdot | x _ {v})\right) \mathrm {d} \pi_ {v} (x _ {v}) \\ = \operatorname {K L} \left(\pi_ {r} \mid \pi_ {r} ^ {0}\right) - \sum_ {\left(v, v ^ {\prime}\right) \in \mathsf {E} _ {r}} \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \log \pi_ {v ^ {\prime} | v} ^ {0} \mathrm {d} \pi_ {v, v ^ {\prime}} - \sum_ {\left(v, v ^ {\prime}\right) \in \mathsf {E} _ {r}} \int_ {\mathbb {R} ^ {d}} \mathrm {H} \left(\pi_ {v ^ {\prime} | v} (\cdot | x _ {v})\right) \mathrm {d} \pi_ {v} (x _ {v}). \\ \end{array}
$$
We finally obtain the result by using the definition of  $\pi^0$  and noticing that  $\int_{\mathbb{R}^d}\mathrm{H}(\pi_{v'|v}(\cdot |x_v))\mathrm{d}\pi_v(x_v) = \mathrm{H}(\pi_{v,v'}) - \mathrm{H}(\pi_v)$  for any  $(v,v')\in \mathsf{E}_r$
Proposition 24. Let  $\varepsilon >0$  and  $\mu_0\in \mathcal{P}$  such that  $\mu_0\ll$  Leb. Assume that  $\pi^0$  is given by (2), where  $r\in \mathsf{V}$  is chosen arbitrarily, and that  $\varphi_r = \mathrm{d}\mu_0 / \mathrm{d}\mathrm{Leb}$ . Also assume A2. Then, the set of marginals of the solution to (TreeSB) is exactly the solution to the entropic-regularized Wasserstein Propagation problem (Solomon et al., 2014, 2015) defined by
$$
\begin{array}{l} \arg \min  \left\{\sum_ {(v, v ^ {\prime}) \in E _ {r}} w _ {v, v ^ {\prime}} W _ {2, \varepsilon / w _ {v, v ^ {\prime}}} ^ {2} \left(\nu_ {v}, \nu_ {v ^ {\prime}}\right) + \varepsilon \sum_ {v \in V} \operatorname {c a r d} \left(C _ {v}\right) H \left(\nu_ {v}\right) + \varepsilon K L \left(\nu_ {r} \mid \mu_ {0}\right): \right. (W P) \\ \left\{\nu_ {v} \right\} _ {v \in \mathsf {V}} \in \mathcal {P} ^ {\ell + 1}, \nu_ {i} = \mu_ {i}, \forall i \in \mathsf {S} \}, \\ \end{array}
$$
Proof. Assume that  $\pi^0$  is given by (2), where  $r\in \mathsf{V}$  is chosen arbitrarily, and that  $\varphi_r = \mathrm{d}\mu_0 / \mathrm{d}\mathrm{Leb}$ . In particular, we have  $\pi_r^0 = \mu_0$ . Moreover, it is clear that  $\pi^0$  verifies A1, and A3 by Proposition 16.
Let  $\{\nu_v\}_{v\in \mathbb{V}}\in \mathcal{P}^{\ell +1}$  and  $\{\nu^{(v,v')}\}_{(v,v')\in \mathsf{E}_r}\in (\mathcal{P}^{(2)})^{|\mathsf{E}_r|}$ . We define
$$
F (\{\nu_ {v} \}) = \sum_ {(v, v ^ {\prime}) \in \mathsf {E} _ {r}} w _ {v, v ^ {\prime}} W _ {2, \varepsilon / w _ {v, v ^ {\prime}}} ^ {2} (\nu_ {v}, \nu_ {v ^ {\prime}}) + \varepsilon \sum_ {v \in \mathsf {V}} \operatorname {c a r d} (\mathrm {C} _ {v}) \mathrm {H} (\nu_ {v}) + \varepsilon \mathrm {K L} (\nu_ {r} \mid \mu_ {0}),
$$
$$
\begin{array}{l} G (\nu_ {r}, \{\nu^ {(v, v ^ {\prime})} \}) = \sum_ {(v, v ^ {\prime}) \in \mathsf {E} _ {r}} \{w _ {v, v ^ {\prime}} \mathbb {E} _ {\nu^ {(v, v ^ {\prime})}} [ \| X _ {v} - X _ {v ^ {\prime}} \| ^ {2} ] - \varepsilon \mathrm {H} (\nu^ {(v, v ^ {\prime})}) \} \\ + \varepsilon \sum_ {(v, v ^ {\prime}) \in E _ {r}} H \left(\nu_ {v} ^ {(v, v ^ {\prime})}\right) + \varepsilon K L \left(\nu_ {r} \mid \mu_ {0}\right). \\ \end{array}
$$
By definition of the regularized Wasserstein distance given in (3), we have for any  $\{\nu_v\}_{v\in \mathbb{V}}\in \mathcal{P}^{\ell +1}$