SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
F \left(\left\{\nu_ {v} \right\}\right) = \min  \left\{G \left(\nu_ {r}, \left\{\nu^ {\left(v, v ^ {\prime}\right)} \right\}\right): \nu^ {\left(v, v ^ {\prime}\right)} \in \mathscr {P} ^ {(2)}, \nu_ {v} ^ {\left(v, v ^ {\prime}\right)} = \nu_ {v}, \nu_ {v ^ {\prime}} ^ {\left(v, v ^ {\prime}\right)} = \nu_ {v ^ {\prime}}, \forall (v, v ^ {\prime}) \in \mathrm {E} _ {r} \right\}. \tag {26}
$$
In particular, we have  $F(\{\pi_v\}) \leq G(\pi_r, \{\pi_{v,v'}\})$  for any  $\pi \in \mathcal{P}^{(\ell + 1)}$ . We now prove the result of Proposition 24 in two steps denoted by Step 1 and Step 2.
Step 1. Let us not assume A2 for now. In this case, we prove in Step 1.a and Step 1.b that solving (WP) is equivalent to solving a modified version of (TreeSB) given by
$$
\pi^ {\star} = \operatorname {a r g m i n} \left\{\mathrm {K L} \left(\pi \mid \pi^ {0}\right): \pi \in \mathcal {P} _ {\mathrm {T} _ {r}}, \pi_ {i} = \mu_ {i}, \forall i \in \mathrm {S} \right\}. \quad \left(\mathrm {T} _ {r} - \text {T r e e S B}\right)
$$
Remark that any solution to  $(\mathsf{T}_r$ -TreeSB) is a solution to (TreeSB), but the converse result may not be true.
Step 1.a:  $(\mathrm{WP}) \Longrightarrow (\mathsf{T}_r$ -TreeSB). Consider a solution  $\{\nu_{v}^{\star}\}_{v \in V}$  to (WP). For any  $(v, v') \in \mathsf{E}_r$ ,  $W_{2,\varepsilon / w_{v,v'}}^2(\nu_v^\star, \nu_{v'}^\star)$  is well defined and thus, there exists  $\nu^{(v, v')} \in \Pi(\nu_v^\star, \nu_{v'}^\star)$  such that
$$
\nu^ {(v, v ^ {\prime})} \in \operatorname {a r g m i n} \left\{\mathbb {E} _ {\pi} \left[ \| X _ {v} - X _ {v ^ {\prime}} \| ^ {2} \right] - \left(\varepsilon / w _ {v, v ^ {\prime}}\right) \mathrm {H} (\pi): \pi \in \Pi \left(\nu_ {v} ^ {\star}, \nu_ {v ^ {\prime}} ^ {\star}\right) \right\}. \tag {27}
$$
Using the gluing lemma, we build the probability measure  $\pi^{\star} = \nu_r^\star \prod_{(v,v')\in \mathsf{E}_r}\nu_{v'|v}^{(v,v')}$  such that (i)  $\pi^{\star}\in \mathcal{P}_{\mathsf{T}_r}$ , and (ii)  $\pi_{v,v'}^{\star}$  and  $\nu^{(v,v')}$  have the same distribution for any  $(v,v')\in \mathsf{E}_r$ . In particular, we have  $\pi_i^\star = \mu_i$  for any  $i\in S$ .
Let us show now that  $\pi^{\star}$  is a solution to  $(\mathsf{T}_r$ -TreeSB). Let  $\pi \in \mathcal{P}_{\mathsf{T}_r}$  such that  $\pi_i = \mu_i$  for any  $i \in S$ . We have
$$
\begin{array}{l} \epsilon \mathrm {K L} (\pi \mid \pi^ {0}) = G \left(\pi_ {r}, \left\{\pi_ {v, v ^ {\prime}} \right\}\right) (Lemma 23) \\ \geq F \left(\left\{\pi_ {v} \right\}\right) \\ \geq F \left(\left\{\nu_ {v} ^ {\star} \right\}\right) \quad (\text {d e f i n i t i o n} \nu^ {\star}) \\ = G \left(\nu_ {r} ^ {\star}, \left\{\nu^ {(v, v ^ {\prime})} \right\}\right) (see(27)) \\ = G \left(\pi_ {r} ^ {\star}, \left\{\pi_ {(v, v ^ {\prime})} ^ {\star} \right\}\right) \quad (\text {d e f i n i t i o n} \pi^ {\star}) \\ = \epsilon \mathrm {K L} \left(\pi^ {\star} \mid \pi^ {0}\right). (Lemma 23) \\ \end{array}
$$
Therefore,  $\pi^{\star}$  is a solution to  $(\mathsf{T}_r$ -TreeSB).
Step 1.b:  $(\mathsf{T}_r\text{-TreeSB})\Rightarrow (\mathsf{WP})$  . Consider now a solution  $\pi^{\star}$  to  $(\mathsf{T}_r\text{-TreeSB})$  . Since  $\pi^{\star}\in \mathcal{P}_{\mathsf{T}_r}$  we have  $\pi^{\star} = \pi_r^{\star}\prod_{(v,v')\in E_r}\pi_{v'|v}^{\star}$  and  $\pi_i^\star = \mu_i$  for any  $i\in S$
Let us show that  $\{\pi_v^\star\}_{v \in \mathsf{V}}$  is a solution to (WP). Let  $\{\nu_v\}_{v \in \mathsf{V}} \in \mathcal{P}^{\ell + 1}$  such that  $\nu_i = \mu_i$  for any  $i \in \mathsf{S}$ .
Let  $\{\nu^{(v,v')}\}_{(v,v')\in \mathsf{E}_r}$  be a family of probability measures such that  $\nu^{(v,v')} \in \mathcal{P}^{(2)}, \nu_v^{(v,v')} = \nu_v, \nu_{v'}^{(v,v')} = \nu_{v'}$  for any  $(v,v') \in \mathsf{E}_r$ .
Using the gluing lemma, we build the probability measure  $\pi = \nu_r\prod_{(v,v')\in \mathsf{E}_r}\nu_{v'|v}^{(v,v')}$ , such that (i)  $\pi \in \mathcal{P}_{\mathsf{T}_r}$  and (ii)  $\pi_{v,v'}$  and  $\nu^{(v,v')}$  have the same distribution for any  $(v,v')\in \mathsf{E}_r$ . We have
$$
\begin{array}{l} \varepsilon \operatorname {K L} \left(\pi \mid \pi^ {0}\right) = G \left(\pi_ {r}, \left\{\pi_ {\left(v, v ^ {\prime}\right)} \right\}\right) (Lemma 23) \\ = G \left(\nu_ {r}, \left\{\nu^ {\left(v, v ^ {\prime}\right)} \right\}\right) \quad (\text {d e f i n i t i o n} \pi) \\ \geq \varepsilon \mathrm {K L} \left(\pi^ {\star} \mid \pi^ {0}\right) \quad (\text {d e f i n i t i o n} \pi^ {\star}) \\ = G \left(\pi_ {r} ^ {\star}, \left\{\pi_ {(v, v ^ {\prime})} ^ {\star} \right\}\right). (Lemma 23) \\ \end{array}
$$
By taking the infimum in the previous inequality over the families  $\{\nu^{(v,v^{\prime})}\}_{(v,v^{\prime})\in \mathsf{E}_r}$ , we obtain by (26) that
$$
F (\{\nu_ {v} \}) \geq G (\pi_ {r} ^ {\star}, \{\pi_ {(v, v ^ {\prime})} ^ {\star} \}) \geq F (\{\pi_ {v} ^ {\star} \}),
$$
and therefore,  $\{\pi_v^\star\}_{v \in \mathsf{V}}$  is a solution to (WP).
Step 2. We now assume A2. By Proposition 3, there exists a unique solution  $\pi^{\star} \in \mathcal{P}^{(\ell + 1)}$  to (TreeSB) such that we  $\pi^0$ -a.s. have  $(\mathrm{d}\pi^{\star} / \mathrm{d}\pi^{0}) = \exp[\bigoplus_{i \in S} \psi_{i}^{\star}]$ , where  $\{\psi_{i}^{\star}\}_{i \in S}$  are measurable potentials with  $\psi^{\star}: \mathbb{R}^{d} \to \mathbb{R}$ . Since  $\pi^0 \in \mathcal{P}_{\mathsf{T}_r}$ , we also have  $\pi^{\star} \in \mathcal{P}_{\mathsf{T}_r}$ , i.e., the potentials  $\{\psi_{i}^{\star}\}_{i \in S}$  do not modify the Markovian nature of  $\pi^0$ . Therefore,  $\pi^{\star}$  is also the unique solution to  $(\mathsf{T}_r$ -TreeSB). Using the equivalence between  $(\mathsf{T}_r$ -TreeSB) and (WP) established in Step 1, we finally obtain the result of Proposition 24.
In particular, Proposition 7 directly derives from Proposition 24 by taking  $r = i_{K - 1}$  and  $\mu_0 = \mu_{i_{K - 1}}$ .
D.4 Comparison with Haasler et al. (2021)
In their work, Haasler et al. (2021) study the static and discrete-state counterpart of our approach. Given a state space  $\mathsf{X}$  such that  $|\mathsf{X}| = n + 1$  with  $n \in \mathbb{N}$ , they establish a correspondence between multi-marginal EOT with a general tree-based cost and discrete-time multi-marginal static Schrödinger bridge, and provide an efficient method to solve these problems. In this section, we provide details on their framework and give a precise comparison between our theory and their results.
To be coherent with the setting of Haasler et al. (2021), we adapt here some of our notation. Let us define  $\mathsf{Z}^{(q)} = \mathbb{R}_{+}^{(n + 1)^q}$ . For any  $q\in \mathbb{N}^*$ , the set of probability measures on  $\mathsf{X}^q$  is defined as  $\mathcal{P}^{(q)} = \{M\in \mathsf{Z}^{(q)}:\langle M,\mathbf{1}\rangle = 1\}$ . We denote  $\mathcal{P} = \mathcal{P}^{(1)}$ . For any tensors  $M,P\in \mathsf{Z}^{(q)}$ , the Kullback-Leibler divergence between  $M$  and  $P$  is defined as  $\mathrm{KL}(M\mid P) = \langle M\log (M / P) - M + P,\mathbf{1}\rangle$  and the
entropy of  $M$  is defined as  $\mathrm{H}(M) = -\mathrm{KL}(M \mid \mathbf{1})$ , where the operations are meant componentwise. In the rest of the section, we consider an undirected tree  $\mathsf{T} = (\mathsf{V}, \mathsf{E})$  with  $|\mathsf{V}| = \ell + 1$  such that  $\mathsf{V}$  may be identified with  $\{0, \dots, \ell\}$ .
Details on the results of Haasler et al. (2021). In their paper, the authors consider a cost tensor  $C \in \mathbb{Z}^{(\ell + 1)}$  that factorizes along  $\mathsf{T}$ , i.e., for any  $\{j_0, \ldots, j_\ell\}$  with for any  $i \in \{0, \ldots, \ell\}$ ,  $j_i \in \{0, \ldots, n\}$ , we have
$$
C _ {j _ {0}, \ldots , j _ {\ell}} = \sum_ {(v, v ^ {\prime}) \in \mathsf {E}} C _ {j _ {v}, j _ {v ^ {\prime}}} ^ {\{v, v ^ {\prime} \}},
$$
where  $C^{\{v,v'\}}\in \mathbb{Z}^{(2)}$  is a cost matrix for transportation between the marginals at vertices  $v$  and  $v^{\prime}$  see (Haasler et al., 2021, Eq. (3.1)). In particular, this cost can be seen as the discrete counterpart of the tree-based cost introduced in (1) in the quadratic setting.
Given a subset  $S \subset V$  with  $|S| = K$  and a set of marginals  $\{\mu_i\}_{i \in S} \in \mathcal{P}^K$ , Haasler et al. (2021) study the EOT problem associated to  $T$ , see (Haasler et al., 2021, Eq. (2.4)), which is given by
$$
\operatorname {a r g m i n} \left\{\langle C, M \rangle - \varepsilon \mathrm {H} (M): M \in \mathcal {P} ^ {(\ell + 1)}, \operatorname {p r o j} _ {i} (M) = \mu_ {i}, \forall i \in \mathsf {S} \right\}. \quad (\text {d i s c r e t e - E m O T})
$$
This problem may be solved with Sinkhorn algorithm (Cuturei, 2013; Knight, 2008; Sinkhorn & Knopp, 1967), for which the authors provide an efficient implementation adapted to the tree-based setting, see (Haasler et al., 2021, Algorithm 3.1). Moreover, they state the convergence of their method in (Haasler et al., 2021, Theorem 3.5), as a direct consequence of the results presented in Luo & Tseng (1992).
In (Haasler et al., 2021, Section 4.2), it is assumed that S corresponds to the set of the leaves of T, as we do, and it is shown an equivalence between (discrete-EmOT) and the discrete-state static SB problem stated in (Haasler et al., 2021, Eq 4.2), which is given by
$$
\operatorname {a r g m i n} \left\{\sum_ {(v, v ^ {\prime}) \in E _ {r}} \mathrm {K L} \left(M ^ {(v, v ^ {\prime})} \mid \operatorname {d i a g} \left(\nu_ {v}\right) A ^ {(v, v ^ {\prime})}\right): \right. \quad (\text {d i s c r e t e - T r e e S B})
$$
$$
M ^ {(v, v ^ {\prime})} \in \mathcal {P} ^ {(2)}, \{\nu_ {v} \} _ {v \in \mathsf {V}} \in \mathcal {P} ^ {\ell + 1}, M ^ {(v, v ^ {\prime})} \mathbf {1} = \nu_ {v}, M ^ {(v, v ^ {\prime})} ^ {\top} \mathbf {1} = \nu_ {v ^ {\prime}}, \nu_ {i} = \mu_ {i}, \forall i \in \mathsf {S} \},
$$
where  $\mathsf{T}_r = (\mathsf{V},\mathsf{E}_r)$  is the directed version of  $\mathsf{T}$  rooted in an arbitrary vertex  $r\in \mathsf{S}$ , and  $A^{(v,v^{\prime})} = \exp (-C^{(v,v^{\prime})} / \varepsilon)\in \mathsf{Z}^{(2)}$  for any  $(v,v^{\prime})\in \mathsf{E}_r$ . Remark that  $A^{(v,v^{\prime})}$  may not necessarily be a transition probability matrix.
Finally, Haasler et al. (2021) provide two main numerical experiments. In (Haasler et al., 2021, Section 5.2), they consider a tree with 15 vertices, 14 edges and 8 leaves, combined to the state-space  $\mathsf{X} = \{0,1\}^{50\times 50}$ , and solve the corresponding (discrete-EmOT) problem for the quadratic cost. In (Haasler et al., 2021, Section 6), they apply their methodology to estimate ensemble flows on a hidden Markov chain. Given  $\tau \in \mathbb{N}^*$ , they consider a tree  $\mathsf{T}$  with  $\tau$  internal vertices (modeling the distribution of  $N$  agents at time  $t \in \{1,\dots ,\tau \}$ ), that are linearly linked, and such that each of these vertices is independently linked to  $S$  leaves of  $\mathsf{T}$  (modeling observations at time  $t \in \{1,\dots ,\tau \}$ ). In this setting, the state space is given by  $\mathsf{X} = \{1,\dots ,100\}^{N}$ . They solve the formulation (discrete-TreeSB) where the reference measure is chosen as a random walk.
Comparison with our results. We now establish remarks on the main differences between our methodology and the work of Haasler et al. (2021).
First of all, the continuous state-space counterpart of (discrete-TreeSB) is given by
$$
\operatorname {a r g m i n} \left\{\mathrm {K L} \left(\pi \mid \pi^ {0}\right): \pi \in \mathcal {P} _ {\mathrm {T} _ {r}}, \pi_ {i} = \mu_ {i}, \forall i \in \mathrm {S} \right\}, \tag {28}
$$
where  $\pi^0$  is a reference measure which factorizes along  $\mathsf{T}_r$ . In this case,  $\pi_{v,v'}$ ,  $\pi_v$  and  $\pi_{v'|v}^0$  in (28) respectively correspond to the continuous version of  $M^{(v,v')}$ ,  $\nu_v$  and  $A^{(v,v')}$  in (discrete-TreeSB). In contrast, our formulation of the multi-marginal Tree Schrödinger Bridge problem given in (TreeSB) is a minimization problem over all probability measures  $\pi \in \mathcal{P}^{(\ell+1)}$ , and is not restricted to the distributions that admit a Markovian factorization along  $\mathsf{T}$  as in (28). Hence, our framework may be considered more general. Remark that under A1, A2 and A3, Proposition 3 states that (TreeSB) admits a unique solution  $\pi^\star \ll \pi^0$  such that  $(\mathrm{d}\pi^\star/\mathrm{d}\pi^0)$  can be written with potentials. Then,  $\pi^\star \in \mathcal{P}_{\mathsf{T}_r}$  since  $\pi^0 \in \mathcal{P}_{\mathsf{T}_r}$ , and (TreeSB) is then equivalent to (28).
Furthermore, (EmOT) is more general than the continuous version of (discrete-EmOT), which we can recover by taking any measure  $\nu$  of the form  $(\mathrm{d}\nu /\mathrm{dLeb}) = \exp [\bigoplus_{i\in \mathbb{S}}\varphi_i]$  in (EmOT), where  $\{\varphi_i\}_{i\in \mathbb{S}}$  is a family of potentials such that  $\left|\int_{\mathbb{R}^d}\varphi_i\mathrm{d}\mu_i\right| < \infty$  for any  $i\in \mathbb{S}$ . As a consequence, our setting allows us to choose the root  $r\in \mathbb{V}\backslash \mathbb{S}$  for the SB problem, whereas Haasler et al. (2021) only consider the case where  $r\in \mathbb{S}$ . In the latter case, we establish in Appendix E that  $r$  can be chosen arbitrarily, as stated by (Haasler et al., 2021, Corollary 4.3).
Finally, TreeDSB deeply differs from the framework of Haasler et al. (2021) due its dynamic nature. Although we solve the same tree-based static SB problem (up to continuous/discrete state-space consideration), our approach consists in computing dynamic iterates (i.e., path measures) using diffusion-based methods instead of static iterates (i.e., distributions) using Sinkhorn algorithm. This paradigm is at the core of the DSB (De Bortoli et al., 2021) methodology, and offers an efficient approach to tackle high-dimensional settings, where Sinkhorn algorithm would fail.
Here, we present some advantages of the method proposed by Haasler et al. (2021) compared to ours. First, Haasler et al. (2021) may choose any kind of tree-based cost in practice, while our methodology only holds for the quadratic cost. This limitation is shared with all approaches based on the DSB (De Bortoli et al., 2021) methodology. Indeed, since the cost is determined by the reference path measure, we often choose quadratic costs associated with Brownian motions or Ornstein-Uhlenbeck processes. Moreover, Haasler et al. (2021) may consider various inhomogeneous (discrete) state spaces for the vertices of  $\mathsf{T}$ , as presented in their numerical experiments. In our case, this approach is not compatible with our diffusion-based method. Finally, unlike Haasler et al. (2021), our method is not scalable with the number of vertices or edges in  $\mathsf{T}$  due to computational limits. This limitation is common to all multi-marginal approaches which rely on neural networks to parameterize the potential and/or the distributions of the multi-marginal OT method, see Li et al. (2020); Fan et al. (2020); Korotin et al. (2022, 2021) for instance.
E Further results on TreeSB
Choice of the root  $r$  in (TreeSB). We recall that the reference measure  $\pi^0$  considered in (TreeSB), which is defined in (2), verifies  $\pi^0\in \mathcal{P}_{\mathsf{T}_r}$  for some fixed root  $r\in \mathsf{V}$  and  $\pi_r^0\ll \mathrm{Leb}$  with density  $\varphi_{r}$ . Moreover, we have  $\pi_{v^{\prime}|v}^{0}(\cdot |x_{v}) = \mathrm{N}(x_{v},\varepsilon /(2w_{v,v^{\prime}})\mathrm{I}_{d})$  for any  $(v,v^{\prime})\in \mathsf{E}_r$ , and thus,  $\pi^0$  is entirely determined by the choice of the root  $r$  and the density on the corresponding vertex  $\varphi_r$ .
As presented in Appendix D.1, we recall that (TreeSB) is equivalent to any multi-marginal Tree-SB problem with a reference measure  $\bar{\pi}^0$  given by (15), i.e.,  $\bar{\pi}^0$  writes as  $(\mathrm{d}\bar{\pi}^0 /\mathrm{d}\pi^0) = \exp [\bigoplus_{i\in \mathbb{S}}\psi_i^0 ]$  where  $\{\psi_i^0\}_{i\in \mathbb{S}}$  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  $i\in \mathbb{S}$ . In the case where  $r$  is chosen as a leaf of  $\mathsf{T}$ , this result implies that (TreeSB) is unchanged if
(a)  $\varphi_r = \mathrm{d}\nu/\mathrm{d}\mathrm{Leb}$  where  $\nu \in \mathcal{P}$  is such that  $\mathrm{KL}(\mu_r|\nu) < \infty$ ,
(b)  $r$  is replaced by  $r' \in \mathbb{S}$ , as long as  $\mathrm{H}(\mu_r) < \infty$  and  $\mathrm{H}(\mu_{r'}) < \infty$ .
Therefore, under A0, the setting chosen in Section 3 is equivalent to any other setting where  $r$  is arbitrarily chosen in S and  $\varphi_r = \mathrm{d}\nu/\mathrm{d}\mathrm{Leb}$  where  $\mathrm{KL}(\mu_r|\nu) < \infty$ .
Consider now the case where  $r \in S^c$ , i.e.,  $r$  is not a leaf of  $T$ . Then, the choice of  $\varphi_r$  can not be made arbitrarily anymore, since it determines a further regularization on the  $r$ -th marginal of the solution to (TreeSB). In this setting, the sequence defined by (mIPF) is unchanged. Hence, TreeDSB proceeds in the same manner as presented in Section 3, except for the first iteration, which we detail now.
Let us define  $\mathsf{P} = \mathrm{path}_{\mathsf{T}_{i_0}}(i_0,r)$ , where  $\mathsf{T}_{i_0} = (\mathsf{V},\mathsf{E}_{i_0})$  is the directed version of  $\mathsf{T}$  rooted in  $i_0$ . We recall that first iterate of (mIPF) is defined by
$$
\pi^ {1} = \operatorname {a r g m i n} \left\{\mathrm {K L} \left(\pi \mid \pi^ {0}\right): \pi \in \mathscr {P} ^ {(\ell + 1)}, \pi_ {i _ {0}} = \mu_ {i _ {0}} \right\}.
$$
Following the proof of Lemma 12, it is clear that
$$
\pi^ {1} = \mu_ {i _ {0}} \bigotimes_ {(v, v ^ {\prime}) \in \mathsf {P}} \pi_ {v ^ {\prime} | v} ^ {0} \bigotimes_ {(v, v ^ {\prime}) \in \mathsf {E} _ {i _ {0}} \backslash \mathsf {P}} \pi_ {v ^ {\prime} | v} ^ {0} = \mu_ {i _ {0}} \bigotimes_ {(v, v ^ {\prime}) \in \mathsf {E} _ {i _ {0}}} \pi_ {v ^ {\prime} | v} ^ {0},
$$
where we emphasize that  $\mathsf{P} = \{(v,v^{\prime})\in \mathsf{E}_{i_0}:(v^{\prime},v)\in \mathsf{E}_r\}$ . Therefore, Proposition 2 still applies between  $r$  and  $i_0$ , by considering  $r$  instead of  $i_{K - 1}$ . In practice, this means that the first iteration of TreeDSB consists in computing the time reversal of the path measures  $\mathbb{P}_{(v',v)}^{0}$  for any  $(v,v^{\prime})\in \mathsf{P}$ .
Extension of the regularized Wasserstein barycenter problem (regWB). Consider the regularized Wasserstein-2 barycenter problem defined as follows
$$
\mu_ {\varepsilon} ^ {\star} = \arg \min  \left\{\sum_ {i = 1} ^ {\ell} w _ {i} W _ {2, \varepsilon / w _ {i}} ^ {2} (\mu , \mu_ {i}) + \ell \varepsilon \mathrm {H} (\mu) + \varepsilon \mathrm {K L} (\mu \mid \mu_ {0}): \mu \in \mathcal {P} \right\}, \quad (\mu_ {0} - \operatorname {r e g W B})
$$
where  $(w_{i})_{i\in \{1,\dots,\ell \}}\in (0, + \infty)^{\ell}$  and  $\mu_0\in \mathcal{P}$  is a reference measure. This formulation admits a further regularization compared to (regWB), which tends to make  $\mu_{\varepsilon}^{\star}$  closer to  $\mu_0$ . In particular, given a Wasserstein barycenter problem onto a star-shaped tree, the formulation ( $\mu_0$ -regWB) may be more adapted than (regWB) if we have an a priori on the form of the regularized barycenter. In the case where  $\mu_0 = \mathrm{N}(0,\sigma_0^2\mathrm{I}_d)$ , letting  $\sigma_0\to \infty$ , we recover the  $(\ell \varepsilon, (\ell - 1)\varepsilon)$  doubly-regularized Wasserstein barycenter problem (regWB). In the same spirit as Proposition 7, we can derive the following result from Proposition 24, which proves that ( $\mu_0$ -regWB) can be solved with TreeDSB.
Proposition 25. Let  $\varepsilon >0$  and  $\mu_0\in \mathcal{P}$  such that  $\mu_0\ll$  Leb. Assume A0. Also assume that  $\mathsf{T}$  is a star-shaped tree with central node indexed by 0, and that the reference measure of (TreeSB) defined in (2) verifies  $r = 0$  and  $\varphi_r = \mathrm{d}\mu_0 / \mathrm{dLeb} > 0$ . Under A2,  $(\mu_0\text{-regWB})$  has a unique solution  $\pi_0^\star$  where  $\pi^{\star}$  is the solution to (TreeSB).
Below, we provide practical guidelines to parameterize  $\mu_0$  when it is chosen as a Gaussian distribution.
Gaussian design of  $\mu_0$  in  $(\mu_0$ -regWB). Consider an undirected star-shaped tree  $\mathsf{T}$  with  $K + 1$  vertices and leaves  $\{1,\ldots ,K\}$ . In order to incorporate the marginal constraints in the penalization brought by  $\mu_0$  when it is a Gaussian distribution, we set its mean to  $\sum_{i = 1}^{K}\mathbb{E}[\mu_i] / K$  and its diagonal covariance matrix as  $\alpha \times (\sum_{i = 1}^{K}\mathrm{diag}(\mathrm{Cov}[\mu_i])^{-1} / K)^{-1}$ , where the inverse operation is componentwise and  $\alpha$  is a positive hyperparameter. This choice of variance helps to correctly explore the state-space at the very first iteration of TreeDSB, which is key to ensure numerical stability. In this setting, (TreeSB) verifies A2 and A3, by Proposition 15 and Proposition 16. In particular, we use this approach for two of our experiments: synthetic Gaussian datasets and Bayesian fusion, see Appendix G.
F Algorithmic techniques
Time discretization in TreeDSB. Denote  $k_{n} = (n - 1) \bmod (K)$  for any  $n \in \mathbb{N}$ . Let  $\mathsf{T} = (\mathsf{V},\mathsf{E})$  be a weighted undirected tree and consider the multi-marginal Schrödinger bridge problem (TreeSB) associated to this tree. We recall that for any  $\{v,v^{\prime}\} \in \mathsf{E}$ , we define  $T_{v,v^{\prime}} = \varepsilon /(2w_{v,v^{\prime}})$ .
Consider the path measures  $\{\mathbb{P}_{(v,v^{\prime})}^{n}\}_{n\in \mathbb{N},(v,v^{\prime})\in E_{k_n}}$  recursively defined by (a) and (b). By combining Proposition 1, Proposition 2 and results on time reversal theory (Haussmann & Pardoux, 1986), we obtain by recursion that for any  $n\in \mathbb{N}$ , any  $(v,v^{\prime})\in E_{k_n}$ ,  $\mathbb{P}_{(v,v^{\prime})}^{n}$  is associated with a Stochastic Differential Equation on  $[0,T_{v,v^{\prime}}]$  given by
$$
\mathrm {d} \mathbf {X} _ {t} = f _ {t, v, v ^ {\prime}} ^ {n} \left(\mathbf {X} _ {t}\right) \mathrm {d} t + \mathrm {d} \mathbf {B} _ {t}, \quad \mathbf {X} _ {0} \sim \pi_ {v} ^ {n}. \tag {29}
$$
Let  $N \in \mathbb{N}^*$ . In order to sample from the dynamics (29) at iteration  $n \in \mathbb{N}$ , we consider its Euler-Maruyama discretization on  $(N + 1)$  time steps,
$$
X _ {m + 1} = X _ {m} + \gamma_ {m + 1} f _ {t _ {m}, v, v ^ {\prime}} ^ {n} \left(X _ {m}\right) + \sqrt {\gamma_ {m + 1}} Z _ {m + 1}, \quad X _ {0} \sim \pi_ {v} ^ {n}, \tag {30}
$$
where  $Z_{m}\sim \mathrm{N}(0,\mathrm{I}_{d})$  for any  $m\in \{1,\ldots ,N\}$ ,  $t_m = \sum_{i = 1}^m\gamma_i$ , and  $\{\gamma_m\}_{m = 1}^N\in (0,\infty)^N$  is a time schedule such that  $\sum_{m = 1}^{N}\gamma_{m} = T_{v,v^{\prime}}$ . This results in approximating the path measure  $\mathbb{P}_{(v,v^{\prime})}^{n}$  by the joint distribution  $\pi_{(v,v^{\prime})}^{n,N}\in \mathcal{P}^{(N + 1)}$  defined by
$$
\pi_ {(v, v ^ {\prime})} ^ {n, N} = \pi_ {v} ^ {n} \bigotimes_ {m = 0} ^ {N - 1} \pi_ {(v, v ^ {\prime}), m + 1 | m} ^ {n, N},
$$
where  $\pi_{(v,v'),m+1|m}^{n,N}(\cdot|x_m) = \mathrm{N}(x_m + \gamma_{m+1}f_{t_m,v,v'}^n(x_m), \gamma_{m+1}\mathrm{I}_d)$  for any  $m \in \{0, \dots, N-1\}$ . If  $N$  is chosen large enough, then  $\pi_{(v,v'),m}^{n,N}$  and  $\mathbb{P}_{(v,v'),t_m}^n$  have approximately the same distribution
for any  $m \in \{0, \ldots, N\}$ . Consequently,  $(\mathbb{P}_{(v,v')}^n)^R$  is naturally approximated by the joint distribution  $\tilde{\pi}_{(v,v')}^{n,N} \in \mathcal{P}^{(N+1)}$  defined by
$$
\tilde {\pi} _ {(v, v ^ {\prime})} ^ {n, N} = \pi_ {v ^ {\prime}} ^ {n} \bigotimes_ {m = 0} ^ {N - 1} \pi_ {(v, v ^ {\prime}), N - m - 1 | N - m} ^ {n, N}.
$$
If  $N$  is chosen large enough, we obtain that
$$
\begin{array}{l} \pi_ {(v, v ^ {\prime}), N - m - 1 | N - m} ^ {n, N} (\cdot | x _ {N - m}) \\ = \mathrm {N} \left(x _ {N - m} - \gamma_ {N - m} f _ {t _ {N - m}, v, v ^ {\prime}} ^ {n} \left(x _ {N - m}\right) + \gamma_ {N - m} \nabla \log p _ {v, v ^ {\prime}, t _ {N - m}} \left(x _ {N - m}\right), \gamma_ {N - m} \mathrm {I} _ {d}\right), \\ \end{array}
$$