SlowGuess/NeurIPS_2023_OCR huggingface dataset:
(a)
Figure 10: (a) Left to right: source low resolution sample, intermediate state and final reconstruction of DSBM-IPF; (b) an unpaired high resolution sample.
Figure 9: DSBM domain transfer results on the AFHQ  $512 \times 512$  dataset.
Data wavenumber
Figure 11:  $\ell_2$  distance between low resolution source and high resolution reconstructed fields.
distances for all frequency classes in the dataset than Diffusion-fb (and thus Random), indicating DSBM is able to reconstruct high resolution fields consistent with the low resolution source.
In this work, we introduce IMF, a new methodology for learning Schrödinger Bridges. IMF is an alternative to the classical IPF and can be interpreted as its dual. Building on this new framework, we present two practical algorithms, DSBM-IPF and DSBM-IMF, for learning SBs. These algorithms mitigate the time-discretization and bias accumulation issues of existing methods. However, DSBM still has some limitations. First, our results suggest DSBM is most effective for solving general transport problems. For generative modeling, we only find minor improvements compared to Bridge and Flow Matching on CIFAR-10 (see Appendix I.6). Second, while DSBM is more efficient than DSB, it still requires sampling from the learned process during the caching step. Finally, the EOT problem becomes more difficult to solve numerically for small values of  $\sigma$ .
In future work, we would like to further investigate the differences between DSBM-IMF and DSBM-IPF. IMF also appears useful for developing a better understanding of the Rectified Flow algorithm (Liu et al., 2023b), as IMF minimizes a clear objective (6) and Rectified Flow can be seen as a limiting case of it. Finally, Rectified Flow has also been extended to solve OT problems with general convex costs by Liu (2022), and it would be interesting to derive a SB version of this extension.
YS acknowledges support from the Huawei UK Fellowship. AC acknowledges support from the EPSRC CDT in Modern Statistics and Statistical Machine Learning (EP/S023151/1). AD acknowledges support of the UK Dstl and EPSRC grant EP/R013616/1. This is part of the collaboration between US DOD, UK MOD and UK EPSRC under the Multidisciplinary University Research Initiative. He also acknowledges support from the EPSRC grants CoSines (EP/R034710/1) and Bayes4Health (EP/R018561/1).
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Outline of the Appendix
In Appendix A, we first clarify the relationship between different methods in the existing literature and our proposed DSBM framework. In Appendix B, we focus on the family of linear SDEs, and draw a link between the parameterization of bridges in this paper and the stochastic interpolant in Albergo et al. (2023). In Appendix C, we give proofs for results in the main text. In Appendix D, we present additional theoretical results for IMF in the Gaussian case. In Appendix E, we derive the discrete-time version of Markovian projection. In Appendix F, we explain the benefits of DSBM compared to DSB in more detail. In Appendix G, we describe a method for learning the forward and backward processes jointly and propose a consistency loss between the forward and backward processes. In Appendix H, we present additional methodological details for a practical scaling of the loss function to reduce variance, similar to standard Denoising Diffusion Models. In Appendix I, we give further details for all experiments and additional experimental results. Finally, we discuss broader impacts of our work in Appendix J.
A Discussion of Existing Works
A.1 Bridge Matching and Flow Matching Models
In this section, we clarify the relationship between variants of Flow Matching and show that they are equivalent under some conditions. We follow the nomenclature of Tong et al. (2023). We refer to the algorithm originally proposed in Lipman et al. (2023) using linear probability paths and described in (Tong et al., 2023, Section 4.1) as Flow Matching (FM), and the algorithm proposed in (Tong et al., 2023, Section 4.2) as Conditional Flow Matching (CFM). There is a small constant parameter  $\sigma_{\mathrm{min}}$  in both algorithms, which controls the smoothing of the modeled distribution. We consider the case  $\sigma_{\mathrm{min}} = 0$ . Then CFM recovers exactly the 1st iteration of Rectified Flow (Liu et al., 2023b). Furthermore, FM, CFM and the 1st iteration of Rectified Flow are all equivalent when performing generative modeling with a standard Gaussian  $\pi_0$ . We refer to them collectively as Flow Matching models (FMMs) as they only differ in the smoothing method. We also present them all under the Bridge Matching framework. These models can also be interpreted in the context of stochastic interpolants (Albergo and Vanden-Eijnden, 2023; Albergo et al., 2023). Finally, we present recent applications of Bridge Matching and show that some of the objectives in Somnath et al. (2023); Liu et al. (2023a); Delbracio and Milanfar (2023) are identical.
Flow Matching and Conditional Flow Matching. In Flow Matching (FM), the objective (Lipman et al., 2023, Equation (21)) is
$$
\mathbb {E} _ {\Pi_ {t, T}} [ \| (\mathbf {X} _ {T} - \mathbf {X} _ {t}) / (T - t) - v _ {\theta} (t, \mathbf {X} _ {t}) \| ^ {2} ],
$$
where  $\Pi_{t,T}$  is given by  $\pi_T(\mathbf{X}_T)\mathrm{N}(\mathbf{X}_t;\frac{t}{T}\mathbf{X}_T,(1 - \frac{t}{T})^2)$ .
In Conditional Flow Matching (CFM),  $\mathbf{X}_t^{0,T} = \frac{t}{T}\mathbf{X}_T + (1 - \frac{t}{T})\mathbf{X}_0$ , with  $\mathbf{X}_0 \sim \mathrm{N}(0, \mathrm{Id})$  and the objective (Tong et al., 2023, Equation (16)) is given by
$$
\mathbb {E} _ {\Pi_ {0, T}} \left[ \left\| \left(\mathbf {X} _ {T} - \mathbf {X} _ {0}\right) / T - v _ {\theta} \left(t, \mathbf {X} _ {t} ^ {0, T}\right) \right\| ^ {2} \right]. \tag {15}
$$
This is the same as (Liu et al., 2023b, Equation (1)). Furthermore,  $(\mathbf{X}_T - \mathbf{X}_0) / T = (1 - \frac{t}{T})(\mathbf{X}_T - \mathbf{X}_0) / (T - t) = (\mathbf{X}_T - \mathbf{X}_t^{0,T}) / (T - t)$ , so the CFM objective is equivalent to
$$
\mathbb {E} _ {\Pi_ {t, T}} \left[ \left\| \left(\mathbf {X} _ {T} - \mathbf {X} _ {t} ^ {0, T}\right) / (T - t) - v _ {\theta} \left(t, \mathbf {X} _ {t} ^ {0, T}\right) \right\| ^ {2} \right]. \tag {16}
$$
The optimal  $v_{\theta}(t,x_t) = (\mathbb{E}_{\Pi_{T|t}}[\mathbf{X}_T\mid \mathbf{X}_t = x_t] - x_t) / (T - t)$ . In the case of generative modeling,  $\pi_0$  is a standard Gaussian distribution and  $\Pi_{0,T}$  is given by  $\mathrm{N}(\mathbf{X}_0;0,\mathrm{Id})\pi_T(\mathbf{X}_T)$ . Thus,  $\Pi_{t,T}$  is also given by  $\pi_T(\mathbf{X}_T)\mathrm{N}(\mathbf{X}_t^{0,T};\frac{t}{T}\mathbf{X}_T,(1 - \frac{t}{T})^2)$ . Therefore, the FM (Lipman et al., 2023) and CFM (Tong et al., 2023) objectives are exactly the same. However, CFM is also applicable when  $\pi_0$  is not Gaussian distributed, so CFM is a generalized version of  $\mathrm{FM^6}$ .
In the case  $\sigma_{\mathrm{min}} > 0$ , FM and CFM are indeed different in how smoothing is performed, and we refer to Tong et al. (2023) for a more detailed analysis.
Stochastic Interpolant. In (Albergo and Vanden-Eijnden, 2023; Albergo et al., 2023), the concept of stochastic interpolant is introduced. In Albergo and Vanden-Eijnden (2023), the interpolation is deterministic (not necessarily linear), of the form  $I_{t}(x_{0}, x_{T}) = \alpha(t)x_{0} + \beta(t)x_{T}$ , while in Albergo et al. (2023), the interpolation is stochastic given by  $I_{t}(x_{0}, x_{T}) = \alpha(t)x_{0} + \beta(t)x_{T} + \gamma(t)\mathbf{Z}$  for  $\mathbf{Z} \sim \mathrm{N}(0, \mathrm{Id})$ . In Albergo and Vanden-Eijnden (2023), an ODE is learned and the associated velocity field  $v_{\theta}$  is obtained by minimizing the following objective (Albergo and Vanden-Eijnden, 2023, Equation (9))
$$
\mathbb {E} _ {\Pi_ {0, T}} \left[ \left\| \partial_ {t} I _ {t} \left(\mathbf {X} _ {0}, \mathbf {X} _ {T}\right) - v _ {\theta} \left(t, \mathbf {X} _ {t} ^ {0, T}\right) \right\| ^ {2} \right].
$$
Hence, if  $I_{t}(x_{0},x_{T}) = \frac{t}{T} x_{0} + (1 - \frac{t}{T})x_{T}$ , we recover (15).
Link with Bridge Matching. When  $\mathbb{Q}$  is associated with the Brownian motion  $(\sigma \mathbf{B}_t)_{t\in [0,T]}$  and  $\sigma \rightarrow 0$  in Bridge Matching, we recover the same objective (5) as the Flow Matching objective (16), since  $\nabla \log \mathbb{Q}_{T|t}(\mathbf{X}_T|\mathbf{X}_t) = (\mathbf{X}_T - \mathbf{X}_t) / (\sigma^2 (T - t))$ . Bridge Matching can also be applied to general distributions  $\pi_0,\pi_T$ ; i.e.  $\pi_0$  does not have to be restricted to a Gaussian. Therefore, Bridge Matching is a generalized version of Flow Matching, see also (Liu et al., 2022b, Equation (10)).
Inverse problems and interpolation. Somnath et al. (2023) and Liu et al. (2023a) present Bridge Matching algorithms between aligned data  $(\mathbf{X}_0, \mathbf{X}_T) \sim \Pi_{0,T}$ . The objectives (Somnath et al., 2023, Equation (8)) and (Liu et al., 2023a, Equation (12)) are equivalent to the Bridge Matching objective (3). The main difference between Liu et al. (2023a) and Somnath et al. (2023) resides in the choice of  $\Pi_{0,T}$ . In the case of Somnath et al. (2023), this choice is motivated by the access to aligned data with applications in biology assuming they are distributed as the true Schrödinger static coupling, i.e.  $\Pi_{0,T} = \Pi_{0,T}^{\mathrm{SB}}$ . In the case of Liu et al. (2023a),  $\Pi_{0,T}$  corresponds to a pairing between clean and corrupted images, e.g. with  $\Pi_0 = \pi_0$  the distribution of clean images and  $\Pi_T = \pi_T$  the distribution of corrupted images obtained from the clean images using the degradation kernel  $\Pi_{T|0}$ .
Finally, in (Delbracio and Milanfar, 2023, Equation (5)) the authors consider a reconstruction process of the form
$$
\mathrm {d} \mathbf {X} _ {t} = \left(\mathbb {E} _ {\Pi_ {0} | t} \left[ \mathbf {X} _ {0} \mid \mathbf {X} _ {t} \right] - \mathbf {X} _ {t}\right) / t \mathrm {d} t, \quad \mathbf {X} _ {T} \sim \Pi_ {T}, \tag {17}
$$
where here we have replaced  $F(x_{t}, t)$  by  $\mathbb{E}_{\Pi_{0|t}}[\mathbf{X}_0 \mid \mathbf{X}_t = x_t]$ . This is justified if the  $\| \cdot \|_p$  norm in (Delbracio and Milanfar, 2023, Equation (4)) is replaced by  $\| \cdot \|_2^2$  (or any Bregman Loss Function, see Banerjee et al. (2005)). In Delbracio and Milanfar (2023),  $\Pi_{0,T}$  corresponds to the joint distribution of clean and corrupted images as in Liu et al. (2023a). Exchanging the role of  $\Pi_0$  and  $\Pi_T$ , (17) can be rewritten equivalently as
$$
\mathrm {d} \mathbf {X} _ {t} = (\mathbb {E} _ {\Pi_ {T | t}} [ \mathbf {X} _ {T} \mid \mathbf {X} _ {t} ] - \mathbf {X} _ {t}) / (T - t) \mathrm {d} t, \qquad \mathbf {X} _ {0} \sim \Pi_ {0}.
$$
We thus obtain the optimal Flow Matching vector field  $v_{\theta}(t,x_{t}) = (\mathbb{E}_{\Pi_{T|t}}[\mathbf{X}_{T} \mid \mathbf{X}_{t} = x_{t}] - x_{t}) / (T - t)$  in (16). Note that Delbracio and Milanfar (2023) also incorporates a stochastic version of their objective (Delbracio and Milanfar, 2023, Equation (7)). It remains an open question whether this objective can be understood as a special instance of the Bridge Matching framework.
A.2 On DSBM and Existing Works
In this section, we show that the DSBM framework recovers the above existing algorithms for different choices of bridges  $\mathbb{Q}_{|0,T}$  and couplings  $\Pi_{0,T}^{0}$  in Algorithm 1. For the independent coupling  $\Pi_{0,T}^{0} = \pi_{0} \otimes \pi_{T}$  and Brownian bridge  $\mathbb{Q}_{|0,T}$  (4) with diffusion parameter  $\sigma_{t} = \sigma$ , the loss function (10) recovers the Brownian Bridge Matching loss (5). Letting  $\sigma \to 0$ , we recover Flow Matching (Lipman et al., 2023). In this case, further iterations repeating lines 7-9 in Algorithm 1 (with only forward projections) recover Rectified Flow (Liu et al., 2023b). If the coupling  $\Pi_{0,T}^{0}$  is given by an estimation of the OT map between  $\pi_{0}$  and  $\pi_{T}$ , then the first iteration recovers OT-CFM (Tong et al., 2023; Pooladian et al., 2023). Finally, for general bridges  $\mathbb{Q}_{|0,T}$ , if we are given the optimal Schrödinger Bridge static coupling  $\Pi_{0,T}^{0} = \Pi_{0,T}^{\mathrm{SB}}$ , then the DSBM procedure converges in one iteration and we recover Somnath et al. (2023).
A.3 DSBM and Rectified Flow
We discuss the differences in more detail between our proposed DSBM method and Rectified Flow. Both Rectified Flow and DSBM are general frameworks for building transport maps between
two general distributions  $\pi_0, \pi_T$ . However, there are a few important theoretical and practical differences. Firstly, we adopt the SDE approach as opposed to the ODE approach in Rectified Flow. This distinction is crucial in theory, as Proposition 5, which guarantees the uniqueness of the characterization of SB, is valid only when  $\sigma_t > 0$ . Consequently, Rectified Flow is not guaranteed to converge to the dynamic optimal transport solution (see e.g. counterexample in Liu (2022)). In a following work, Liu (2022) established formal connections between Rectified Flow and OT when restricting the class of vector fields to gradient fields. In DSBM, the connection to OT is obtained by considering its entropy-regularized version. Furthermore, by adopting the SDE approach, we observe significant improvements of sample quality in our experiments when performing transport between two general distributions. This is in line with the theoretical analysis in Albergo et al. (2023). On the other hand, while Bridge Matching also achieves high sample quality using the SDE approach, the transported samples are much more dissimilar to the input data (see e.g. Figures 14, 15, 19). Lastly, Rectified Flow also performs Markovian projections iteratively, but only in the forward direction. Consequently, the bias in the learned marginals  $\mathbb{P}_T^n$  is accumulated and cannot be corrected in later iterations, i.e. the first iteration of RF will achieve the most accurate marginal  $\mathbb{P}_T^1$ . Subsequent iterations can improve the straightness of the flow, but at the cost of sampling accuracy of  $\mathbb{P}_T^n$ . We observe in practice that this becomes particularly problematic if the first iteration of Rectified Flow (which is equivalent to CFM) fails to provide a good transport and learn an accurate  $\mathbb{P}_T^1$ , e.g. in the case of moons-8gaussians (Table 2), Gaussian transport (Figure 3), and MNIST, EMNIST transfer (Figure 5 and Figure 13). As Rectified Flow cannot recover from this issue, we observe the accuracy of  $\mathbb{P}_T^n$  only deteriorates in further iterations as  $n$  increases. In our methodology, we leverage Proposition 9 to perform forward and backward Bridge Matching, and we observe that the marginal accuracy is able to improve with iteration.
B The Design Space of Brownian Bridges
B.1 Relationship to Stochastic Interpolants
From stochastic interpolants to Brownian bridges. In this section, we draw a link between our parameterization of bridges and the one used in Albergo et al. (2023). In Albergo et al. (2023), a stochastic interpolant is defined as
$$
\mathbf {X} _ {t} = \bar {\alpha} _ {t} x _ {0} + \bar {\beta} _ {t} x _ {T} + \bar {\gamma} _ {t} \mathbf {Z}, \tag {18}
$$
where  $\mathbf{Z} \sim \mathrm{N}(0, \mathrm{Id})$ . Since their methodology and analysis mainly relies on the probability flow, they work with (18), which is easier to analyse. In our setting, as we deal mostly with diffusions, it is natural to parameterize Brownian bridges as follows
$$
\mathrm {d} \mathbf {X} _ {t} = \left\{- \alpha_ {t} \mathbf {X} _ {t} + \beta_ {t} x _ {T} \right\} \mathrm {d} t + \gamma_ {t} \mathrm {d} \mathbf {B} _ {t}. \tag {19}
$$
The goal of this section is to derive explicit formulas between the parameters  $\bar{\alpha}_t, \bar{\beta}_t$  and  $\bar{\gamma}_t$  of (18) and the parameters  $\alpha_t, \beta_t$  and  $\gamma_t$  of (19). Consider  $(\mathbf{X}_t)_{t \in [0,T]}$  given by (19). We have that for any  $t \in [0,T]$
$$
\mathbf {X} _ {t} = \exp [ - A _ {t} ] x _ {0} + \int_ {0} ^ {t} \beta_ {s} \exp [ A _ {s} - A _ {t} ] \mathrm {d} s x _ {T} + \int_ {0} ^ {t} \gamma_ {s} \exp [ A _ {s} - A _ {t} ] \mathrm {d} \mathbf {B} _ {s},
$$
where  $A_{t} = \int_{0}^{t}\alpha_{s}\mathrm{d}s$ . Therefore, we have that
$$
\bar {\alpha} _ {t} = \exp \left[ - \int_ {0} ^ {t} \alpha_ {s} \mathrm {d} s \right], \quad \bar {\beta} _ {t} = \int_ {0} ^ {t} \beta_ {s} \exp \left[ - \int_ {s} ^ {t} \alpha_ {u} \mathrm {d} u \right] \mathrm {d} s, \quad \bar {\gamma} _ {t} ^ {2} = \int_ {0} ^ {t} \gamma_ {s} ^ {2} \exp \left[ - 2 \int_ {s} ^ {t} \alpha_ {u} \mathrm {d} u \right] \mathrm {d} s, \tag {20}
$$
$$
\alpha_ {t} = - \frac {\bar {\alpha} _ {t} ^ {\prime}}{\bar {\alpha} _ {t}}, \quad \beta_ {t} = \bar {\beta} _ {t} ^ {\prime} + \bar {\beta} _ {t} \alpha_ {t}, \quad \gamma_ {t} ^ {2} = \left(\bar {\gamma} _ {t} ^ {2}\right) ^ {\prime} + 2 \bar {\gamma} _ {t} ^ {2} \alpha_ {t} = 2 \bar {\gamma} _ {t} \bar {\gamma} _ {t} ^ {\prime} + 2 \bar {\gamma} _ {t} ^ {2} \alpha_ {t}. \tag {21}
$$
Using this relationship, we get that the Markovian projection, see Definition 1, is given by
$$
\mathrm {d} \mathbf {X} _ {t} ^ {\star} = f _ {t} ^ {\star} (\mathbf {X} _ {t}) \mathrm {d} t + \gamma_ {t} \mathrm {d} \mathbf {B} _ {t}, \quad \quad f _ {t} ^ {\star} (x _ {t}) = \mathbb {E} _ {\Pi_ {T | t}} [ - \alpha_ {t} \mathbf {X} _ {t} + \beta_ {t} \mathbf {X} _ {T} \mid \mathbf {X} _ {t} = x _ {t} ].
$$
$$
\begin{array}{l} f _ {t} ^ {\star} \left(x _ {t}\right) = \mathbb {E} _ {\Pi_ {T \mid t}} \left[ - \alpha_ {t} \mathbf {X} _ {t} + \beta_ {t} \mathbf {X} _ {T} \mid \mathbf {X} _ {t} = x _ {t} \right] \\ = \mathbb {E} _ {\Pi_ {0, T | t}} [ - \alpha_ {t} (\bar {\alpha} _ {t} \mathbf {X} _ {0} + \bar {\beta} _ {t} \mathbf {X} _ {T} + \bar {\gamma} _ {t} \mathbf {Z}) + \beta_ {t} \mathbf {X} _ {T} | \mathbf {X} _ {t} = x _ {t} ]. \\ \end{array}
$$
Using (21), we get that