SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Using assumption (c), we have  $\varphi_{i}(x_{i})\in [-\infty , + \infty)$  for any  $i\in \mathbb{S}$ . Let  $k\in \{0,\dots ,K - 2\}$ . We have by definition of  $F^n$
$$
\varphi_ {i _ {k}} ^ {n} (x _ {i _ {k}}) = F ^ {n} (x ^ {i _ {k}}) - \sum_ {\stackrel {m = 0} {m \neq k}} ^ {K - 1} \varphi_ {i _ {m}} ^ {n} (x _ {i _ {m}} ^ {\star}) = F ^ {n} (x ^ {i _ {k}}) - F ^ {n} (x ^ {\star}),
$$
where we used assumption (b) in the last equality. Since  $x^{i_k} \in \mathsf{A}^0$  and  $x^{\star} \in \mathsf{A}^{0}$ , we have by assumption (c),
$$
\lim _ {n \to \infty} \varphi_ {i _ {k}} ^ {n} (x _ {i _ {k}}) = F (x ^ {i _ {k}}) - F (x ^ {\star}) = \varphi_ {i _ {k}} (x _ {i _ {k}}).
$$
Furthermore, by combining the definition of  $F^n$  with assumption (b), we have
$$
\lim _ {n \to \infty} \varphi_ {i _ {K - 1}} ^ {n} (x _ {i _ {K - 1}}) = F (x ^ {i _ {K - 1}}) = \varphi_ {i _ {K - 1}} (x _ {i _ {K - 1}}),
$$
In what follows, before proving Proposition 3, we respectively show in Proposition 15 and Proposition 16 how A2 and A3 can be satisfied in the case where  $\pi^0\in \mathcal{P}_{\mathrm{T}_r}$ , as in (2), that is
$$
\pi^ {0} = \pi_ {r} ^ {0} \bigotimes_ {(v, v ^ {\prime}) \in \mathsf {E} _ {r}} \pi_ {v ^ {\prime} | v} ^ {0}.
$$
Proposition 15. Let  $\pi^0\in \mathcal{P}_{\mathsf{T}_r}$ . Assume that  $\pi_r^0 = \mu_r$  if  $r\in \mathsf{S}$  or  $\pi_r^0 = \mathrm{N}(m_r,\sigma_r\mathrm{Id})$ , with  $m_{r}\in \mathbb{R}^{d}$  and  $\sigma_r > 0$  if  $r\in \mathsf{S}^{\mathrm{c}}$ . In addition, assume that for any  $(v,v^{\prime})\in \mathsf{E}_r$ ,  $\pi_{v^{\prime}|v}^{0}(\cdot |x_{v}) = \mathrm{N}(x_{v},\sigma_{v,v^{\prime}}\mathrm{Id})$  with  $\sigma_{v,v^{\prime}} > 0$ . Finally, assume that for any  $i\in \mathsf{S}$ ,  $\int_{\mathbb{R}^d}\| x\|^2\mathrm{d}\mu_i(x) < +\infty$  and  $\mathrm{H}(\mu_i) < +\infty$ . Then A2 is satisfied.
Proof. Let  $\pi = \otimes_{i\in \mathsf{S}}\mu_i\otimes_{i\in \mathsf{S}^c}\nu_i$  with  $\nu_{i}$  any Gaussian measure with positive definite covariance matrix. First, we have that
$$
\mathrm {K L} (\pi \mid \pi^ {0}) = \mathrm {K L} (\pi_ {r} \mid \pi_ {r} ^ {0}) + \sum_ {(v, v ^ {\prime}) \in \mathsf {E} _ {r}} \int_ {\mathbb {R} ^ {d}} \mathrm {K L} (\pi_ {v ^ {\prime} | v} | \pi_ {v ^ {\prime} | v} ^ {0}) \mathrm {d} \pi_ {v}.
$$
For any  $(v,v^{\prime})\in \mathsf{E}_r$  , there exists  $C_{v,v^{\prime}}\geq 0$  such that
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} \mathrm {K L} \left(\pi_ {v ^ {\prime} | v} \mid \pi_ {v ^ {\prime} | v} ^ {0}\right) \mathrm {d} \pi_ {v} \leq C _ {v, v ^ {\prime}} - \mathrm {H} \left(\pi_ {v ^ {\prime}}\right) + \int_ {\mathbb {R} ^ {d} \times \mathbb {R} ^ {d}} \| x _ {v} - x _ {v ^ {\prime}} \| ^ {2} / \left(2 \sigma_ {v, v ^ {\prime}} ^ {2}\right) \mathrm {d} \pi_ {v} \otimes \pi_ {v ^ {\prime}} \left(x _ {v}, x _ {v ^ {\prime}}\right) \\ \leq C _ {v, v ^ {\prime}} - \mathrm {H} \left(\pi_ {v ^ {\prime}}\right) + \left(1 / \sigma_ {v, v ^ {\prime}} ^ {2}\right) \int_ {\mathbb {R} ^ {d}} \| x _ {v} \| ^ {2} \mathrm {d} \pi_ {v} \left(x _ {v}\right) + \left(1 / \sigma_ {v, v ^ {\prime}} ^ {2}\right) \int_ {\mathbb {R} ^ {d}} \| x _ {v ^ {\prime}} \| ^ {2} \mathrm {d} \pi_ {v ^ {\prime}} \left(x _ {v ^ {\prime}}\right) <   + \infty . \\ \end{array}
$$
We conclude the proof upon remarking that  $\mathrm{KL}(\pi_r\mid \pi_r^0) < + \infty$
Proposition 16. Let  $\pi^0\in \mathcal{P}_{\mathbb{T}_r}$ . Assume that  $\pi_r^0 = \mu_r$  if  $r\in \mathsf{S}$  or  $\pi_r^0 = \mathrm{N}(m_r,\sigma_r\mathrm{Id})$ , with  $m_{r}\in \mathbb{R}^{d}$  and  $\sigma_r > 0$  if  $r\in \mathsf{S}^{\mathrm{c}}$ . In addition, assume that for any  $(v,v^{\prime})\in \mathsf{E}_r$ ,  $\pi_{v^{\prime}|v}^{0}(\cdot |x_{v}) = \mathrm{N}(x_{v},\sigma_{v,v^{\prime}}\mathrm{Id})$  with  $\sigma_{v,v^{\prime}} > 0$ . Finally, assume that for any  $i\in \mathsf{S}$ ,  $\mu_{i}$  admits a positive density w.r.t. the Lebesgue measure. Then A3 is satisfied.
Proof. We have that  $\pi^0$  admits a positive density w.r.t the Lebesgue measure. Letting  $\tilde{\pi}^0 = \bigotimes_{i\in \mathbb{S}}\mu_i\bigotimes_{j\in \mathbb{S}^c}\tilde{\mu}_j$  where  $\tilde{\mu}_j$  which admits a positive density w.r.t. the Lebesgue measure for any  $j\in S^c$ , we get that  $\tilde{\pi}^0$  admits a positive density w.r.t. the Lebesgue measure and therefore  $\pi^0\sim \tilde{\pi}^0$  which concludes the proof.
Using the preliminary results presented above, we are now ready to prove Proposition 3.
Proof of Proposition 3. Assume A1 and A2. Since  $\mathcal{P}_{\mathsf{S}}$  is convex and closed in total-variation norm, there exists a probability distribution  $\pi^{\star}$  solution to (7), or equivalently to (static-mSB), by using A2 with (Csiszár, 1975, Theorem 2.1.). Moreover, this solution is unique by strict convexity of  $\mathrm{KL}(\cdot \mid \pi^0)$ .
We now turn to the proof of existence of potentials defining  $(\mathrm{d}\pi^{\star} / \mathrm{d}\pi^{0})$ , by adapting the arguments of (Nutz, 2021, Section 2.3.). Define  $\nu^n = \operatorname*{argmin}\{\mathrm{KL}(\pi \mid \pi^0):\pi \in \mathcal{P}_S^n\}$  for any  $n\in \mathbb{N}^*$ . Since  $\{\mathcal{P}_S^n\}_{n\in \mathbb{N}^*}\subset \mathcal{P}^{(\ell +1)}$  is a decreasing sequence of sets that are convex and closed in total-variation norm such that (6) holds, we get from (Nutz, 2021, Proposition 1.17.) with A2 that
$$
\lim _ {n \to \infty} \| \nu^ {n} - \pi^ {\star} \| _ {\mathrm {T V}} = 0,
$$
$$
\lim  _ {n \rightarrow \infty} \| \left(\mathrm {d} \nu^ {n} / \mathrm {d} \pi^ {0}\right) - \left(\mathrm {d} \pi^ {\star} / \mathrm {d} \pi^ {0}\right) \| _ {\mathrm {L} ^ {1} \left(\pi^ {0}\right)} = 0. \tag {8}
$$
Following (Nutz, 2021, Example 1.18), there exists a family of bounded measurable functions  $\{\varphi_i^n\}_{n\in \mathbb{N}^*,i\in \mathbb{S}}$  with  $\varphi_i^n:\mathbb{R}^d\to \mathbb{R}$  such that for any  $n\in \mathbb{N}^*$
$$
\left(\mathrm {d} \nu^ {n} / \mathrm {d} \pi^ {0}\right) = \exp \left[ \bigoplus_ {i \in \mathbb {S}} \varphi_ {i} ^ {n} \right]. \tag {9}
$$
We consider such family  $\{\varphi_i^n\}_{n\in \mathbb{N}^*,i\in \mathbb{S}}$  for the rest of the proof. By combining (8) and (9), we obtain, up to extraction,
$$
\left(\mathrm {d} \pi^ {\star} / \mathrm {d} \pi^ {0}\right) = \lim  _ {n \rightarrow \infty} \exp \left[ \bigoplus_ {i \in S} \varphi_ {i} ^ {n} \right] \quad \pi^ {0} - a. s.. \tag {10}
$$
We now define the following sets
$$
\mathsf {A} ^ {\star} = \left\{x \in \left(\mathbb {R} ^ {d}\right) ^ {\ell + 1}: \lim  _ {n \rightarrow \infty} \bigoplus_ {i \in \mathbb {S}} \varphi_ {i} ^ {n} (x _ {i}) \in [ - \infty , + \infty) \right\},
$$
$$
\mathsf {B} ^ {\star} = \{x \in (\mathbb {R} ^ {d}) ^ {\ell + 1}: \lim  _ {n \to \infty} \bigoplus_ {i \in \mathsf {S}} \varphi_ {i} ^ {n} (x _ {i}) > - \infty \} \subset \mathsf {A} ^ {\star}
$$
Using (10), we have  $\pi^0 (\mathsf{A}^\star) = 1$ . Using A3, it comes  $\tilde{\pi}^0 (\mathsf{A}^\star) = 1$ . Moreover, we also get that  $\pi^{\star}(\mathsf{B}^{\star}) = 1$  by (10). Thus, it comes  $\pi^0 (\mathsf{B}^\star) > 0$ , and  $\tilde{\pi}^0 (\mathsf{B}^\star) > 0$  using A3.
We then apply Lemma 13 to  $\tilde{\pi}^0$  and  $\mathsf{A} = \mathsf{A}^{\star}$ . Since  $\tilde{\pi}^0 (\mathsf{B}^\star) > 0$ , it implies that there exists  $x^{\star}\in \mathsf{B}^{\star}$  and a measurable set  $\mathsf{A}^0\subset \mathsf{B}^\star$  verifying the properties (a) and (b). Following (Nutz, 2021, Corollary 2.12), we may assume without loss of generality in the statement of Lemma 13 that the sets  $\mathsf{X}_m^0$  are measurable with  $\prod_{m = 0}^{\ell}\mathsf{X}_m^0\subset \mathsf{A}$ . In this case, we obtain that  $\mu_i(\mathrm{proj}_i(\mathsf{A}^0)) = 1$  for any  $i\in S$ .
We now aim at applying Lemma 14 to the set  $\mathsf{A}^0$ . Remark that  $\mathsf{A}^0$  directly satisfies assumption (a). For any  $n \in \mathbb{N}^*$ , consider the following transformation of the functions  $\{\varphi_i^n\}_{i \in S}$
$$
\varphi_ {i _ {k}} ^ {n} \leftarrow \varphi_ {i _ {k}} ^ {n} - \varphi_ {i _ {k}} ^ {n} (x _ {i _ {k}} ^ {\star}), \quad \forall k \in \{0, \dots , K - 2 \},
$$
$$
\varphi_ {i _ {K - 1}} ^ {n} \gets \varphi_ {i _ {K - 1}} ^ {n} + \sum_ {k = 0} ^ {K - 2} \varphi_ {i _ {k}} ^ {n} \left(x _ {i _ {k}} ^ {\star}\right).
$$
For any  $i\in \mathbb{S}$  , we restrict  $\varphi_{i_k}^n$  to  $\mathsf{X}_{i_k}^0$  , so that the family  $\{\varphi_i^n\}_{n\in \mathbb{N}^*,i\in \mathbb{S}}$  now verifies assumption (b). Finally, since  $\mathsf{A}^0\subset \mathsf{A}^\star$  and  $x^{\star}\in \mathsf{B}^{\star}$  , we directly obtain assumption (c).
Therefore, Lemma 14 may be applied. It provides us with the family of functions  $\{\varphi_i\}_{i\in \mathbb{S}}$  defined by  $\varphi_{i}:\mathrm{X}_{i}^{0}\to [-\infty , + \infty)$  with  $\varphi_{i} = \lim_{n\to \infty}\varphi_{i}^{n}\mu_{i}$ -a.s. for any  $i\in \mathbb{S}$ . Since  $\mu_{i}(\operatorname {proj}_{i}(\mathsf{A}^{0})) = 1$  for any  $i\in \mathbb{S}$ , we may extend the functions  $\varphi_{i}$  to  $\mathbb{R}^d$ . In particular, we can find a family of functions  $\{\psi_i^\star \}_{i\in \mathbb{S}}$  with  $\psi_i^\star :\mathbb{R}^d\to [-\infty , + \infty)$  such that  $\psi_i^\star = \varphi_i$ $\mu_{i}$ -a.s. Note that these functions are measurable as limits of measurable functions.
Since  $\pi^0\sim \tilde{\pi}^0$  by A3, (10) turns into
$$
\left(\mathrm {d} \pi^ {\star} / \mathrm {d} \pi^ {0}\right) = \exp \left[ \bigoplus_ {i \in \mathbf {S}} \psi_ {i} ^ {\star} \right] \quad \pi^ {0} - \mathrm {a . s .}. \tag {11}
$$
Finally, we show that the functions  $\psi_i^\star$  are  $\mu_{i}$ -a.s. finite. Let  $i\in \mathbb{S}$ . Let us define  $\mathsf{A}_i = \{x_i\in \mathbb{R}^d:$ $\psi_i^\star (x_i) = -\infty \}$ . Using (11), we obtain  $(\mathrm{d}\pi^{\star} / \mathrm{d}\pi^{0})(\mathsf{A}_{i}\times (\mathbb{R}^{d})^{\ell}) = 0$ . Since  $\pi_i^\star = \mu_i$ , we have
$$
\mu_ {i} (\mathsf {A} _ {i}) = \pi^ {\star} (\mathsf {A} _ {i} \times (\mathbb {R} ^ {d}) ^ {\ell}) = \int_ {\mathsf {A} _ {i} \times (\mathbb {R} ^ {d}) ^ {\ell}} (\mathrm {d} \pi^ {\star} / \mathrm {d} \pi^ {0}) \mathrm {d} \pi^ {0} = 0,
$$
We now turn to the proof of Corollary 4, which states that the iterates of (mIPF) can be expressed via potentials, in the same manner as the solution  $\pi^{\star}$  to (static-mSB).
Proof of Corollary 4. Assume A1, A2 and A3. We prove the result of this corollary by recursion on  $n \in \mathbb{N}^*$ . First take  $n = 1$ . In this case, the first iteration of (mIPF) is a multi-marginal SB problem of the form (static-mSB) where  $S = \{i_0\}$  with reference measure  $\pi^0$ . Therefore, using A2 and A3, we can apply Proposition 3 and obtain existence of  $\psi_{i_0}^1: \mathbb{R}^d \to \mathbb{R}$  such that
$$
\left(\mathrm {d} \pi^ {1} / \mathrm {d} \pi^ {0}\right) = \exp [ \psi_ {i _ {0}} ^ {1} ] \quad \pi^ {0} \mathrm {- a . s .}
$$
By taking  $\psi_{i_k}^0 = 0$  for  $k\in \{1,\ldots ,K - 1\}$ , we thus obtain the result at step  $n = 1$ .
Now assume that the result is verified for some  $n \in \mathbb{N}^*$ , with  $k_n = (n - 1) \bmod(K)$ . We define  $k_n + 1 = n \bmod(K)$  and  $q_n \in \mathbb{N}$  as the quotient of the Euclidean division of  $n$  by  $K$ . In this case, the  $(n + 1)$ -th iteration of (mIPF) is a multi-marginal SB problem of the form (static-mSB) where  $S = \{i_{k_n + 1}\}$  with reference measure  $\pi^n$ . Using (13), we have that A2 is satisfied for this new (static-mSB) problem. A1 and A3 are satisfied for this problem, given the form of  $\pi^n$ . Therefore, we can apply Proposition 3 and obtain existence of  $\psi_{i_{k_n + 1}}^{q_n + 1}: \mathbb{R}^d \to \mathbb{R}$  such that
$$
\left(\mathrm {d} \pi^ {n + 1} / \mathrm {d} \pi^ {n}\right) = \exp \left[ \psi_ {i _ {k _ {n} + 1}} ^ {q _ {n} + 1} \right] \quad \pi^ {n} - \mathrm {a . s}. \tag {12}
$$
By assumption, we have that  $\pi^n\ll \pi^0$  . Hence, we obtain  $\pi^{n + 1}\ll \pi^0$  and thus,
$$
\left(\mathrm {d} \pi^ {n + 1} / \mathrm {d} \pi^ {0}\right) = \left(\mathrm {d} \pi^ {n + 1} / \mathrm {d} \pi^ {n}\right) \left(\mathrm {d} \pi^ {n} / \mathrm {d} \pi^ {0}\right) \quad \pi^ {0} - \mathrm {a . s}.
$$
By combining (12) with the result of the recursion at step  $n$ , we directly obtain the result at step  $n + 1$ , which achieves the proof.
Proofs of Proposition 5 and Proposition 6. In this part of the section, we establish the proofs of results related to the convergence of (mIPF), respectively Proposition 5 and Proposition 6, which can be seen as a natural extension of (Ruschendorf, 1995, Proposition 2.1.) and (Ruschendorf, 1995, Theorem 3.1.).
Proof of Proposition 5. Under A1 and A2, we obtain by Proposition 3 existence and uniqueness of a solution to (static-mSB), which we denote by  $\pi^{\star}$ . Since  $\pi^{\star} \in \mathcal{P}_{\mathbb{S}}$ , using recursively (Csiszár, 1975, Theorem 3.12.), the fact that  $\{\pi_{i_k} = \mu_{i_k} : \pi \in \mathcal{P}^{(|\mathsf{V}|)}\}$  is convex for any  $k \in \{0, \dots, K - 1\}$  and (mIPF), we obtain
$$
\operatorname {K L} \left(\pi^ {\star} \mid \pi^ {0}\right) = \operatorname {K L} \left(\pi^ {\star} \mid \pi^ {n}\right) + \sum_ {i = 1} ^ {n} \operatorname {K L} \left(\pi^ {i} \mid \pi^ {i - 1}\right). \tag {13}
$$
Therefore, we have  $\sum_{i=1}^{\infty} \mathrm{KL}(\pi^i \mid \pi^{i-1}) \leq \mathrm{KL}(\pi^\star \mid \pi^0) < \infty$  and thus,
$$
\lim  _ {i \rightarrow + \infty} \mathrm {K L} \left(\pi^ {i} \mid \pi^ {i - 1}\right) = 0. \tag {14}
$$
Let  $n \in \mathbb{N}^*$  with  $n > 2K$ ,  $k \in \{0, \ldots, K - 1\}$  and let  $q_n \in \mathbb{N}$  be the quotient of the Euclidean division of  $n - 1$  by  $K$ . We define  $n_k = q_n K + k + 1$  with  $(n_k - 1) = k \bmod(K)$  if  $n_k \leq n$ . Otherwise, we set  $n_k = (q_n - 1)K + k + 1$  with  $(n_k - 1) = k \bmod(K)$ . Note that we always have  $|n - n_k| \leq 2K$ . In particular, we have  $\pi_{i_k}^{n_k} = \mu_{i_k}$  by definition of (mIPF). Therefore, we obtain
$$
\begin{array}{l} \| \pi_ {i _ {k}} ^ {n} - \mu_ {i _ {k}} \| _ {\mathrm {T V}} \leq \| \pi^ {n} - \pi^ {n _ {k}} \| _ {\mathrm {T V}} \\ \leq \left\| \pi^ {n} - \pi^ {n - 1} \right\| _ {\mathrm {T V}} + \dots + \left\| \pi^ {n _ {k} + 1} - \pi^ {n _ {k}} \right\| _ {\mathrm {T V}} \quad (\text {t r i a n g l e i n e q u a l i t y}) \\ \leq \left(2 \mathrm {K L} \left(\pi^ {n} \mid \pi^ {n - 1}\right)\right) ^ {1 / 2} + \dots + \left(2 \mathrm {K L} \left(\pi^ {n _ {k} + 1} \mid \pi^ {n _ {k}}\right)\right) ^ {1 / 2}, \quad (\text {P i n s k e r ' s i n e q u a l i t y}) \\ \end{array}
$$
where each term goes to 0 as  $n \to +\infty$  in the last inequality by (14), which achieves the proof.
We now turn to the proof Proposition 6, which requires several preliminary technical results. For the rest of this section, we define, for any  $n \in \mathbb{N}$ ,  $q_{n}$  as the quotient of the Euclidean division of  $n - 1$  by  $K$  (in particular,  $q_{0} = -1$ ).
Schrodinger equations. Under A1, A2 and A3, we know from Proposition 3 that the unique solution  $\pi^{\star}$  to (static-mSB) can be  $\pi^0$ -a.s. written as  $(\mathrm{d}\pi^{\star} / \mathrm{d}\pi^{0}) = \exp [\bigoplus_{i\in S}\psi_{i}^{\star}]$ , where  $\{\psi_i^\star \}_{i\in S}$  are measurable potentials, referred to as Schrodinger potentials. These functions are determined by the fixed-point Schrodinger equations
$$
\psi_ {i} ^ {\star} (x _ {i}) = \log \left[ r _ {i} (x _ {i}) / \int_ {(\mathbb {R} ^ {d}) ^ {\ell}} \exp \left[ \sum_ {j \in \mathsf {S} \backslash \{i \}} \psi_ {j} ^ {\star} (x _ {j}) \right] h (x _ {0: \ell}) \mathrm {d} \nu_ {- i} (x _ {- i}) \right] \quad \mu_ {i} \text {- a . s .}, \quad \forall i \in \mathsf {S},
$$
which are obtained by marginalising  $\pi^{\star}$  along its constrained marginals. This family of potentials is not unique. Indeed, for any family of real numbers  $\{\lambda_{i_k}\}_{k\in \{0,\dots ,K - 2\}}$  , we have
$$
\left(\mathrm {d} \pi^ {\star} / \mathrm {d} \pi^ {0}\right) = \exp [ \bigoplus_ {i \in S} \tilde {\psi} _ {i} ],
$$
where  $\tilde{\psi}_{i_k} = \psi_{i_k}^* +\tilde{\lambda}_{i_k}$  for any  $k\in \{0,\ldots ,K - 1\}$  with  $\tilde{\lambda}_{i_k} = \lambda_{i_k}$  if  $k\in \{0,\dots ,K - 2\}$  and  $\tilde{\lambda}_{i_{K - 1}} = -\sum_{i = 0}^{K - 2}\lambda_{i_k}$ .
Remark on the initialisation of (mIPF). Consider a probability measure  $\bar{\pi}^0\in \mathcal{P}^{(\ell +1)}$  of the form
$$
\left(\mathrm {d} \bar {\pi} ^ {0} / \mathrm {d} \pi^ {0}\right) = \exp \left[ \bigoplus_ {i \in S} \psi_ {i} ^ {0} \right], \tag {15}
$$
where  $\{\psi_i^0\}_{i\in S}$  is a family of measurable potentials with  $\psi_i^0:\mathbb{R}^d\to \mathbb{R}$  such that  $\left|\int_{\mathbb{R}^d}\psi_i^0\mathrm{d}\mu_i\right| < \infty$  for any  $i\in S$ . Then, for any  $\pi \in \mathcal{P}_S$ , we have
$$
\mathrm {K L} (\pi \mid \pi^ {0}) = \mathrm {K L} (\pi \mid \bar {\pi} ^ {0}) + \int_ {(\mathbb {R} ^ {d}) ^ {K}} \bigoplus_ {i \in \mathsf {S}} \psi_ {i} ^ {0} \mathrm {d} \pi = \mathrm {K L} (\pi \mid \bar {\pi} ^ {0}) + \sum_ {i \in \mathsf {S}} \int_ {\mathbb {R} ^ {d}} \psi_ {i} ^ {0} \mathrm {d} \mu_ {i}.
$$
Hence, (static-mSB) is equivalent to the multi-marginal SB problem
$$
\operatorname {a r g m i n} \left\{\mathrm {K L} \left(\pi | \bar {\pi} ^ {0}\right): \pi \in \mathcal {P} ^ {(\ell + 1)}, \pi_ {i} = \mu_ {i}, \forall i \in \mathrm {S} \right\}.
$$
We refer to (Peyre et al., 2019, Proposition 4.2) for the EOT counterpart of this result. This means that the solutions of the multi-marginal Schrödinger Bridge problem are invariant by multiplication of the reference measure by potentials on the fixed marginals. Consequently, the initialisation of the (mIPF) sequence may be chosen as  $\bar{\pi}^0$  instead of  $\pi^0$ .
For sake of clarity, we now refer to the reference probability measure of (static-mSB) as  $\bar{\pi}$  or  $\pi^{-1}$  and to the initialisation of the (mIPF) iterates as  $\pi^0$ .
Solving (mIPF) with potentials. To prove the convergence of the (mIPF) iterates to the solution  $\pi^{\star}$  given by Proposition 3, we first rewrite these iterates with potentials, following the form of  $\pi^{\star}$ .
To do so, we recursively define the sequence of potentials  $\{\psi_i^n\}_{n\in \mathbb{N},i\in S}$  by
$$
\psi_ {i _ {0}} ^ {0} = \dots = \psi_ {i _ {K - 2}} ^ {0} = 0, \tag {16}
$$
$$
\psi_ {i _ {K - 1}} ^ {0} (x _ {i _ {K - 1}}) = \log (r _ {i _ {K - 1}} (x _ {i _ {K - 1}}) / \int_ {(\mathbb {R} ^ {d}) ^ {\ell}} h (x _ {0: \ell}) \mathrm {d} \nu_ {- i _ {K - 1}} (x _ {- i _ {K - 1}})),
$$
and for any  $n\in \mathbb{N}^*$  and  $k\in \{0,\dots ,K - 1\}$
$$
\begin{array}{l} \psi_ {i _ {k}} ^ {q _ {n} + 1} \left(x _ {i _ {k}}\right) = \log \left[ r _ {i _ {k}} \left(x _ {i _ {k}}\right) / \int_ {\left(\mathbb {R} ^ {d}\right) ^ {\ell}} \exp \left[ \bigoplus_ {\ell = 0} ^ {k} \psi_ {i _ {\ell}} ^ {q _ {n} + 1} \left(x _ {i _ {\ell}}\right) \bigoplus_ {m = k + 1} ^ {K - 1} \psi_ {i _ {m}} ^ {q _ {n}} \left(x _ {i _ {m}}\right) \right] \right. \\ \left. \times h \left(x _ {0: \ell}\right) \mathrm {d} \nu_ {- i _ {k}} \left(x _ {- i _ {k}}\right) \right], \tag {17} \\ \end{array}
$$
recalling that  $q_{n}$  is the quotient of the Euclidean division of  $n - 1$  by  $K$ .
We now define the sequence of probability measures  $\{\pi^n\}_{n\in \mathbb{N}}$  by
$$
\mathrm {d} \pi^ {n} / \mathrm {d} \bar {\pi} = \exp \left[ \bigoplus_ {\ell = 0} ^ {k _ {n}} \psi_ {i _ {\ell}} ^ {q _ {n} + 1} \bigoplus_ {m = k _ {n} + 1} ^ {K - 1} \psi_ {i _ {m}} ^ {q _ {n}} \right], k _ {n} = (n - 1) \operatorname {m o d} (K), n = q _ {n} K + k _ {n} + 1. \tag {18}
$$
In particular, we have  $\left(\mathrm{d}\pi^{0} / \mathrm{d}\bar{\pi}\right) = \exp [\oplus_{\ell = 0}^{K - 1}\psi_{i_{\ell}}^{0}] = \exp [\psi_{i_{K - 1}}^{0}]$ , and thus  $\int_{\mathbb{R}^d}\psi_{i_{K - 1}}^0\mathrm{d}\mu_{i_{K - 1}} = \mathrm{KL}(\mu_{i_{K - 1}}\mid \bar{\pi}_{i_{K - 1}})$ . Consequently,  $\pi^0$  can be chosen as the initialisation of (mIPF), following the previous remark, if we assume that  $\mathrm{KL}(\mu_{i_{K - 1}}\mid \bar{\pi}_{i_{K - 1}}) < \infty$ . In (TreeSB) with  $r = i_{K - 1}$ , the latter assumption is directly verified since we choose  $\bar{\pi}_{i_{K - 1}} = \mu_{i_{K - 1}}$ .
Let  $n \in \mathbb{N}$ , with  $k_n = (n - 1) \bmod(K)$ ,  $k_n + 1 = n \bmod(K)$ . Using (16) and (17), we get that  $\pi_{i_{k_n}}^n = \mu_{i_{k_n}}$ . Moreover, we have
$$
\mathrm {d} \pi^ {n} / \mathrm {d} \pi^ {n - 1} = \exp \left[ \psi_ {i _ {k _ {n}}} ^ {q _ {n} + 1} - \psi_ {i _ {k _ {n}}} ^ {q _ {n}} \right], \tag {19}
$$
with the convention that  $\psi_{i_{K - 1}}^{-1} = 0$ . In particular, we obtain that  $\pi_{|i_{k_n + 1}}^{n + 1} = \pi_{|i_{k_n + 1}}^n$ .
In conclusion, the sequence  $\{\pi^n\}_{n\in \mathbb{N}}$  defined in (18) verifies  $\pi^{n + 1} = \mu_{i_{k_n + 1}}\pi_{|i_{k_n + 1}}^n$  for any  $n\in \mathbb{N}$ . By decomposition property of the Kullback-Leibler divergence, this sequence solves (mIPF) with initialisation  $\pi^0$ . We consider such iterates in the following.
Since  $\pi_{i_{kn}}^n = \mu_{i_{kn}}$ , we have that
$$
\operatorname {K L} \left(\pi^ {n} \mid \pi^ {n - 1}\right) = \int_ {\mathbb {R} ^ {d}} \left(\psi_ {i _ {k _ {n}}} ^ {q _ {n} + 1} - \psi_ {i _ {k _ {n}}} ^ {q _ {n}}\right) \mathrm {d} \mu_ {i _ {k _ {n}}}. \tag {20}
$$
Before proving a multi-marginal counterpart to (Ruschendorf, 1995, Lemma 4.1), we state and prove the following result.
Proposition 17. Let  $\pi_0, \pi_1$  two probability measures on  $\mathbb{R}^d$  such that  $\pi_0 \ll \pi_1$ . Then, denoting  $f = \mathrm{d}\pi_0 / \mathrm{d}\pi_1$ , the following assertions are equivalent:
(a)  $\mathrm{KL}(\pi_0\mid \pi_1) <   + \infty$
(b)  $\int_{\mathbb{R}^d}\left|\log (f)(x)\right|\mathrm{d}\pi_0(x) <   + \infty$
(c)  $\int_{\mathbb{R}^d}\log (f)(x)\mathbb{1}_{f(x) > 1}\mathrm{d}\pi_0(x) <   + \infty$
If one of these conditions is satisfied then  $\int_{\mathbb{R}^d}|\log (f)(x)|\mathrm{d}\pi_0\leq \mathrm{KL}(\pi_0\mid \pi_1) + 2 / \mathrm{e}$ .
$$
\int_ {\mathbb {R} ^ {d}} | \log (f) (x) | \mathbb {1} _ {f <   1} \mathrm {d} \pi_ {0} (x) \leq \int_ {\mathbb {R} ^ {d}} | \log (f) (x) f (x) | \mathbb {1} _ {f <   1} \mathrm {d} \pi_ {1} (x) \leq 1 / \mathrm {e}, \tag {21}
$$
where we have used that for any  $u \in [0,1]$ ,  $|u\log (u)| \leq 1 / \mathrm{e}$ . We have that (b) implies (c). Using the previous result we have that (c) implies (b). Hence (c) and (b) are equivalent. In addition, it is clear that (b) implies (a). Finally (this is more of a convention), we have that  $\mathrm{KL}(\pi_0 \mid \pi_1) = \int_{\mathbb{R}^d}\log (f)(x)\mathbb{1}_{f(x) > 1}\mathrm{d}\pi_0(x) + \int_{\mathbb{R}^d}\log (f)(x)\mathbb{1}_{f(x) < 1}\mathrm{d}\pi_0(x) < +\infty$ . Using (21) this implies (c). Finally, we have
$$
\begin{array}{l} \int_ {\mathbb {R} ^ {d}} | \log (f) (x) | \mathrm {d} \pi_ {0} (x) = \int_ {\mathbb {R} ^ {d}} \log (f) (x) \mathrm {d} \pi_ {0} (x) - 2 \int_ {\mathbb {R} ^ {d}} \log (f) (x) \mathbb {1} _ {f (x) <   1} \mathrm {d} \pi_ {0} (x) \\ \leq \operatorname {K L} \left(\pi_ {0} \mid \pi_ {1}\right) + 2 / \mathrm {e}, \\ \end{array}
$$
We begin with the following lemma which controls the integral of the potentials uniformly w.r.t.  $n\in \mathbb{N}$ . It can be seen as the multi-marginal counterpart of (Ruschendorf, 1995, Lemma 4.1).
Lemma 18. Assume A4. There exist  $\{c_i\}_{i\in \mathbb{S}}\in (0, + \infty)^K$  such that for any function  $f:(\mathbb{R}^d)^{\ell +1}\to$ $\mathbb{R}$  of the form  $f = \bigoplus_{i\in \mathbb{S}}f_{i}$ , we have
$$
c _ {i} \| f \| _ {\mathrm {L} ^ {1} \left(\pi^ {*}\right)} \geq \| f _ {i} \| _ {\mathrm {L} ^ {1} \left(\mu_ {i}\right)}, \quad \forall i \in S. \tag {22}
$$
For any  $n\in \mathbb{N}^{\star}$  , we have
(a)  $\sum_{i\in \mathbb{S}}\int_{\mathbb{R}^d}\psi_i^n\mathrm{d}\mu_i\leq \mathrm{KL}(\pi^\star \mid \bar{\pi}) <   \infty ,$
(b)  $\int_{(\mathbb{R}^d)^{\ell +1}}(\bigoplus_{i\in \mathsf{S}}\psi_i^\star -\bigoplus_{i\in \mathsf{S}}\psi_i^n)\mathrm{d}\pi^\star \leq \mathrm{KL}(\pi^\star \mid \bar{\pi}) <   \infty ,$
(c)  $\sup_{n\in \mathbb{N}}\int_{\mathbb{R}^d}\left|\psi_i^n\right|\mathrm{d}\mu_i <   \infty$ $\forall i\in S$