Transport with Support: Data-Conditional Diffusion Bridges
Authors: Ella Tamir
Martin Trapp
Arno Solin
Venue: Submitted to ICLR 2023 (2023)
Keywords: diffusion models
optimal transport
particle filtering
stochastic control
sequential Monte Carlo

Abstract: The dynamic Schrödinger bridge problem provides an appealing setting for posing optimal transport problems as learning non-linear diffusion processes and enables efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg, in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are typically limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We combine learning diffusion models with Bayesian filtering and optimal control, allowing for constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data and show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times. 

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TRANSPORT WITH SUPPORT: DATA-CONDITIONA DIFFUSION BRIDGES
Anonymous authors Paper under double-blind review
ABSTRACT
The dynamic Schrödinger bridge problem provides an appealing setting for posing optimal transport problems as learning non-linear diffusion processes and enables efficient iterative solvers. Recent works have demonstrated state-of-the-an results (e.g., in modelling single-cell embryo RNA sequences sampling from com- plex posteriors) but are typically limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We combine learning diffusion models with Bayesian fil- tering and optimal control, allowing for constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data and show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates the marginals at intermediate and terminal times.
INTRODUCTION
Generative diffusion models have gained increasing popularity and achieved impressive results in a variety of challenging application domains, such as computer vision (e.g., Ho et al., 2020; Song al., 2021a; Dhariwal & Nichol, 2021), reinforcement learning (e.g. Janner al., 2022), and time series modelling (e.g., Rasul et al., 2021; Vargas et al., 2021; Tashiro et al., 2021; Park et al., 2022). Recent works have explored connections between denoising diffusion models and the dynamic Schrödinger bridge problem (SBP, e.g., Vargas et al., 2021: De Bortoli et al., 2021; Shi et al., 2022) to adopt iterative schemes for solving the dynamic optimal transport problem more efficiently. The solution of the SBP that correspond to denoising diffusion models is then given by the finite-time process, which is the closest in Kullback-Leibler (KL) divergence to the forward noising process of the diffusion model under marginal constraints. Data is then generated by time-reversing the process.
In many applications, the interest not purely in modelling transport between an initial and terminal state distribution In naturally occurring generative processes, we typically observe snapshots of realizations along intermediate stages of individual sample trajectories (see Fig. 1). Such problems arise in medical diagnosis (e.g., tissue changes and cell growth), demographie modelling, environ- mental dynamics, and animal movement modelling-See wintering patterns. Recently, constrained optimal control problems have been explored by adding additional fixed path constraints (Maoutsa et al., 2020; Maoutsa & Opper, 2021) or modifying the prior processes (Fernandes al., 2021). However, defining meaningful fixed path constraints or prior processes for the optimal control problems can be challenging, while sparse observational data are accessible in many real-world applications.
Fig. 4 for modelling bird migration and
In this work, we propose the Iterative Smoothing Bridge (ISB), an iterative method for solving control problems under sparse observational data constraints and constraints on the initial and terminal distribution. We perform the conditioning by leveraging the iterative pass idea from the Iterative Proportional Fitting procedure (IPFP) (Kullback, 1968; De Bortoli et al., 2021) procedure and applying differentiable particle filtering (Reich, 2013; Corenflos et al., 2021) within the outer loop. Integrating sequential Monte Carlo methods (e.g., Doucet et al., 2001: Chopin & Papaspiliopoulos. 2020) into the IPFP framework in such way is non-trivial and can be understood as a novel iterative version of the algorithm by Maoutsa & Opper (2021) but with more general terminal constraints and path constraints defined by data.

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Time. 1
Sparse observations
~ N((10,)) 0)",)
N((10,0)
N)
N(O,
(a) Unconstrained transport (Schrödinger bridge)
(b) Constrained transport (Iterative smoothing bridge)
Figure I: Illustrative example transport between initial unit Gaussian and shifted unit Gaussian at the terminal time T. Unconstrained transport on the left and the solution constrained by sparse observations (m) on the right. Colour coding of the initial points is only for distinguishing the paths.
We summarize the contributions as follows. (i) We propose a novel method for solving constrained op- timal transport as bridge problem under sparse observational data constraints. (ii) Thereof, we utilize the strong connections between the constrained bridging problem and particle filtering in sequential Monte Carlo, extending them from pure inference to learning. Additionally, (iii demonstrate prac- tical efficiency and show that the iterative smoothing bridge approach scales to high-dimensional data.
1.1 RELATED WORK
Schrödinger bridges The problem of learning stochastic process moving samples from one distribution to another can be posed as a type of a transport problem known as a dynamic Schrödinger bridge problem (SBP, e.g., Schrödinger, 1932; Léonard, 2014), where the resulting marginal densities are desired to resemble a given reference measure. In machine learning literature, the problem has been studied through learning the drift function of the dynamical system (De Bortoli al., 2021; Wang et al., 2021; Vargas et al., 2021; Bunne et al., 2022). When an SDE system also defines the reference measure, the bridge problem becomes synonymous with constrained optimal control problem (e.g., Caluya & Halder, 2022; 2021; Chen et al., 2021), which has been leveraged in learning Schrödinger bridges by Tianrong Chen (2022) through norward-backwe
SDEs. An optimal control problem with both constraints on the initial and terminal distribution and fixed path constraint has been studied in Maoutsa et al. (2020) and Maoutsa & Opper (2021), where particle filtering is applied to continuous path constraints but the boundary constraints are defined by single point. Furthermore, the combination of Schrödinger bridges and state-space models has been studied by Reich (2019), 9), in a setting where Schrödinger bridges are applied to the transport problem between filtering distributions.
Diffusion models in machine learning The recent advances in diffusion models in machine learning literature have been focused in generating samples from complex distributions defined by data through transforming samples from an easy-to-sample distribution by a dynamical system (e.g., Ho et al., 2020; Song al., 2021b;a bb;; Nichol & Dhariwal, 2021). The concept of reversing SDE trajectories via score-based learning (Hyvärinen & Dayan, 2005; Vincent, 2011) has allowed for models scalable enough to be applied based diffusion models have been applied to problems where the dynamical system itself is of interest, for example, for the problem time series amputation Tashiro et (2021) and inverse problems in imaging in Song et al. (2022). Other dynamical models parametrized by neural networks have been applied to modelling latent time-series based on observed snapshots of dynamics (Rubanova et al., 2019; Li et al., 2020), but without further constraints the initial terminal distributions.
high-dimensional data sets directly in the data space. In earlier work, score-
State-space models their general form, state-space models combine a latent space dynamical system with an observation (likelihood) model. Evaluating the latent state distribution based on observational data can be performed by applying particle filtering and smoothing (Doucet et al., 2000) by approximations of the underlying state distribution of non-linear state-space model by a specific model family, for instance, a Gaussian (see Särkkä, 2013, for an overview). Speeding up parameter inference and leamnnng in state-space models has been widely studied (e.g., Schön
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al., 2011; Svensson & Schön, 2017; Kokkala et al., 2014). Particle smoothing be connected to Schrödinger bridges via the two-filter smoother (e.g., Bresler, 1986; Briers et al., 2009; Hostettler, 2015), where the smoothing distribution is estimated by performing filtering both forward from the initial constraint and backward from the terminal constraint. We refer to Mitter (1996) and Todorov (2008) for more detailed discussion the connection of stochastic control and filtering and to Chopin & Papaspiliopoulos (2020) for an introduction to particle filters.
2 BACKGROUND
- R denote the space of continuous functions from [0, 7') to R' and BCC)) denote the Borel o-algebra on C. Let (TO, TT) denote the space of probability measures on (c, BC)) such that the marginals divergence from measure Q to measure IP is written as DEL [QQ || P), where we assume that Q < P. For modelling the time dynamies, we assume (continuous- a non-linear latent Itô SDE (see, e.g., Øksendal, 2003; Särkkä & Solin, 2019) in T] x IRd with drift function fo and diffusion function g(-). and Gaussian observation model, ie.,
coincide with probability densities To and æy, respectively. The KL
state-space model consisting of
X0 ~ no.
dx; = fo( t) dt + g(t dB,,, Yk ~ N(y xx, Ia) t=te
(1)
where the drift fo Rd 10, T Rd is mapping modelled by neural network (NN) parameterized by 0 E + diffusion g denotes the latent stochastic process and y, denotes the observation -space process. practice, we con- sider the continuous-disered time setting, where the process is observed at discrete time instances & such that observational data can be given in terms of a collection of input-output pairs ((t,, yke))
0 + I. and B, denotes standard d-dimensional Brownian motion. X;
M
2.1 SCHRÖDINGER BRIDGES AND OPTIMAL CONTROL
The Schrödinger bridge problem (SBP, Schrödinger, 1932; Léonard, 2014) can be described as an entropy-regularize optimal transport problem where the optimality is measured through the KI. divergence from a reference measure to the posterior measure Q, with fixed initial and final densities To and πT, i.e.,
min (To.
DKL
QQ P).
In this work, we consider only the case where the measures P and Q are constructed as the marginals of an SDE, ie., Q is the probability measure of the marginal of the SDE in Eq. (1) at time f, whereas P. corresponds to the probability measure of the marginal of reference SDE dx, fx,, t) dt + g() d,G,, at time t, where we call f the reference drift. Under the optimal control formulation of the SBP (Caluya & Halder, 2021) the KL divergence in Eq. (2) reduces
V." 2g(t)² 1l0(xx) - f(x,, de
(3)
where the expectation is over paths from Eq. (1). Rüschendorf & Thomsen (1993) and Ruschendorf (1995) showed that a solution to the SBP can be obtained by iteratively solving two half-bridge problems using the Iterative Proportional Fitting procedure (IPFP) for 0, L,...,, steps:
Q21+1 = arg min DEL (.9())))
[Q Qz] and Q2+
arg min DRL Q (
[ Q2411)
(4)
where Qo is set as the reference measure, and (mT. and ( RT) denote the sets of probability measures with only either the marginal at time 0 or time T coinciding with no or Tr, respectively. Recently the IPFP to solving Schrödinger bridges has been adapted as a machine learning problem (Bernton et al., 2019; Vargas et al., 2021; De Bortoli et al., 2021). In practice, the interval [0, 7] discretized and the forward drift fo and the backward drift by of the corresponding reverse-timg time process (Haussmann & Pardoux, 1986; Föllmer, 1988) are modelled by NNs. Under the Gaussian transition approximations, the resulting discrete-time time diffusion model can be reversed by applying a mean-matching based objective.
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Figure 2: Sketch of diffusion bridge between 2D data distribution (m²) and isotropic Gaussian (TT) constrained by sparse observations (x). The forward diffusion at the first iteration (ISB I) learns to account for the sparse observations but does not converge the correct terminal distribution Under review as a conference paper at ICLR 2023
00
Forward
S
S
1SB
J
S S
8
Backward 3
Forward
S
S
ISB 6
J
Sparse observations
S
S
Backward
3
time
(t= 7), and the backward diffusion vice versa. After iterating (ISB 6), the forward and backward diffusions converge to the correct targets and are able to account for the sparse observational data.
METHODS
Given initial and terminal distribution To and Tr. are interested in learning data-conditional bridge between and my- Let DD = [(,, y; )) be a set of M sparsely observed values, iee. only few no observations are made at each point in time, and let the state-space model of interest be given by Eq. (1). Note that we deliberately use v,) (instead of (tx, yk)) to highlight that we allow for multiple observations at the same time point fg. Our aim is to find parameterization
the drift function fo such that evolving N particles x with xi ~ (with = 1, 2, ., .N) according to Eq. (1) will result in samples xi. from the terminal distribution лт. Inspired by the IPFP by De Bortoli et al. (2021), which decomposes the SBP into finding two half- bridges, we propose to iteratively solve the two half-bridge problems while accounting for the additional sparse observations simultaneously. For this, let
dx; = fx,e(xe, t) dt + g(t) dBe dz, (2, 1) dt g(t) d
Xg
(5)
2g TT:
denote the forward and backward SDE iteration / = 1, 1,,, where 3r is the reverse-time Brownian motion. For simplicity, we denote Br Br when the direction of the SDE is clear.
To learn the conditiood bridge, we iteratively employ the following steps: I evolve and filter forward particle trajectories according to Eq. (5) with drift fi-1,0 and observations ((I,, Mr
learn the drift function bis for the reverse-time SDE, 3 evolve and filter backward particle trajectories according to Eq. (6) with the drift bt learned in step 2 and observations {(tx, vx) kmm M and 4 learn the drift function fLo for the forward SDE based the backward particles. Fig. illustrates the forward and backward process of our iterative scheme for data-conditioned denoising diffusion bridge. Next, we will go through steps 4 in detail and introduce the Iterative Smoothing Bridge method for solving data-conditional diffusion bridges.
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3.1 THE ITERATIVE SMOOTHING BRIDGE
The Iterative Smoothing Bridge (ISB) method iteratively generates particle filtering trajectories (steps 1 and 3 in Fig. 2) and learns the parameterizations of the forward and backward drift functions fi and bi (steps and 4) ) by minimizing a modified version of the mean meaanmmatching objective presented by De Bortoli et al. (2021). Note that steps 2 and 4 are dependent applying differential resampling in the particle filtering steps and 3 for reversing the generated trajectories. We will now describe the forward trajectory generating step and the backward drift learning step 2 in detail. Steps 3 and 4 are given by application of and 2 on their reverse-tine time counterparts.
Step (and 3 ): Given a fixed discretization of the time interval [0. T] denoted as (1x) with t 0 and kkg = T, denote the time step lengths as 4& fx+1 fg.- By truncating the Itô-Taylor series of the SDE, we can consider an Mulerr-aruyyaaa (e.g., Ch. in Särkkä & Solin, 2019) type of discretization for the continuous-time problem. We give the time-update of the 2th particle at time fx evolved according to Eq. (5) before conditioning on the observational data as
xi Xtx + f1-1,0(Xt,, Ldtx
1.t 1)4a + g((x- 1) A El
(7)
where E/ ~ N(O, I). In step 3 the particles ži, of the backward SDE Eq. (6) are similarly obtained. The SDE dynamics sampled in steps I and 3 apply the learned drift functions fi-1,9 and bi from the previous step and do not require sampling from the underlying SDE model. For times tę at which no observations are available, we set x = x; (and z = ž,, respectively) and otherwise compute the particle filtering weights wit based on the observations {(ty y3) E D Dity = fx} } for resampling, See Sec. 3.2 for details on the particle filtering proposal density and calculation of the particle weights.
For resampling, we employ a differentiable resampling procedure, where the particles and weights (Kt, , w are transported uniformly weighted particles x by solving an entropy-regularize optimal transport problem (Cuturi, 2013; Peyré & Cuturi, 2019; Corenflos et al., 2021), see App. D for further details. Through application of the e-regularized optimal transport map T (c)) E N (see
Corenflos et al., 2021, for details) the particles are resampled via the map to x X T(e). where
Xt. E IR xxd denotes the stacked particles {x1, 1 at time ty before resampling. The resampled particles for the backward process are given similarly.
Step 2 (and 4 ):
K.N Given the particles {x], R= we now aim to learn the drift function for the respective reverse-time process. In case no observation is available at time ts. we apply the mean- matching loss based on a Gaussian transition approximation proposed in De Bortoli et al. (2021):
fs+1,no obs = ||br, ( {x ( (1,14411444 X - (x² (x).,, (x) +x ( 21 (8)
In case an observation is available at time tu the particle values Xr, will be coupled through the optimal transport map. Therefore, the transition density is a sum of Gaussian variables (see App. A for details and derivation), and the mean-matchin loss is therefore given by
st 1,t
|lb (xx,(xxi... (xi... A++1)A Xiym fi-10(xi tx)
+NN T(E), (x2, + f1-1,0 +x) 1|2. (9)
The overall objective function is a combination of both loss functions, with the respective mean- matching loss depending on whether & is an observation time. The final loss function is written as:
(())= Σ Ek=1 oi (6)) #0 Ex= on ( (.))
(10)
where Tound denotes an indicator function that returns '1' iff the condition is true, and '0' otherwise. Consequently, the parameters & of by,s are learned by minimizing Eq. (10) through gradient descent. In practice, a cache of trajectories {xi, ]==1.1=1 is maintained through training of the drift functions, and refreshed at fixed number of inner loop iterations, as in De Bortoli et al. (2021), avoiding differen- tiation over the SDE generation computational graph. The calculations for step follow similarly.
K.N
The learned backward drift br.s can be interpreted an analogy of the backward drift in Maoutsa & Opper (2021), connecting our approach to solving optimal control problems through Sumilton-Jacole equations, see App. A.2 A.2 for analysis of the backwards SDE and the control objective. While we are generally considering problem settings where the number of observations is low, we propose that letting M - co yields the underlying marginal distribution, see Prop. 2 in App. A.3.
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3.2 COMPUTATIONAL CONSIDERATIONS
The ISB algorithm is generic approach to learn conditioon diffusion bridges under various choices of, egg., the particle filter proposal density or the reference drift. Next, we cover practical con- siderations for the implementation of the method and highlight the model choices in the experiments.
Multiple observations per time step Naturally, we can make more than one observation at single point in time fg. denoted as Di,- {G) D Diej = =. To compute particle weights wt, for the tth particle we consider only the H -nnarest neighbours of xi, in Di, instead of all observations in Di. By restricting to the H-nearest neighbours, denoted as D we introduce an additional locality to the proposal density computation which can be helpful in case of multimodality. On the other hand, letting H > results in weights which take into account the local density of the observations, not only the distance to the nearest neighbour. In experiments with few observations, we set H = 1, the choice of is discussed when we have set the value higher.
Particle filtering proposal The proposal density chosen for the ISB is the bootstrap filter, where the proposal matches the Gaussian transition density p(x,, (x,,)). Assuming Gaussian noise model N(0, I)),, the unnormalized log-weight for the 1th particle at time ty are given by:
log wi,
Σ M,ED
|x - -,,1
(11)
20²
Observational noise schedule In practice, using constant observation noise o² variance can result in an iterative scheme which does not have a stationary point as L + oc. Even if the learned drift function Si, was optimal, the filtering steps and 3 would alter the trajectories unless all particles would have uniform weights. Thus, we introduce noise schedule xl) which ensures that the observa- tion noise increases in the number of ISB iterations, causing ISB to converge to the IPFP (De Bortoli al., 2021) as L + We found that letting the observation noise first decrease and then increase (in the spirit of simulated annealing) often outperformed a strietly increasing observation noise schedule. The noise schedule is studied in App. C, where derive the property that letting L yields IPFP.
Drift initialization
Depending on the application, one may choose to incorporate additional information by selecting an appropriate initial drift. A possible choice includes pre-trained neural network drift learned to transport mo to Ry without accounting for observations. However, starting from drift for the unconstrained SBP can be problematic in cases where the observations are far away from the unconstrained bridge. To encouraged exploration. one may choose fo = 0 for the initial drift. In various problem settings, we found both, zero drift and starting from the SBP, be successful in the experiments, see App. C for discussion.
EXPERIMENTS
To assess the properties and performance of the ISB, we present a range of experiments that demon- strate how the iterative learning procedure can incorporate both observational data and terminal constraints. We start with simple examples that build intuition (ef. Fig. and Fig. 2) and show stan- dard ML benchmark tasks. For quantitative assessment, we design an experiment with non-linear SDE for which the marginal distributions are available in closed-form. Finally, we demonstrate our model both in a highly multimodal bird migration task, conditioned image generation, and in single-cell embryo RNA modelling problem. Ablation studies are found in App. C.
Experiment setup In all experiments, the forward and backward drift functions fo and by are parametrized as neural networks. For low-dimensional experiments we apply the MLP block design as in De Bortoli et al. (2021), and for the image experiment an U-Net as in Nichol & Dhariwal (2021). The latent state SDE was simulated by Fuler-Maruyann with fixed time-step of 0.01 over 100 steps and 1000 particles if not otherwise stated. All low-dimensional (at most 5) experiments were run on a MacBook Pro laptop CPU, whereas the image experiments used single Nvidia A100 GPU and ran for 5 h 10 min. Notice that since ISB only performs particle filtering outside the stochastic gradient training loop, the training runtime is in the same order as in the earlier Schrödinger bridge image generation experiments of De Bortoli et al. (2021). Thus we omit any wall-clock timings. Full details for all the experiments are included in App. B.

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(a) 0.00
(b) 0.25
(c) 0.50
(d) 0.75
(c) t 0.99
Figure 3: 2D toy experiments from scikit-learn with both cases starting from a Gaussian: The TWO CIRCLES (top) and TWO MOONS (bottom) data sets, with observations (red markers) constraining the problem. For the circles, the 10 circular observations at ( 0.5 first force the method to create a circle that then splits into two; in the lower plot the observations / E [0.25, 0.5, 0. 75) split the data into clusters before joining them into two moons. See Fig. 6 in the Appendix for the IPFP result.
All experiment settings include number of hyperparameter choices, some of them typical to all diffusion problems and some specific to particle filtering and smoothing. The diffusion g is a pre-determined function not optimized in the training. We divide the experiments to two main subsets: problems of 'sharpening to achieve a data distribution' and 'optimal transport problems'. In the former, the initial distribution has support overlapping with the terminal distribution and the process noise level g() goes from high to low as time progresses. Conversely the latter setting, the particles sampled from the initial distribution must travel reach the support the terminal distribution, and we chose to use a constant process noise level. Perhaps the most significant choice of hyperparameter the observational noise level, as imposes a preference on how closely should the observational points be followed, see App. C.1 for details.
2D toy examples We show illustrative results for the TWO MOONS and CIRCLES from scikit-leand We add artificial observation data to bias the processes. For the circles, the observational data consists of 10 points, spaced evenly on the circle. The points are all observed simultaneously, at halfway through the process, forcing the marginal density of the generating SDE to collapse to the small circle, and then to expand. For the two moons, the observational data is collected from 10 trajectories of diffusion model which generates the two moons from noise, and these 10 trajectories are then observed at three points in time. Results are visualized in Fig. (see videos in supplement). For Fig. 6.
reference, we have included plots of the IPFP dynamics in the supplement,
Quantitative comparison on the Beneš SDE In order to quantify how observing process in between its initial and terminal states steers the
Table 1 Results for the Beneš experiment.
Negative log predictive density AVeRaGE MIDDLE
ISB model to areas with higher likelihood, we test its performance on a Beneš SDE model (see, e.g. Särkkä & Solin, 2019). The Beneš SDE is non-linear one-dimensional SDE of form dr, tanh(x;) dt + d,6, with To 0, but its marginal density available closed-form, allowing for negative log-likelihood evaluation. We simulate trajectories from the Beneš SDE and from the reverse drift and stack the reversed trajectories. The terminal distribution shifted and scaled so that the Beneš SDE itself does not solve the transport problem from To to IT, App. B.2 for details and visualizations of the processes.
METHOD
END
Schrödinger B
4.787
3.565
0.1919
Iterative smoothing B
3.557
2.985
0.1567
As baseline, we fit Schrödinger bridge model with no observational data, using the Beneš SDE drift as the reference model. The ISB model is initialized with a zero-drift model (not with the Beneš reference), thus making learning more challenging. We compare the models in terms of negative log predictive density in Table 1, where we see that the ISB model captures the process well on average (over the entire time-horizon) and at selected marginal times.
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Summer
Example sightings during migration
Winter
Bird observatios
ISB result
Figure 4: Bird migration example. The top row describes nesting and wintering areas for the birds as well as example sightings during migration. At the bottom, we show the marginal densities of the ISB model from the initial terminal distribution that match the sightings along the migration.
Bird migration Bird migration can be seen as a regular seasonal transport problem, where birds move (typically North-South) along flyway, between breeding and wintering grounds. We take this as a motivating example of constrained optimal transport, where the geographical and constraints and preferred routes accounted for by bird sighting data (see Fig. 4 top). By adapting data from Am- brosini et al. (2014) and Pellegrino et al. (2015), we propose a simplified data set for geese migration in Europe (OIBMD: ornithologically implausible bird migration data; available in the supplement). We applied the ISB for 12 iterations, with a linear observation noise schedule from to 0.2, and constant diffusion noise 0.05. The drift function was initialized method did not rely on a separately fit model optimized for generating the wintering distribution based on the breeding distribution. For comparison, we include the Schrödinger bridge results in App. B.3.
crrnffuntioon, and thus the
Constraining an image generation process We demonstrate that the ISB approach scales well nigh-dimensional inputs by studying proof-of-concept image generation task. We modify the diffusion generative process of the MNIST (LeCun et al., 1998) digit 8 by artificial observations steering the dynamical system in the middle of the generation process. While the concept of observations in case of image generation is somewhat unnatural, it showcases the scalability of the method to high-dimensional data spaces. Here, the drift is initialized using a pre-trained neural network obtained by first running Schrödinger bridge model for image generation. The process then given an observation in the form of a bottom-hall dynamical process. As the learned model uses information from the observation both before and after the observation time, the lower half of the image is sharper than the upper half, We provide further details this experiment and sampled trajectories in App. B.4.
a MNIST digit 8 in the middle of the
Single-cell embryo RNA-seq Lastly, we evaluated our approach an Em- bryoid body scRNA-seq time course (Tong et al., 2020). The data consists of RNA measurements collected over five time ranges from developing human embryo system. No trajectory informa- tion available, instead we only have access to snapshots of RNA data. This leads to data set over 5 time ranges, the first from days 0-3 and the last from days 15-18. In the experiment, we followed the protocol by Tong et al. (2020), reduced the data dimensionality d 5 using PCA, and used the first and last time ranges the initial and terminal constraints. All other time ranges are considered observational data. Contrary to the other ex- periments, intermediate data imprecise (only time range of multiple days is known) but abundant.
Table 2: Results for single-cell embryo RNA experiment.
Earth mover's 's distance
METHOD
t=0
f=1
t=2
f=3
t=T
TrajectoryNet (Tong et al., 2020)
0.62
1.15
1.49
1.26
0.99
IPML. (Vargas al., 2021)
0.34
1.13
1.35
1.01
0.49
IPFP (no observations) ISB (single-cell observations)
0.57
1.53
1.86
1.32
0.85
0.57
1.04
1.24
0.94
0.83
We learned the ISB using a zero drift and compared it against an unconditional bridge obtained through the IPFP (De Bortoli et al., 2021) see Fig. 5. The ISB learns to generate trajectories
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t=T
Princpal axds a1
D
Principal axts #1
My me
(a) Schrödinger bridge (via IPFP)
(b) Iterative Smoothing Bridge
Figure 5: Illustration of the trajectories of the high-dimension single-cell experiment for the Schrödinger bridge (a) and the ISB (b), projected onto the first two principal components. The first five trajectories are highlighted in colour, and intermediate observation densities visualized as slices.
with marginals closer to the observed data while performing comparably to the IPFP at the initial and terminal stages. This improvement is also verified numerically in Table 2, showing that the ISB obtains a lower Earth mover's 's distance between the generated marginals and the observational data than IPFP. Additionally, Table 2 lists the performance of previous works that do not use the intermediate data during training (Tong et al., 2020) or only use it to construct an informative reference drift (Vargas et al., 2021), see App. B.5 for details. In both cases, ISB outperforms the other approaches w.r.t. the intermediate marginal distributions (t = 1, 2, 3), while IPML (Vargas et al., 2021) outperforms ISB the initial and terminal stages due to its data-driven reference drift. Notice that while we reduced the dimensionality via PCA to 5 for fair comparisons to Vargas et al. (2021), the ISB model would also allow modelling the full state-space model, with observations in the high-dimensional gene space and a latent SDE.
5 DISCUSSION AND CONCLUSION
The dynamic Schrödinger bridge problem provides an appealing setting for posing optimal transport problems learning non-linear diffusion processes and enables efficient iterative solvers. However, while recent works have state-of-ther the- performance in many complex application domains, they are typically limited to learning bridges with only initial and terminal constraints dependent on observed data. In this work, we have extended this paradigm and introduced the Iterative Smoothing Bridge (ISB), an iterative algorithm for learning cata-conditional smoothing bridges. For this, we leveraged the strong connections between the constrained bridging problem and particle filtering in sequential Monte Carlo, extending them from pure inference to learning. We thoroughly assessed the applica- bility and flexibility of our approach in various experimental settings, including synthetic data sets and complex real-world scenarios (e.g., bird migration, conditional image generation, and modelling single-cell RNA-sequencing time-series). In our experiments, we showed that ISB generalizes well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals intermediate and terminal times.
Accurately modelling the dynamics of complex systems under both path constraints induced by sparse observations and initial and terminal constraints is a key challenge in many application domains. These include biomedical applications, demographic modelling, and environmental dynamics, but also machine learning specific applications such as reinforcement learning, planning. and time-series modelling. All these applications have in common that the dynamic nature of the problem is driven by progression of time, and not only progression of a generative process as often the case in, e.g., gen- erative image models. Thus, constraints over intermediate stages have natural role and interpretation in this wider set of dynamic diffusion modelling applications. We believe the proposed ISB algorithm opens up new avenues for diffusion models in relevant real-world modelling tasks and will be stim- ulating for future work. For example, more sophisticated observational models, alternative strategies to account for multiple observations, and different noise schedules could be explored. Furthermore, the proposed approach could be extended to other types of optimal transport problems, such as the Wasserstein barycenter, frequently employed case of the multi-marginal optimal transport problem.

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Proof sketch. Our objective is to find a backward drift function Notice that at observation times ty, this is not equivalent to finding the reverse drift of the SDE forward transition and differential resampling combined, since the drift function fi-1.e alone does not map the particles {x{ Ji=1 to the particles {x/ J We will derive a loss function for learning the backward drift as in Eq. (13) below, leaving the discussion why it is meaningful choice of Bortoli Under review as a conference paper at ICLR 2023
A METHOD DETAILS
We present the details of the objective function derivation in App. A. and explain the connection of the backward drift function Hamilton-Jacobi equations in App. A.2. App. A.3, we discuss the behaviour of our model at the limit M + oc, that is, when the observations fully represent the marginal densities of the stochastic process.
A.1 DERIVING THE MEAN- MATCHIGG LOss AT OBSERVATION TIMES
Proposition 1. Define the forward SDE as
dx; = f1,o(X1, t) dt + g(t) dB,,
Xg ~ TV.
(12)
and a backward SDE drift as
b (x+1)
fi- f ())))))) g(lx+1) 1 In Ptat
(13)
where Piz-i is the particle filtering density after differential resampling at time txt1- (k+1) minimizes the loss function
Then
C 141000 = I1b.,o 1 tx+1)As - Xismi - f1-1,((x 1b
DN Tejeon (x% f1-1,0(2,,, tx)ss)))))))))))))) II? (14)
Ce.
where we denote Ce. glfc+1)=Ar before resampling.
Var (Enzn T(E)da and (x) IN are the particles
(x+1) 1 as in Eq. (13).
backward drift to App. A.2. Our derivation closely follows the proof of Proposition et al. (2021), ), but provide the details here for the sake of completeness.
First, we give the transition density Px,, [x (xx) and apply it to derive the observation time loss
oooss The derivation for the loss f obs is skipped, since it is as in proof of Proposition 3 in De Bortoli et al. (2021). Suppose that at lg, there observations. By definition, the particles before resampling {x 1.2 are generated by the Gaussian transition density
p(Rte xxx)) = N(X xxx + (x.f( (x), ggggg (1(()))))())A A²
(15)
Recall that the resampled particles are defined as a weighted average of all the particles, x = E XL, Tejaan Thus, the transition density from {x, the particles (x), }. is also Gaussian,
p(x (x(=== N(8
Σ Teer + A (().)) n=1
(16)
We will derive the loss function Eq. (9) by modifying the mean matching proof De Bortoli
al. (2021) by the transition mean Eq. (16) and the backward drift definition Eq. (13). Using the particle filtering approximation, the marginal density can be decomposed as Ptz (X++1) =
N ΣM Pts xx))x.. xxt (xx+1). By substituting the transition density Eq. (16) follows that
Plesi (Xte4z)
N
( 11 (E (x) + fi-1 fi-10(xxee1 (xte X1e4il" 112 2g(te+2) (ootut))2Cc.cc Ce,
Σ
Pta (xx)) exp
(17)
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where Z is the normalization constant of Eq. (16). As in the proof of Proposition of De Bor- toli et al. (2021), we derive an expression for the score function. Since In Pie+1 (Xte.1) = Vxy. PiR) (Xtx11) we first manipulate Exept Pia+i (Xestz)
Picts (86411
V Seipa Pla+i (Xt++1)
(18)
N
|| [Z Tes (xtx + 2g(lx+1) )(el:t))CC,,iss
f1-1,((X0 (w))) X 112
V Xeg++ p(x),) exp
exp
(19)
EE Ulln+1)2ACCa EEuininAAACC
p(xix)
Σ
(x 11-10 (x)) Xtx12)
(20)
n-=
E (xs)em(x)) fi-11 (x 2g(tx+1 Ac,
Xlets 112) 1))
exp
(21)
P4 (X50 Paets lnf (2411)
Substituting Pte (1/b)=
to the equation above gives
Pxt 1xe11 (x1)
V Xezz4 Pea+i (Xte+1)
(Te))a + 11-11,0xx (x)) X(xni) ) (22)
= =Pict1 (Xtemr)
Peesi xx (xx)
g(tx+1)2AxCa
and dividing by Ptess yields
V Vln 12 (2)
(Tie)i (x), + f1 tu)) - X(mxn)
(23)
Σ Px. Necke (xi:)
g(4++1)2&CC
Substituting Eq. (23) to the definition of the optimal backward drift Eq. (13) gives
bi, ( (x+1)
f1 (1 - g(tx+1)) In Pre+s (XE+1)
=
-1-1100X0mn tx)
(24)
N
(T + 1fi-10(X1,, Xlat1)
o(ta-1) 1)2
E
Pris |x1411 (Xte.)) )
g((u+1)2ARCo
x=1
where taking fi-1, fi-1,0(Xeau1 the inside the sum yields
(x.((xtu1s (x+1)= =
Px.g Ixcess (Xturz)
(25)
N Σ
Txl, + 10 (ka))
8tet1
A
1 AAx))
Ce.i
Ce.
Multiplying the equation above by Ae gives
(xbr, (2+1)
X 2 tx).
(26)
Σ Tor + fi-1,((x
Ce.
Thus may set the objective for finding the optimal backward drift bi
Xi Si- 11-1110 (x (x)
11 = |
(++1)da
Ce.
(27)
+ Ce,
DN
(1) (x%2 + S1-1,0 Ax) 11?.
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Notice that if the weights before resampling are uniform, then [(e) = IN. and for all i E 1,2,...,
holds that Ce., = 1, since all but one of the terms in the sum g(fe+1)² Var (>N> (()),,, vanish. Similarly, for one-hot weights Ce. In practice, we set the constant Ce.t = in Eq. (9)
and observe good empirical performance with the simplified loss function.
A.2 CONNECTION TO HAMILTON-JACOM EQUATIONS
We connect the backward drift function (x.((xx-11+4+)) 4k+1)
f (2)
g(tx+1) In Ptu4 (x)) 1 to the Hamilton-Jacobi equations for stochastic control through following the setting of Maoutsa & Opper (2021), which applies the drift f1-1,0 2 In (x())) for backwards SDE initialized at Tr.
Consider stochastic control problem with a path constraint U Uxx,, t), optimizing the following loss function,
J
[1)()) f(x,, (x),,)) xxxr),
(28)
N i=1
2g(t)²
with the paths x sampled as trajectories from the SDE
Xo Au.
dx, fi t) dt + g(t) d3;,
(29)
and the loss In xxxx) measures distance from the distribution Ty- Since we set the path constraint via observational data, our method resembles setting Uxx]. t) = 0 when is not an observation time, and U (xx) = log pyy xx).. where p(y x)) is the observation model.
Let (x) denote the marginal density the controlled (drift fa) SDE at time t. Maoutsa Opper (2021), the marginal density is decomposed as
qe (x(x) = (x),,
(30)
where (x) is a solution to a backwards Siieer-Planck-oolieee equation starting from (x) Wr, and the density evolves as in
(FPK) partial differential
dx.(x)
+ (x,
(31)
where CJ is the adjoint FPK operator to the uncontrolled system. The density pe(x) corresponds to the forward filtering problem, initialized with To.
dpr(x) = Ly(p)(x)) - U(x, Oxx(x);
(32)
dt
where Cy is the FPK operator of the uncontrolled SDE (with drift f). The particle filtering trajectories (xt, 3' generated in our method samples from the density defined by Eq. (32). In the context of our method, the path constraint matches the log-weights of particle filtering at observation times and zero elsewhere.
In Maoutsa & Opper (2021), a backward evolution for is applied, using the backwards time -r-r (x) = gr (x), yielding a backwards SDE starting from ğo (x) = 1- reweighted according to Tr. The backward samples from are generated following the SDE dynamics
dx (f(xi, r) +
dt g(t) d,,.
(33)
We have thus selected the backward drift by to match the drift of (x), the backward controlled density. Intuitively, our choice of 6, is a drift which generates the smoothed particles when initialized at H the terminal state of the forward SDE. The discrepancy between गा and the distribution induced by ( then motivates the use of an iterative scheme after learning to simulate from (x)).
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Proof sketch. We drop the a and H(M) from the weight notation for simplicity notation, but remark that the particle filtering weights are dependent of both quantities. Consider the number of particles N fixed, and denote the d-dimensional sphere centered at x as B(x(,, r). Since each particle xi, lies the support of the true underlying marginal density Pt.* then for any radius r Under review as a conference paper at ICLR 2023
A.3 OBSERVING THE FULL MARGINAL DENSITY
Suppose that at time lx. we let the number of observations grow unbounded. We analyse the behaviour of our model at the resampling step, the limit M + oo for the number of observations and for the observation noise. When applying the bootstrap proposal, recall that we combined the multiple observations compute the tog-weight
log wix
Σ YEE
|x - -,,
(34)
20
which works well in practice for the sparse-data settings we have considered. Below we analyse the behaviour of an alternative way to combine the weights and show that given an infinite number of observations, it creates samples from the true underlying distribution.
Proposition 2. Let {xi, Viz be set of particles and { the observations at time iy- Assume that the observations have been sampled from density Pt, and that for all i it holds that x, supplpt.). Define the particle weights as
log wt, O.M = log
Σ
exp(- - -
(35)
ZID
H(M)
4,6
YJED
where Z is the normalization constant of the observation model Gaussian ply xtx)): Then for each particle Xhe its weight satisfies
lim lim o. = PPx (I(r) a-0 +200
(36)
such that B(x E supp(pt.). and H > O, we may choose M high enough so that the points y, E D.H. satisfy Yj E B(x,, r). It follows from Eq. (35) that
w//
exp( -1x] - -2112/20²).
(37)
ZID
(MM)
YJED M(M0 4.th
For any > 0 and with observation noise a cr, we may set H(M) so that the sum above approximates the integral
B(x r) I
B(xt ply x (
(38)
By applying the Lebesque differentiation theorem, we obtain that for almost every X we have lim wi Pt, (x², since - O. the density p(y x collapses to the Dirac delta of x
Prop. 2 can be interpreted as the infinite limit of a kernel density estimate of the true underlying distribution. Resampling accurately reweights the particles so that the probability resampling particle x is proportional to the density Pt. compared to the other particles. Notice that the result does not guarantee that the particles will cover the support of Pt., since we did not assume that the drift initialization generates marginal density time tę covering its support.
B EXPERIMENTAL DETAILS
B. 2D TOY DATA SETS
For the constrained transport problem for two-dimensional scikit-learn, the observational data we chose to was different for each of the three data presented; moons, circles and the
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(a) 0.00
(b) 0.25
0.50
(d) t 0.75
(c) 0.99
Figure 6: The IPFP result for the experiment in Fig. 3 in the main paper.2I toy experiments, where observations (red markers) not used while training but included in the figure for reference. The dynamics learned by IPFP are clearly different from the ISB learned dynamics.
S-shape. All three experiments had the same discretization (t E [0, 0.99). A 0.01), learning rate 0.001, and differentiable resampling regularization parameter 0.01. The process noise g(t)² follows a linear schedule from 0.001 to 1, with low noise at time = 0 and high noise at = 0.99, and each iteration of the ISB method trains the forward and backward drift networks each for 5000 iterations, with batch size 256. Other hyperparameters are explained below.
Two moons The observational data consists of 10 points selected from the Schrödinger bridge trajectories, all observed at E [0.25, 0.5, 0.75] with an exponential observation noise schedule n(l) 1.25/-1, The ISB was ran for 6 epochs, and initialized with drift from the pre-trained Schrödinger bridge model from the unconstrained problem.
Two circles The observational data consists of 10 points which lie evenly distributed on circle, observed at = 0.5 with an exponential observational noise schedule sal)) 0.5 1.25'-1. The ISB ran for 6 epochs, and initialized with drift from the pre-trained Schrödinger bridge model from the unconstrained problem.
S-shape The observational data consists of 6 points, with pairs being observed times & [0.4, 0.5, 0.6). We used bilinear observational noise schedule with linear decay for the first half of the iterations from x()) 4 x((22)² I and linear ascend for the second half of the iterations from k( ((/22)2 = 1 to 1s (L)2 = 4. The ISB ran for 6 epochs, with zero drift initialization.
True ør
ISB
Schrtkinggrr
0
Figure kernel density estimate of the Beneš SDE terminal state. We compare कт to the Schrödinger bridge and ISB terminal states. Both unconstrained Schrödinger bridge and ISB terminal states succeed representing गт well, with the Schrödinger bridge terminal state more closely matching रy near its mean.
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10
12
12
12
Time,
Time, É
Time,
(a) Trajectories of the Beneš SDE
(b) SBP trajectories
(c) ISB trajectorics (ours)
Figure 8: Comparison of the solution for the SBP (with Beneš SDE reference drift) and the ISB (with zero initial drift) on the Beneš SDE under sparse observations (o). The target distribution Tr slightly shifted and scaled from the Beneš SDE. Even if the SBP has the true model as reference drift,
its trajectories degenerate into a unimodal distribution, while the ISB manages cover both modes even if only sparse observations available.
B.2 THE BENES SDE
In the Beneš SDE experiment, obtain the sparse observational data from sampled Benes SDE trajectories while the terminal state is shifted and scaled (3 + 5xT) version of Beneš marginal density. As the Beneš trajectories were first generated by simulating the SDE until t and then in reverse from = 6 to / O, set T = 11.97. We apply the analytical expression for the Beneš marginal density for computing log px(xx.)..
(x)
cosh(x) V2rt cosh(x,)
exp
exp
1/2 2f
(xx-x0)²
x0)²
(39)
See the Beneš SDE trajectories in Fig. 8a. As expected, the transport model with observations performs well in the generative task, but its trajectories cover also some low-likelihood space around i = 6 (in the middle part in Fig. 8b). The observations for the ISB model were sampled from the generated trajectories, 10 observations at 10 random time-instances (see Fig. 8c)
Both the unconstrained Schrödinger bridge model and the ISB model a learning rate of 0.001 for the neural networks. Likely due to the fact that the problem was only one-dimensional, convergence of the Schrödinger bridge to process which matches the desired terminal state was fast, and we chose not to run the model for higher number of ISB iterations, see Fig. for comparison of the trained model marginal densities and the true terminal distribution TT. We set the observation noise schedule to the constant 0.7, and at each iteration of the ISB or the unconstrained Schrödinger bridge the drift neural networks were trained for 5000 iterations each with the batch size 256, and the trajectories were refreshed every 500 iterations with cache size of 1000 particles. The number of nearest neighbours to compare to was H 10.
for iterations, using
B.3 THE BIRD MIGRATION DATA SET
The ISB model learned bird migration trajectories which transport the particles from the Northern Europe summer habitats to the southern winter habitats, see Fig. 10 for a comparison of Schrödinger bridge and ISB. Since the problem lies on sphere, Schrödinger bridge methods adjusted for learning on Riemannian manifolds could have been applied here. For simplicity we mapped the probelm to two-dimensional plane using Mercator projection, and solved the problem on [0, 5) x [0.5] square. The SDE had the discretization t E [0, 0.99), A 0.01 and a constant process noise gt) 2 = 0.05. The model was trained for iterations, and initialized with a zero drift, while the observational data was chosen by the authors to promote learning trajectories clearly different from the unconstrained transport trajectories. The observation noise schedule was piecewise linear (starting at 2, going to 0.1 at iteration 6, then rising linearly to reach 2 at iteration 12). At each ISB iteration, the neural networks were trained for 5000 iterations each, and the trajectories refreshed every 1000 iterations. We used batch size of 256 and learning rate 0.001.
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Figure 9: Model trajectories for MNIST digit "8' conditioned on a lower-loop of single '8' at
1 = 0.38 to bias the lower half of the digits to look alike, with the effect still visible at terminal time T.
B.4 THE MNIST GENERATION TASK
Applying state-space model approaches such as particle filtering and smoothing to generative diffusion models directly in the observation space (that is, not in a lower-dimensional latent space) has to our knowledge not been explored before. Some experimental design choices had great impact into the training objectives sensibility, as the observational data is completely artificial and its timing during the process modifies the filtering distribution significantly. As the MNIST conditional generative model was trained to display the scalability of our method beyond low-dimensional toy examples, we did not further explore optimizing the hyperparameters or the observation model. To avoid the background noise in MNIST images in the middle of the generative process impacting the particle filtering weights excessively, the observation model is a Gaussian with masked inputs equal to zero in pixels where the observation image is black, see Fig. 9 for sampled trajectories. The figure shows the progression of seven samples, where the lower half of the eights resemble the observation target.
The SDE was for time E [0, 0.5), with the digit eight observed t 0.38. The ISB method was applied for 10 iterations, with a discretization E [0, 0.495], As = 0.005, and the process noise g(()² followed linear schedule from 0.0001 to At each iteration of the method, the forward and backward drift neural networks were trained for 5000 iterations with a batch size of 256, and the trajectory cache regenerated every 1000 iterations. The observational data consisted of a single sample of lower half of the digit eight, observed at time I. = 0.38. The observation noise schedule was a constant x(I)= 0.3.
B.5 SINGLE- CELL DATA SET
We directly use the preprocessed data from the TrajectoryNet (Tong al., 2020) repository. A major difference between our implementation and Vargas et al. (2021) is the reference drift. We set the reference drift to zero, which means that we utilize the intermediate data only as observations the state-space model. On the contrary, Vargas et al. (2021) fits a mixture model of 15 Gaussians on the combined data set (across all measurement times) and sets the reference drift the gradient of the log likelihood of the mixture model. Effectively, such a reference drift aids in keeping the SDE trajectories within the support of the combined data set. We remark that if the intermediate observed marginals had clearly disjoint support, combining all the data would cause the mixture model to have "gaps' and could cause an unstable reference model drift. Thus we consider our approach of setting the reference drift to zero more generally applicable.
As in Vargas et al. (2021), we set the process noise to (t) = and model the SDE between time t E [0,4] 4]. The learning rate is set to 0.001 with batch size 256 and number of neural network training iterations 5000, and we apply the ISB for 6 iterations. We filter using 1000 points from the
intermediate data sets, but compute the Earth mover's distance by a comparison to all available data. As the observational data at T = 1, 2, 3 consists of high number of data points, the parameters H (number of nearest neighbours) and (observation noise) need to be carefully set. We set H 10
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to only include the close neighbourhood of each particle, and set the observation noise schedule as constant 0.7.
C COMPUTATIONAL CONSIDERATIONS
In Sec. 3.2, we raised a number of important computational considerations for the constrained transport problem. Below we discuss them in detail, analyzing the limit J, - co from the perspective of setting the observation noise schedule in App. C.I, and presenting ablation results on modifying the initial drift in the bird migration experiment in App. C.2.
C.I DISCUSSION ON OBSERVATION NOISE
We briefly mentioned Sec. 3.2 that when letting -> oc, the choice of observation noise should be carefully planned in order for the ISB procedure have stationary point. Here explain why unbounded observation noise schedule s(l) implies convergence to the IPF method for uncontrolled Schrödinger bridges (De Bortoli et al., 2021), when using a nearest neighbour bootstrap filter as the proposal density.
Proposition 3. Let Q E TR. be bounded domain where both the observations and SDE trajectories lie, and let the particle filtering weights {wi, Ji=1 be in Eq. (11), but after normalization. If the schedule s(l) is unbounded with respect to I, then for any 8 there exists l' such that for the normalized weights it holds
1
(40)
N
Proof sketch. Since we set the proposal density to be the bootstrap filter, the observation weights at ISB iteration are equal to
log wi,
Σ
| ysl1..
(41)
2x(1)2
Since x({) is unbounded, for any >> 3 such that m(()) 2 S. We choose the value of S so that the following derivation yields Eq. (40).
Let S = 0.5R | diam (52)2, and apply the property that ||x| -y,11² s diam(?))² to Eq. (41),
log wh 1 2
1
Σ
|x
25²
E ! 4c
(42)
Σy Σy,eDH |xx] - 11
Eyseg diam (2)²
R- diam -R
R ! di)
The bound above is for the unnormalized weights, and the normalized log-weights are defined
N
log wh.re = log wh.n. - llgg Σ
exp(log w/ ..))
(43)
where for the normalizing constant it holds that
N Σ
log Σ exp(log w/ <
log
-log(N),
(44)
j=1
since w/ wp.,, is the value of probability density and thus always wHt < 1. Combining Eq. (43),
Eq. (42) and Eq. (44), it follows that
log wf - (-logg() 2 R,
(45)
where taking exponentials on both sides gives
1
1
Wp
2 101-11000 --xx(pX(R)
(46)
N
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Under review as a conference paper at ICLR 2023
Summer
Intermediate during migrationn
Winter
Bird observations
Unconditional
Conditional tzero drift)
Conditional (tramsport drift)
Figure 10: Top row: The first map image on the left describes the initial position of the birds, and the final on the right their position after migration. The observational data in the middle are bird observations during migration, at given timestamps. Second row: Marginal densities of Schrödinger bridge model from the initial to terminal distribution, without using the observations. Third row: Marginal densities of our model, using both initial and terminal distributions and observational data and zero drift initialization. Bottom row: Same as third row, but with the second row dynamics as initialization.
Since the weights normalized, even the largest particle weight w/ .a. can differ from as much as every smaller weight in total lies under +.
1 N
+ + (N-1) (1(1 exp((ppp)) N )
(47)
implying that for any weight i it holds that
10 S (N (N-1) (LN--u--))(-))))) < exp(-R),
(48)
and selecting - log(1 5) sufficient for 1.
Effectively, the above derivation implies that for an unbounded observation noise schedule k<(), the particle weights will converge to uniform weights. Since performing differentiable resampling on uniform weights implies that T (e) IN. the ISB method trajectory generation step and the objective in training Nthe backward drift converge to those of the IPF method for solving unconstrained Schrödinger bridges. Intuitively, this means that at the limit L + oc, our method will focus reversing the trajectories and matching the terminal distribution while not further utilizing information from the observations.
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C.2 ABLATION ON INITIAL DRIFT
We conducted an ablation study on drift initialization for the bird migration problem. As the distributions mo and тт (as pictured in Fig. 10) are complex, we consider the problem setting to be interesting for setting fo as the unconstrained transport problem drift. To this end, we trained a Schrödinger bridge model for 10 epochs, and trained an ISB model with the same hyperparameter selections as explained in App. B.3, using the Schrödinger bridge as the initialization. Compare the two bottom rows of Fig. 10 to see selection of marginal densities of the two processes. Based on visual analysis of the densities, it seems that the zero drift and pre-trained diffusion model initializations produce similar results around the observations, although the Schrödinger bridge initialization gave slightly sharper results at terminal time.
D DIFFERENTIABLE RESAMPLING
In the ISB model steps and 3 presented in Sec. 3.1, we applied differentiable resampling (see Corenflos et al., 2021). Resampling itself is a basic block of particle filtering. A differentiable resampling step transports the particles and weights (xi, w to an uniform distribution over set of particles through applying the differentiable ensemble transport map (e)). that is
(x)
w/o) (X ((c), 1/N) (xt, 1/N),
(49)
where X,, E Nxd denotes the stacked particles (xf te J at time lg before resampling and x denotes the particles post resampling. Here we give the definition of the map T(e) and review the regularized optimal transport problem which has to be solved to compute it. We partly follow the presentation in Sections and of Corenflos et al. (2021), but directly apply the notation we use for particles and weights and focus explaining the transport problem rather than the algorithm used to solve it.
The standard particle filtering resampling step consists of sampling N particles from the categorical distribution defined by the weights (wt, J resulting in the particles with large weights heing most likely to be repeated multiple times. A result from Reich (2013) gives the property that the random resampling step can be approximated by deterministic ensemble transform T. In heuristic terms, the ensemble transform map will be selected so that the particles { Jimi will be transported with minimal cost, while allowing all the weights to be uniform.
Let and be atomic measures, E'' w dzg and v = ΣE N N where 6, is the Dirac delta at æ. Then y is the particle filtering distribution before resampling. Define the elements of cost matrix E N as Ci.j = || X I12, and the 2- 2-ssseerttein distance between two atomic measures
(2,
min PES(u.) 1=1
E.C.P...
(50)
Above the optimal matrix P is to be found within S(p, v). which is space consisting of mixtures of N particles to N particles such that the marginals coincide with the weights of μ and v, formally
S(u,v) - { PE 1o. Σ
wig
P
(51)
The entropy-regularizee Wasserstein distance with regularization parameter e is then
(Cus+ P.j wi, N
(Cus + Elg
2
min PES(
P.s
(52)
The unique minimizing transport map of the above Wasserstein distance is denoted by POPT and the ensemble transport map then set as T = NPOPT, This means that we can find the matrix Tee) via minimizing the regularized Wasserstein distance, which is done by applying the iterative Sinkhorn algorithm for entropy regularreg optimal transport (Cuturi, 2013).
23


Reviews:
[{"review_id": "ymxFJsOhRnm", "reviewer": "Reviewer_JvCw", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Authors present computationally efficient framework for learning data conditional diffusion bridges using Iterative Smoothing Bridge. The proposed framework is assessed by experimental results on both synthetic and real world data.\n", "summary_of_the_review": "The paper is well written, presents novel framework with experimental assessment of it. Proofs of propositions on which the framework relies are not full but it seems that they are more or less true based on provided sketches. ", "strength_and_weaknesses": "Strength:\n1. strong theoretical formulation of the proposed approach\n2. well explained algorithm\n3. well performed experiments\n\nWeaknesses:\n1. lack of rigorous proofs of main propositions (proof sketches of proposition 1 and 2)", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "b3FW_7ioQMB", "reviewer": "Reviewer_V4yF", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "In this paper, the authors propose to add sparse constraints to the original Schrodinger bridges through optimal control. Specifically, the paper assumes that there exist some intermediate sparse samples during the diffusion process. By modifying the Iterative Proportional Fitting procedure (IPFP) method with spare intermediate constraints, the Iterative Smoothing Bridge (ISB) method is proposed. Experiments show that the ISB method can help the forward and backward drift functions successfully evolve toward the intermediate observations.", "summary_of_the_review": "Generally, the paper proposes an interesting problem. But the ambiguity in writing and implementation makes it hard to follow.", "strength_and_weaknesses": "Strength\n- The proposed problem may be important in many different applications with sparse intermediate observations, especially in the medical area.\n- It is reasonable to use $L^2$ loss to handle the forward and backward drifts.\n- Experimental results are convincing. \n\nWeakness\n- No convergence guarantee of the proposed method.\n- For step 2 and 4, since there only exists sparse intermediate observations, to make the algorithm converge, it seems that a large number of samples is needed to make the method converge.\n- It is reasonable to assume both $g$ and $\\beta$ the same in both equation (5) and (6)?\n- I may miss this part in the paper. Empirically, how to define $g$?\n- The second paragraph of Step 1 in Sec. 3.1 is not very clear.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "SNJ2fqLqTJ", "reviewer": "Reviewer_aFhC", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors proposed the iterative smoothing bridge by leveraging ideas from Schr\\\"{o}dinger bridge and particle filter. Such a method allows us to learn constrained stochastic processes governed by sparse observations at intermediate stages. The algorithm is evaluated on synthetic data, real data, and small-scale high-dimensional data. This is an interesting problem and worth a deeper investigation and the proposed methodology seems to be a reasonable candidate.", "summary_of_the_review": "An interesting method to solve an important problem, not clear/scalable enough though.", "strength_and_weaknesses": "\n**Pros:** \n\n1. This paper is proposed to address an interesting problem, a path-constrained Schrodinger bridge, where a particle filter kicks in to tackle the sparse observations; \n\n**Cons:**\n\nWhile I am not an expert in the field of particle filters, my biggest concern is the clarity: \n\n1. more background introduction on particle filters may be needed for non-expert readers in the appendix; \n2. what is the optimal transport map **$\\boldsymbol{T}_{\\varepsilon}$**;\n3. why do we need the H-nearest neighbours?\n4. maybe I am wrong, i think simulated annealing only proposed to gradually decrease the noise. Why does decreasing and then increasing the noise scale resemble simulated annealing? \n5. the resampling steps may be detailed.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "FDPBC-s1PT", "reviewer": "Reviewer_3a4i", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper studies the problem of learning a dynamical process given sparse observations of the process at intermediate times using a modification of the Schrodinger bridge process. In particular, the authors propose a method they term Iterative Smoothing Bridge (ISB) which alternates fitting forward and backward drifts parametrized by neural networks with a particle filtering and update step. This latter step is included to incorporate potentially sparse observations of the intermediate time dynamics. The authors discuss connections between their methods and stochastic control and give some theoretical claims. They then give an empirical study of their method.\n", "summary_of_the_review": "While this work proposes a new method -- ISB -- for the important problem of learning dynamical processes with sparse observations, it does not achieve strong empirical performance and has limited conceptual novelty. For these reasons I think it is slightly too weak to merit acceptance.", "strength_and_weaknesses": "Strengths:\n- They study an important problem: that of learning dynamical processes given some sparse observations at intermediate times\n\n- The ISB method is novel\n\n- They achieve some improved empirical results when compared against related methods in the literature\n\n- The work is fairly well written\n\nWeaknesses:\n- While their method does offer some improvement compared to related methods, it is not a significant improvement, see for example table 2, figure 6, figure 7\n\n- The authors give little theoretical justification for their method, beyond Prop. 3 which is not rigorously proved and in any case is essentially well-known. \n\n- The core elements of the method appeared in prior works, namely De Bortoli et al 2021 and Corenflos et al 2021. The novelty of their method is in the combination of these approaches.\n\nWriting feedback:\n- Typos in background paragraph on page 3: should be $\\mathcal{C} = C([0, T]; \\mathbb{R}^d)$. There's an extra \"to\" in the sentence that begins with $x_t$\n- The differentiable re-sampling procedure is hardly explained at all, and only a passing reference to Corenflos et al is given. Comprehension would be greatly aided by giving some discussion of this method, if only in an appendix. Also, it seems that the differentiability of the re-sampling method is not used in your method - is this correct?\n\nDe Bortoli, V., Thornton, J., Heng, J., & Doucet, A. (2021). Diffusion Schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34, 17695-17709.\n\nCorenflos, A., Thornton, J., Deligiannidis, G., & Doucet, A. (2021, July). Differentiable particle filtering via entropy-regularized optimal transport. In International Conference on Machine Learning (pp. 2100-2111). PMLR.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject