Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior

## Abstract

Abstract Trajectory inference seeks to recover the temporal dynamics of a population from snapshots of its (uncoupled) temporal marginals, i.e. where observed particles are not tracked over time. Lavenant et al. [ 33 ] addressed this challenging problem under a stochastic differential equation (SDE) model with a gradient-driven drift in the observed space, introducing a minimum entropy estimator relative to the Wiener measure. Chizat
et al. [ 14 ] then provided a practical grid-free mean-field Langevin (MFL) algorithm using Schrödinger bridges. Motivated by the overwhelming success of observable state space models in the traditional paired trajectory inference problem (e.g. target tracking), we extend the above framework to a class of latent SDEs in the form of observable state space models .
In this setting, we use partial observations to infer trajectories in the latent space under a specified dynamics model (e.g. the constant velocity/acceleration models from target tracking). We introduce PO-MFL to solve this latent trajectory inference problem and provide theoretical guarantees by extending the results of [ 33 ] to the partially observed setting. We leverage the MFL framework of [ 13 ] , yielding an algorithm based on entropic OT between dynamics-adjusted adjacent time marginals.
Experiments validate the robustness of our method and the exponential convergence of the MFL dynamics, and demonstrate significant outperformance over the latent-free method of [ 13 ] in key scenarios.

## 1 Introduction

Estimating the temporal dynamics and trajectories of a population from collections of unpaired observations at specific time points is a challenging fundamental problem with many potential applications and a recent flurry of interest in the community [, , , ].
Previous work focused on the fully observed setting, where all variables that are important to the underlying dynamics are directly observed with no hidden states. This setting is relevant to single-cell genomic data analysis, where the goal is to understand the trajectories of a population of cells at unobserved times and reconstruct the trajectories of individual cells in gene space. Here, note that physics properties such as momentum do not apply. That said, research in signal processing and control theory has overwhelmingly shown the importance of being able to handle latent states in dynamics modeling more generally [, ]. Even linear state space models have enjoyed a recent resurgence for modeling text sequence data with large language models (e.g. []).

Figure: (a) Ground truth.
Refer to caption: extracted/5644114/figures/velocity_ground_truth.png

While in general the problem of recovering a hidden state is not identifiable, systems theory has developed observability conditions on the underlying dynamics model that does allow for such recovery []. For instance, in target tracking, oftentimes only a position variable is observed, yet the tracking algorithm uses a state space model that includes a hidden velocity state [, ]. This hidden state allows for better predicting the future position of the target, improving the final trajectory inference by not only better modeling the dynamics, but making it easier to identify which of several targets observed at any given time correspond to the current track [, ]. The hidden states themselves, when interpretable, may also be of direct interest for downstream applications.

The trajectory inference problem has many similarities to the tracking problem. In particular, at any given time, a cloud of points is observed, but these points are not labeled, i.e. there is no indication of which points at time $t_{1}$ match to the points at time $t_{2}$. Inferring these “matches” is the task of trajectory inference, and closely parallels the data association problem in target tracking [, ]. As a result, we are motivated to introduce latent state space modeling to the trajectory inference problem.

While itself being a fully observed framework without incorporating a known dynamics, the optimal transport (OT)-based method of [] is particularly amenable to our aims. It proposes to optimize collections of particles at each time step to minimize a data fit term (a cost between the particle cloud and the observed data points) and a trajectory fit term consisting of the entropic Wasserstein distance between sets of particles at adjacent time points. The entropic OTframework arises naturally from the SDE model (as we will see later), and provides an explicit and robust procedure for obtaining inferred trajectories from unpaired time series data by following the OT plan between time points. Representing the inferred time marginal densities as particles is also particularly amenable to our partially observed framework, as we can have the particles be in the hidden state space and form a data fit term to the observations using a specified (stochastic) observation model. In many ways this parallels the observation model/hidden state particle setup used by the particle filter [] and other sequential Monte Carlo methods [] for the paired-observation trajectory inference setting.

Applications While our focus is on the theoretical underpinnings of the latent trajectory inference problem and MFL-based algorithm, we envision our approach being broadly applicable to a variety of real-world tasks. For instance, the extra smoothing induced by introducing a hidden velocity state could improve trajectory inference in the genomic data analysis setting mentioned above [], or any application where trajectory inference is relevant. Another possible application domain could be survey or medical data. Specifically, when trajectory data is needed, *longitudinal studies* are often designed where individuals are followed over time and continue to be re-interviewed, but significant logistical challenges are involved in such a procedure []. Trajectory inference could allow for different individuals to be sampled at each time point, significantly easing the burden on researchers. Our approach for introducing hidden states (e.g. velocity) would be particularly impactful, as in both social science and medical data, often individual’s trajectories (e.g. preferences or health) do carry significant momentum. Finally, our approach could have advantages in private learning of time series models or private synthetic data generation for time series. This is because in (differential) privacy [, ], the goal is to preserve the statistics of an individual, and trajectory inference allows for each individual’s data to be limited to a single point in time, not requiring a full trajectory record that may be difficult to privatize. Our partial observation framework would allow for even more privacy to be maintained as some variables could remain hidden if an appropriate dynamics model was available.

Related works
There have been many works in the mathematical and computational biology community on trajectory inference: see [] for a survey and comparison of single-cell trajectory inference methods. [] introduces the use of OT for trajectory inference; however, the method generates paths that are generally not smooth. [] uses OT to construct measure-valued splines, which yields smooth paths, [] models population dynamics as a Jordan-Kinderlehrer-Otto (JKO) flow [], and [] uses OT to analyze gene trajectories.

[] consider the limits of trajectory inference from single-cell snapshots in the equilibrium setting. However, as far as we are aware, [] was the first work to provide theoretical guarantees of any estimator for trajectory inference. They introduce a min-entropy estimator for gradient-driven drift models and prove convergence to the ground truth in the limit as the number of observations become dense in the observation period. [] extends the entropy minimization formulation of [] by considering a different fitting functional, reducing optimization space, and using MFL dynamics. [] considers the application of these OT frameworks to the steady-state setting with known cell birth and death rates.

There has been much work on latent space for generative models, many of which use OT. [] uses score-based generative modeling in latent space. [] uses a pre-trained encoder and decoder, consider diffusion in latent space, and prove theoretical guarantees that the output distribution is close to the ground truth. [, ] consider learn latent manifold structures using OT, [] considers gradient flow in latent space to study equivariant networks, [] studies the latent space of generative models using OT, and[] considers the nonlinear filtering problem using partial observations using OT.

There are also some works considering concrete applications of Schrödinger bridges with non-Wiener reference measures. For example, [] considers Schrödinger bridges where the prior is any Markov evolution for control theory and [] shows that Schrödinger bridges between Gaussians against reference measures induced by linear SDEs have a closed forms.

Notation
For probability measures $\mu,\,\nu$, the relative entropy (e.g. the KL divergence) is $H(\mu|\nu)=\int\log(d\mu/d\nu)d\mu$ if $\mu\ll\nu$ and $+\infty$ otherwise. For $n\in\mathbb{N}$, let $[n]:=\{1,\dots,n\}$. We use $\mathcal{X}$ to denote the *latent* space and $\mathcal{Y}$ to denote the *observation* space. We use the notation $\mathcal{P}(\cdot)$ to denote the probability distributions over a space. The path space is $\Omega=C([0,1]:\mathcal{X})$, the set of continuous $\mathcal{X}$-valued paths. In our theoretical discussion, as in [], we assume without loss of generality that the end time interval is $t=1$. If $\mathbf{R}\in\mathcal{P}(\Omega)$ is a probability measure on the space of paths, its marginal at time $t$ is denoted as $\mathbf{R}_{t}\in\mathcal{P}(\mathcal{X})$. We generally use the Greek letters $\mu,\rho$ to denote probability distributions on $\mathcal{X},\mathcal{Y}$, respectively. We use $\delta_{x}$ to denote a Dirac delta at $x$. By an abuse of notation, we use $|\cdot|^{2}=\langle\cdot,\cdot\rangle\in\mathbb{R}$ for both the squared norm of a vector and the quadratic variation of a stochastic process. For clarity, we use the notation $g_{\sharp}\cdot$ when applied to measures and $g(\cdot)$ when applied to random variables (similarly for $\xi$). We use $\mathrm{vol}$ to denote the uniform measure on $\mathcal{X}$.

## 2 Latent Trajectory Inference

Let $X_{t}\in\mathcal{X}$ be an unobserved state vector evolving according to the following SDE for $t\in[0,1]$:

$$ $dX_{t}=-\Xi(t,X_{t})dt-\nabla\Psi(t,X_{t})dt+\sqrt{\tau}dB_{t},$ (1) $$

where $\{B_{t}\}$ is a Brownian motion, $\tau$ is the *known* diffusivity parameter, $\Xi\in C([0,1]\times\mathcal{X}:\mathcal{X})$ is a *known* driving vector field, and $\Psi\in C^{2}([0,1]\times\mathcal{X})$ is an *unknown* potential function. Let $\mathbf{P}$ be the law of the SDE with initial condition $\mathbf{P}_{0}$ where $\mathbf{P}_{t}\in\mathcal{P}(\mathcal{X})$ are the marginals of $\mathbf{P}$ at time $t\in[0,1]$.(^1^11Our SDE differs from that of [, ] with the addition of non-zero $\Xi$.)

Consider a smooth function $g:\mathcal{X}\rightarrow\mathcal{Y}$ transforming $X_{t}$ into the observation space $\mathcal{Y}$: $Y_{t}=g(X_{t})$.
Suppose we have $T$ observation times with $0\leq t_{1}^{T}<\cdots<t_{T}^{T}\leq 1$, and we observe $N_{i}^{T}$ i.i.d. samples from the marginal distribution of $Y_{t_{i}^{T}}$:

$$ $\{Y_{i,j}^{T}\}_{j=1}^{N^{T}_{i}}\overset{\text{i.i.d.}}{\sim}g_{\sharp} \mathbf{P}_{t_{i}^{T}}:=\mathbf{Q}_{t_{i}^{T}},$ $$

forming empirical distributions $\hat{\rho}_{i}^{T}=\sum_{j=1}^{N^{T}_{i}}\delta_{Y_{i,j}^{T}}$.

The goal is to recover $\mathbf{P}$ given the snapshots $(\hat{\rho}_{1}^{T},\dots,\hat{\rho}_{T}^{T})$. In general, to make this problem well-posed and tractable, we make several assumptions on this very general setup. The first of these is a notion of observability that generalizes(^2^22We include stochasticity ($\tau>0$) and allow for unknown $\Psi$.) the ensemble observability introduced in []:

###### Definition 1 ( $\mathcal{C}_{\Psi}$ -ensemble observability) .

Assume that $\Psi$ is unknown but restricted to a class $\mathcal{C}_{\Psi}$. We say the tuple $(g,\Xi,\mathcal{C}_{\Psi})$ is *$\mathcal{C}_{\Psi}$-ensemble observable* if, given $g$, $\Xi$, $\tau$, and all marginals $\mathbf{Q}_{t}=g_{\sharp}\mathbf{P}_{t}$ of $Y_{t}$ for all $t\in[0,1]$, the marginals $\mathbf{P}_{t}$ of $X_{t}$ are uniquely determined for all $t\in[0,1]$.

With this observability assumption, we can infer the latent dynamics solely from the marginals $\mathbf{Q}_{t}$. A discussion of the relationship between this condition and that of classical/ensemble observability is present in Appendix A. There, we also verify the conditions of Definition 1 for several important setups, e.g., the key velocity-based dynamics model we use in our experiments.

Theoretical assumptions
Let $\mathcal{X},\mathcal{Y}$ be Polish spaces, where $\mathcal{X}$ is a smooth and compact Riemannian manifold or a compact and convex subset of $\mathbb{R}^{d}$. In the manifold setting, we assume its Ricci curvature $K$ is bounded from below, e.g. $K>-\infty$.

The path space $\Omega=C([0,1]:\mathcal{X})$ is equipped with the uniform topology and its Borel $\sigma$-algebra. The probability space on paths $\mathcal{P}(\Omega)$ is equipped with the weak topology, e.g. convergence against bounded, continuous functions. Assume our probability space $(\Omega,\mathcal{F},\mathbb{P})$ is complete and filtered, where the filtration is with respect to the process $\{X_{t}\}$. $\mathbb{P}$ is a probability measure, and if it is not specified, expectations are taken with respect to $\mathbb{P}$. Let $\mathbf{W}^{\Xi,\tau}$ be the measure induced by the SDE $dZ_{t}=-\Xi(t,Z_{t})\,dt+\sqrt{\tau}\,dB_{t}$.

###### Assumption 2.1 (Dynamics and Observation Model) .

Assume $\Xi:C([0,1]\times\mathcal{X}:\mathcal{X})$ is *known*, divergence-free, Lipschitz continuous, and satisfies $\|\Xi\|_{L^{\infty}}<+\infty$. Assume the observation function $g:\mathcal{X}\to\mathcal{Y}$ is smooth, measurable, bounded, and time invariant.

The divergence-free assumption is required
so that the time marginals of $\mathbf{W}^{\Xi,\tau}$ remain $\mathrm{vol}$ for all time if the initial condition is $\mathrm{vol}$(^3^33Note that if $\mathcal{X}$ has a boundary, we need a zero flux condition on $\Xi$, but we do not consider this in our theoretical analysis.). Lipschitz continuity and $\|\Xi\|_{L^{\infty}}<+\infty$ are technical conditions necessary for our proofs, and there are also some necessary mild technical conditions on the pair $(\Xi,\Psi)$. These assumptions are discussed in Appendix B. Finally, by bounded for $g$, we mean the image of a set of finite measure also has finite measure.

###### Definition 1 ( 𝒞 Ψ subscript 𝒞 Ψ \mathcal{C}_{\Psi} caligraphic_C start_POSTSUBSCRIPT roman_Ψ end_POSTSUBSCRIPT -ensemble observability) .

###### Assumption 2.1 (Dynamics and Observation Model) .

## 3 PO-MFL: Approximate Minimum Entropy Estimation

Inspired by [], we use *minimum entropy estimation* as the fundamental tool for connecting temporal snapshots into continuous trajectories, where the entropy is the relative entropy (KL divergence) of the estimated trajectory distribution with respect to the known portion of the SDE. In other words, we estimate the trajectory distribution by maximizing its log-likelihood with respect to the distribution induced by the SDE, subject to matching the observed marginals. We will show that the optimal point of the minimum entropy objective function converges to the ground truth trajectory distribution.

It is not practical, however, to directly work with the trajectory distribution as it is an infinite-dimensional object. In what follows, we will show that the minimum entropy objective in trajectory space can be reduced to an OT-based objective, where marginals at adjacent time points are connected via entropic OT.

This reduction allows us to perform trajectory inference using only representations of the latent space time marginals, which can be accomplished via sets of particles. These particles can then be optimized via MFL dynamics.

### 3.1 Minimum entropy objective function

In this section, we specify the minimum entropy objective function on the trajectory distribution. Let $\{t_{i}^{T}\}\subset[0,1]$ be our observation times, where $\Delta t_{i}:=t_{i+1}^{T}-t_{i}^{T}$. Recall that in general, we do not have exact measurement of the temporal marginals, we will only have samples from them. As a result, we must introduce a *fit function* to measure the discrepancy between the observation space time marginals of the estimated trajectory distribution $\mathbf{R}$ and the observed samples.

Let the observed empirical distribution smoothed by the $h$-wide heat kernel $\Phi_{h}$ be $\hat{\rho}_{i}^{T,h}:=\Phi_{h}\left(\frac{1}{N_{i}^{T}}\sum_{i=1}^{N_{i}^{T}}
\delta_{Y_{i,j}^{T}}\right)\in\mathcal{P}(\mathcal{Y})$ for $i\in[T]$.(^4^44This smoothing $h$ aids the proofs and will be taken to a limit of zero in the following theoretical results.)
Consider the fit function $\mathrm{Fit}^{\lambda,\sigma}:\mathcal{P}(\mathcal{Y})^{T}\to\mathbb{R}$:

$$ $\mathrm{Fit}^{\lambda,\sigma}(\mathbf{Q}_{t_{1}^{T}},\dots,\mathbf{Q}_{t_{T}^{ T}}):=\frac{1}{\lambda}\sum_{i=1}^{T}\Delta t_{i}\mathrm{DF}^{\sigma}(g_{ \sharp}\mathbf{R}_{t^{T}_{i}},\hat{\rho}_{i}^{T,h}),$ $$

with data-fitting term introduced by [] augmented by observation function $g$ to be

$$ $\displaystyle\mathrm{DF}^{\sigma}(g_{\sharp}\mathbf{R}_{t^{T}_{i}},\hat{\rho}_ {i}^{T,h})$ $\displaystyle:=\int_{\mathcal{Y}}-\log\left[\int_{\mathcal{X}}\exp\left(-\frac {\|g(x)-y\|^{2}}{2\sigma^{2}}\right)d\mathbf{R}_{t^{T}_{i}}(x)\right]d\hat{ \rho}_{i}^{T,h}(y)$ $\displaystyle=H(\hat{\rho}_{i}^{T,h}|g_{\sharp}\mathbf{R}_{t_{i}^{T}}*\mathcal {N}_{\sigma})+H(\hat{\rho}_{i}^{T,h})+C,$ $$

where $\mathcal{N}_{\sigma}$ is the Gaussian kernel with variance $\sigma^{2}$, $C>0$ is a constant, and we use the substitution $\mathbf{Q}_{t_{i}^{T}}=g_{\sharp}\mathbf{R}_{t_{i}^{T}}$.(^5^55Here, note that the inner integral is over $\mathcal{X}$ as the optimization will occur on the latent space, while the inner integral is over $\mathcal{Y}$ as the observations are over $\mathcal{Y}$.) Note that this is the negative log-likelihood under the noisy observation model $\hat{Y}_{i,j}^{T}=g(X_{i,j}^{T})+\sigma Z_{i,j}$, where $\hat{Y}_{i,j}^{T}$ is the observation and $Z_{i,j}\overset{i.i.d.}{\sim}\mathcal{N}(0,I)$. It is easy to see that $\mathrm{DF}^{\sigma}$ is jointly convex in $(\mathbf{R}_{t_{i}^{T}},\hat{\rho}_{i}^{T,h})$ and linear in $\hat{\rho}_{i}^{T,h}$. The main difference compared to [] is that our data-fitting term is in observation space, e,g. the addition of the function $g$. We briefly mention that the data-fitting term in [] is $H(\mathcal{N}_{\sigma}*\hat{\mu}_{i}^{T,h}|\mathbf{R}_{t_{i}^{T}})$. The data-fitting term introduced by [] (and thus ours also) is computationally more effective when $\mathbf{R}_{t_{i}^{T}}$ is a discrete measure and thus is more amenable for the MFL dynamics. This difference in data-fitting term does not yield major changes for the theoretical results, which can be seen in Appendix B.

The minimum entropy estimator introduced in [] is the minimizer of the functional $\mathcal{F}:\mathcal{P}(\Omega)\to\mathbb{R}$

$$ $\mathcal{F}(\mathbf{R}):=\mathrm{Fit}^{\lambda,\sigma}(\mathbf{Q}_{t_{1}^{T}}, \dots,\mathbf{Q}_{t_{T}^{T}})+\tau H(\mathbf{R}|\mathbf{W}^{\Xi,\tau}).$ (2) $$

Recall that our key novelties are the fit term in *observation space* and entropy minimization in path space with respect to *divergence-free, Markov* path measures. Nonetheless, we show that we can still recover the ground truth in the limit as the number of observations becomes dense.

###### Theorem 3.1 (Consistency (informal, see Thm. B.1 )) .

Suppose $\mathbf{P}$ is the SDE given in (1) with initial condition $\mathbf{P}_{0}\in\mathcal{P}(\mathcal{X})$ such that $H(\mathbf{P}_{0}|\mathrm{vol})<+\infty$. Let $\mathbf{R}^{T,\lambda,h}\in\mathcal{P}(\Omega)$ be the unique minimizer of (2):

$$ $\mathbf{R}^{T,\lambda,h}:=\underset{{\mathbf{R}\in\mathcal{P}(\Omega)}}{\arg \min}\,\mathcal{F}(\mathbf{R}).$ $$

Then, we have the weak convergence

$$ $\lim_{h\to 0,\lambda\to 0}\left(\lim_{T\to\infty}\mathbf{R}^{T,\lambda,h} \right)=\mathbf{P}.$ $$

This result parallels Theorem 2.3 in [], which provides a consistency result in the fully observed setting where the entire state vector $X_{t}$ is observed and $\Xi=0$ identically.
Due to these differences, the result in [] cannot be directly applied to our setting, and while our proof is able to follow a similar overall structure, dense and nontrivial changes throughout the extensive proof are required. These arguments can be found in Appendix B.

###### Theorem 3.1 (Consistency (informal, see Thm. B.1 )) .

### 3.2 The reduced problem

As in [], we need to “reduce” our problem over the space $\mathcal{P}(\mathcal{X})^{T}$ to use the MFL dynamics []. As before, let $\Delta t_{i}:=t_{i+1}^{T}-t_{i}^{T}$ and $\tau_{i}:=\Delta t_{i}\cdot\tau$.
Consider the following entropic OT cost, defined for some $\tau_{i}>0$, as

$$ $T_{\tau_{i},\Xi}(\mu,\nu):=\min_{\gamma\in\Pi(\mu,\nu)}\int c_{\tau_{i}}^{\Xi} (x,y)\,d\gamma(x,y)+\tau_{i}H(\gamma|\mu\otimes\nu)=\min_{\gamma\in\Pi(\mu,\nu )}\tau_{i}H(\gamma|p_{\tau_{i}}^{\Xi}\mu\otimes\nu),$ (3) $$

where $p_{t}^{\Xi}$ is the transition probability density of $\mathbf{W}^{\Xi,\tau}$ over the time interval $[0,t]$ and the cost function is $c_{\tau_{i}}^{\Xi}(x,y):=-\Delta t_{i}\log(p_{\tau_{i}}^{\Xi}(x,y))$. Note that in general, $p_{\tau_{i}}^{\Xi}$ cannot be found explicitly, so we will discuss how to approximate this below. We recall from [, App. A] that this optimization problem is a Schrödinger bridge problem.(^6^66The classical Schrödinger bridge problem [] gives the most likely evolution of a probability distribution following a Brownian motion, conditioned on the endpoints $\mu$ at time $t=0$ and $\nu$ at time $t=1$, e.g. the reference measure is the Wiener measure. [, ] introduce the Schrödinger bridge problem as an entropy minimization problem against general reference measures. This is the formulation we use.) Define the functional $G:\mathcal{P}(\mathcal{X})^{T}\to\mathbb{R}$ for $\boldsymbol{\mu}=(\boldsymbol{\mu}^{(1)},\dots,\boldsymbol{\mu}^{(T)})$ that represents a family of reconstructed temporal marginals, by

$$ $G(\boldsymbol{\mu}):=\mathrm{Fit}^{\lambda,\sigma}(g_{\sharp}\mathbf{ \boldsymbol{\mu}})+\sum_{i=1}^{T-1}\frac{1}{\Delta t_{i}}T_{\tau_{i},\Xi}( \boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(i+1)}).$ (4) $$

We consider the *reduced objective* $F:\mathcal{P}(\mathcal{X})^{T}\to\mathbb{R}$, defined as

$$ $F(\boldsymbol{\mu}):=G(\boldsymbol{\mu})+\tau H(\boldsymbol{\mu}),$ (5) $$

where $H(\boldsymbol{\mu})=\sum_{i=1}^{T}\int\log(\boldsymbol{\mu}^{(i)})d\boldsymbol
{\mu}^{(i)}$ is the minus differential entropy of the family of measures $\boldsymbol{\mu}$. Similar to [], we have an equivalence of minimizing $\mathcal{F}$ (2), the objective in path space $\mathcal{P}(\Omega)$, and $F$ (5), the reduced objective over $\mathcal{P}(\mathcal{X})^{T}$. The proof is provided in Appendix C.

###### Theorem 3.2 (Representer theorem) .

Let $\mathrm{Fit}:\mathcal{P}(\mathcal{Y})^{T}\to\mathbb{R}$ be *any* function and let $\Xi$ be bounded and divergence-free.

- (i)
If $\mathcal{F}$ admits a minimizer $\mathbf{R}^{*}$ then $(\mathbf{R}_{t_{1}^{T}}^{*},\dots,\mathbf{R}_{t_{T}^{T}}^{*})$ is a minimizer for $F$.
- (ii)
If $F$ admits a minimizer $\boldsymbol{\mu}^{*}\in\mathcal{P}(\mathcal{X})^{T}$, then a minimizer $\mathbf{R}^{*}$ for $\mathcal{F}$ is built as
$\mathbf{R}^{*}(\cdot)=\int_{\mathcal{X}^{T}}\mathbf{W}^{\Xi,\tau}(\cdot|x_{1},
\dots,x_{T})\,d\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}(x_{1},\dots,x_{T}),$
where $\mathbf{W}^{\Xi,\tau}(\cdot|x_{1},\dots,x_{T})$ is the law of $\mathbf{W}^{\Xi,\tau}$ conditioned on passing through $x_{1},\dots,x_{T}$ at times $t_{1}^{T},\dots,t_{T}^{T}$, respectively and $\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}$ is the composition of the optimal transport plans $\gamma_{i}$ that minimize $T_{\tau_{i},\Xi}(\boldsymbol{\mu}^{*(i)},\boldsymbol{\mu}^{*(i+1)})$, for $i\in[T-1]$.

Note that the composition of the transport plans is obtained as:

$$ $\mathbf{R}_{t_{i},\dots,t_{T}}(dx_{1},\dots,dx_{T})=\gamma_{1}(dx_{1},dx_{2}) \gamma_{2}(dx_{3}|x_{2})\cdots\gamma_{T-1}(dx_{T}|x_{T-1}),$ (6) $$

where the OT plans $\gamma_{i}(dx_{i},dx_{i+1})=\gamma_{i}(dx_{i+1}|x_{i})\mu_{i}(dx_{i})$ are conditional probabilities (or “disintegrations”).
As in [], the “reduction” of the optimization space from $\mathcal{P}(\Omega)$ to $\mathcal{P}(\mathcal{X})^{T}$ is enabled by the Markov property of $\mathbf{W}^{\Xi,\tau}$, which holds for us due to the Lipschitz continuity assumption on $\Xi$ and that $\mathbf{W}^{\Xi,\tau}$ remains the uniform measure at all time.
Theorem 3.2 allows us to compute a minimizer for $\mathcal{F}$ from a minimizer for $F$ and its associated OT plans.

###### Theorem 3.2 (Representer theorem) .

### 3.3 Approximating the entropic OT cost

Although now we have a reduced problem over $\mathcal{P}(\mathcal{X})^{T}$, we still cannot solve the objective function $F$ (5), as the entropic OT cost $T_{\tau_{i},\Xi}(\mu,\nu)$ is defined with the transition function $p_{\tau_{i}}^{\Xi}$, which is generally not available in closed form. Here, we give an approximation of $T_{\tau_{i},\Xi}(\mu,\nu)$ by considering an Euler-Maruyama discretization []. Let $\mu\in\mathcal{P}(\Omega)$ be a stochastic process following the SDE $dX_{t}=-\Xi(t,X_{t})\,dt+\sqrt{\tau}\,dB_{t}$ with an arbitrary initial distribution. Let $\Delta t:=t_{2}-t_{1}$ and suppose $\mu_{t_{1}},\mu_{t_{2}}$ are two time marginals of $\mu$.
Recall that if $X_{t_{1}}\sim\mu_{t_{1}}$, and

$$ $X_{t_{2}}:=X_{t_{1}}-\int_{t_{1}}^{t_{2}}\Xi(s,X_{s})\,ds+\sqrt{\tau}\int_{t_{ 1}}^{t_{2}}dB_{s},$ (7) $$

then $X_{t_{2}}\sim\mu_{t_{2}}$.
For small $\Delta t=t_{2}-t_{1}$, since $\Xi$ and $\mu_{s}$ are smooth, the integrand of the second term will be approximately constant over the integration interval. Thus, we can approximate the first two terms of (7) as

$$ $\xi^{\Delta t}(X_{t_{1}}):=X_{t_{1}}-\Xi(t_{1},X_{t_{1}})\cdot\Delta t.$ $$

Finally, note that the last term of (7) is an isotropic Gaussian with variance $\tau\Delta t$.

This suggests approximating the transition kernel $p_{\tau_{i}}^{\Xi}$ as a deterministic drift given by the current $\Xi$, followed by isotropic Gaussian noise. For $\Xi=0$, this would reduce to the kernel used by the MFL method, i.e. the Brownian motion transition kernel.

This provides an intuition for why our approach is more robust than that of the MFL method when the true $\Xi$ is non-zero, since $\xi^{\Delta t}(X_{t_{1}})-X_{t_{2}}\approx\mathcal{N}(0,\tau\Delta t)$ for small $\Delta t$, while the $X_{t_{1}}-X_{t_{2}}$ used by the MFL algorithm has non-zero expectation, is non-Gaussian, and often has significantly higher variance.(^7^77In a sense, our Euler-Maruyama approximation can be considered a *first order* approximation method, while MFL corresponds to a *zeroth order* method. Higher order methods could be an avenue for future work.)

Let $\operatorname{TV}(\cdot,\cdot)$ denote the total variation distance. We have the following result for using the above approximation in the transition kernel, which is a special case of [, Thm. 2.1]. Note that this implies that the difference in probability between the approximate kernel and true kernel for any event is of order $O((\Delta t)^{1/3})$, which converges to 0 as $T\to\infty$.

###### Proposition 3.3 .

Let $X_{t_{2}}$ be as in (7) and $\tilde{X}_{t_{2}}:=\xi^{\Delta t}(X_{t_{1}})+\sqrt{\tau}(B_{t_{2}}-B_{t_{1}})$, where $X_{t_{1}}=\delta_{x}$. Then,

$$ ${\operatorname{TV}}(X_{t_{2}},\tilde{X}_{t_{2}})\leq C_{1}e^{C_{2}|x|^{2}}( \Delta t)^{1/3},$ $$

where the constants $C_{1},C_{2}>0$ depend only on $\operatorname{dim}\mathcal{X}$ and the Lipschitz constant of $\Xi$.

Applying this approximate transition kernel yields updated, tractable OT terms in the objective function. Specifically, instead of $T_{\tau_{i},\Xi}(\boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(i+1)})$ in (4), we use $T_{\tau_{i}}(\xi_{\sharp}^{t_{i+1}-t_{i}}\boldsymbol{\mu}^{(i)},\boldsymbol{
\mu}^{(i+1)})$, where

$$ $T_{\tau_{i}}(\mu,\nu):=\min_{\gamma\in\Pi(\mu,\nu)}\tau_{i}H(\gamma|p_{\tau_{i }}\mu\otimes\nu)$ $$

is the entropic OT cost in [, Eq. 6] and $p_{t}(x,y)$ is the transition probability density of the Brownian motion on $\mathcal{X}$ over the time interval $[0,t]$. This cost is easily computed as $p_{\tau_{i}}$ is the Gaussian kernel.

Above, we consider the pushforward of the first marginal for ease of analysis. In our implementation and experiments, we use $T_{\tau_{i}}(\xi_{\sharp}^{\Delta t/2}\boldsymbol{\mu}^{(i)},\xi_{\sharp}^{-
\Delta t/2}\boldsymbol{\mu}^{(i+1)})$, e.g. we pushforward the first marginal and pullback the second marginal. This entropic OT problem has the corresponding cost function $\tilde{c}_{\tau_{i}}^{\Xi}(x,y):=-\Delta t_{i}\log(p_{\tau_{i}}(\xi^{\Delta t/
2}(x),\xi^{-\Delta t/2}(y)))$, and we use Varadhan’s approximation []

$$ $\tilde{c}_{\tau_{i}}^{\Xi}(x,y)\approx\frac{1}{2}\left\|y-\frac{\Delta t}{2} \Xi(t_{2},y)-x+\frac{\Delta t}{2}\Xi(t_{1},x)\right\|^{2},$ $$

which holds for $\tau_{i}$ small, e.g. see Algorithm 1. To wrap up our discussion here, it is important to highlight that we require a generalization of [, Thm. 2.3] using our path measure $\mathbf{W}^{\Xi,\tau}$,
Theorem 3.1, to justify convergence of our estimator when including $\Xi$ in our cost function in the entropic OT problem (3).

###### Proposition 3.3 .

### 3.4 Mean-field Langevin dynamics and exponential convergence

We provide a brief description of the MFL dynamics. See Appendix D for a more complete discussion. MFL dynamics are designed to minimize functionals of the form $F_{\epsilon}=G+(\tau+\epsilon)H$, where $G:\mathcal{P}(\mathcal{X})\to\mathbb{R}$ is smooth and $H$ is minus the differential entropy.
As in [], we increase the entropy factor by $\epsilon>0$ because $G$ is not convex, but $F_{0}=G+\tau H$ is. Using the first-variation $V[\boldsymbol{\mu}]$ of $G$ given in Proposition D.1, the MFL dynamics is defined as the solution of the following non-linear McKean-Vlasov SDE, for $s\geq 0$:

$$ $\begin{cases}dX_{s}^{(i)}=-\nabla V^{(i)}[\boldsymbol{\mu}_{s}](X_{s}^{(i)})\, ds+\sqrt{2(\tau+\epsilon)}\,dB_{s}^{(i)}+d\Phi_{s}^{(i)},&\mathrm{Law}(X_{0}^{ (i)})=\boldsymbol{\mu}_{0}^{(i)}\\ \boldsymbol{\mu}_{s}^{(i)}=\mathrm{Law}(X_{s}^{(i)}),&i\in[T],\end{cases}$ (8) $$

where $d\Phi_{s}^{(i)}$ is the boundary reflection in the sense of the Skorokhod problem, e.g. []. Here, $\Xi$ *does not, and should not* show up in (8) as we already consider $\Xi$ for the entropic OT problem (3) that induces the Schrödinger potentials. Thus, the McKean-Vlasov SDE is exactly the same as that of []. For computation, we discretize the MFL dynamics via noisy particle gradient descent, which we discuss in Appendix D.2.

In [, ], it is shown that the MFL dynamics in (8) converges at an exponential rate to the minimizer, and this convergence also holds in relative entropy and in Wasserstein distance. We provide the proof for the following similar result for our partially observed setting in Appendix D.

###### Theorem 3.4 (Convergence) .

Assume $\mathcal{X}$ is the $d$-torus.
Let $\boldsymbol{\mu}_{0}\in\mathcal{P}(\mathcal{X})^{T}$ be such that $F(\boldsymbol{\mu}_{0})<+\infty$. Then for $\epsilon\geq 0$, there exists a unique solution $(\boldsymbol{\mu}_{s})_{s\geq 0}$ to the MFL dynamics (8). Let $\epsilon>0$ and assume that $\boldsymbol{\mu}_{0}$ has a bounded absolute log-density, it holds

$$ $F_{\epsilon}(\boldsymbol{\mu}_{s})-\min F_{\epsilon}\leq e^{-Cs}(F_{\epsilon}( \boldsymbol{\mu}_{0})-\min F_{\epsilon}),$ $$

where $C=\beta e^{-\alpha/\epsilon}$ for some $\alpha,\beta>0$ independently of $\boldsymbol{\mu}$ and $\epsilon$. Moreover, taking a smooth time-dependent $\epsilon_{s}$ that decays asymptotically as $\tilde{\alpha}/\log s$ for some $\tilde{\alpha}>\alpha$, it holds $F_{0}(\boldsymbol{\mu}_{s})-F_{0}(\boldsymbol{\mu}^{*})\lesssim\log\log s/\log
s\to
0$ and $\boldsymbol{\mu}_{s}$ converges weakly to the min-entropy estimator $\boldsymbol{\mu}^{*}$.

###### Theorem 3.4 (Convergence) .

### 3.5 PO-MFL

We summarize our proposed latent trajectory inference method in Algorithm 1. We recall that discussion on the cost function (line 4) and MFL dynamics (line 7) can be found in Sections 3.3 and 3.4, respectively. We use the Sinkhorn algorithm [] for entropic OT.

Figure: Algorithm 1 PO-MFL: framework for latent trajectory inference

Figure: Figure 2: Average $W_{2}$ distance between ground truth and PO-MFL recovered positions in “constant velocity” model from Appendix E. (left) Number of time points. (middle) Number of observations. (right) Velocity.
Refer to caption: extracted/5644114/figures/output.png

Given the set of observed temporal marginal samples in the *observation* space $\mathcal{Y}$, Algorithm 1 yields a set of $m$ particles at each time step $i$ representing the temporal marginal distributions in the *latent* space $\mathcal{X}$. Simulated trajectories may be recovered by sampling from the composition of entropic transport plans as shown in (6).

## 4 Experiments

In this section, we briefly provide synthetic experiments that demonstrate the advantages of having a dynamics prior. All experiments were run on an M1 Macbook Air with 16 GB of RAM, and each experiment takes no more than a few minutes each to run. More details on the experimental setup can be found in Appendix E.

Figure: Figure 3: (left) Velocity of one particle at end of optimization. (right) Population velocity at beginning of optimization, showing exponential convergence.
Refer to caption: extracted/5644114/figures/particle_velocity.png

“Constant velocity” model We compare the behavior of our method, PO-MFL, to that of MFL, using the “constant velocity” model popular in target tracking [], see Appendix A.1 for further details of this model and its ensemble observability.(^8^88This model can be interpreted as introducing velocity as a hidden state to be inferred, in order to build *momentum* into the dynamics (an object in motion tends to stay in motion). This is an extremely generic model and makes minimal assumptions on the underlying data, as evidenced by its use in target tracking.) In this model, the state space is
$X=(x,y,\dot{x},\dot{y})\in\mathbb{R}^{4}$, with $\Xi$ given in the appendix and observations $g(X)=[I_{2},0_{2\times 2}]X$. Note that due to non-zero process noise $\tau$, despite the name, this model does not imply that the velocity is constant in time. The particles are initialized at the origin with velocities set as $\dot{x}=5$ and $\dot{y}=7$, i.e. $X_{0}=(0,0,5,7)$. The ground truth is shown in Figure 1(a).

Figure: (a) Ground truth.
Refer to caption: extracted/5644114/figures/bifurcated_ground_truth.png

Our optimization method observes only the positions of the particles, i.e. $g(x,y,\cdot,\cdot)=(x,y)$, but it uses $\Xi$ as being a constant velocity prior. Results shown in Figure 1(b) show that PO-MFL is able to successfully reconstruct the paths trajectories, while MFL fails to converge. Furthermore, in Figure 1(c), we verify that the population average of the particles’ velocities matches with the ground truth.

Figure 3 (left) displays the $y$ velocity of one particle for the last 500 iterations of optimization. Although at each iteration, the velocity is stochastic, we can see that the mean is at $7$. In Figure 3 (right), we plot the average $y$ velocity in the first 400 iterations of optimization, providing empirical evidence of the exponential convergence of our algorithm guaranteed by Theorem 3.4.

Figure 4 shows a crossing paths experiment where the population is divided into two groups, one moving right and down, and the other right and up, with their paths crossing in the middle. In this particularly illuminating regime, PO-MFL leverages the “constant velocity” model used in this section to distinguish the downward moving group from the upward moving group. Note that PO-MFL is not told a priori which samples belong to which group. While MFL here collapses to the centroid, we point out that even if its optimization was successful, the MFL would prefer $U$-shaped trajectories here rather than the correct straight-line trajectories, as it does not retain a hidden velocity state and only seeks to match adjacent time points by their relative position via entropic OT.

We provide a variety of additional experiments in Appendix E illustrating how performance changes as the number of observed particles, the spacing of time points, and the underlying ground truth initial velocity affects performance. Figure 2 shows the average $W_{2}$ distance between the ground truth positions and recovered positions (averaged by time point) across these experiments. Our approach remains significantly more robust as these parameters are varied compared to MFL.

Figure: (a) Ground truth.
Refer to caption: extracted/5644114/figures/ground_truth_circle.png

Circular motion model In our second model, the particles $(\theta,\dot{\theta},\ddot{\theta})\in\mathbb{S}\times\mathbb{R}^{2}$ represent a constant acceleration model on the unit circle, starting from the initial condition $(0,0.5,1)$. Here, we use angular velocity and angular acceleration. In this experiment, we only observe the position, e.g. $g(\theta,\cdot,\cdot)=\theta$. In Figure 5(a), we show the ground truth with position on the left and angular velocity on the right. In Figure 5(b), we show that PO-MFL successfully reconstruct the positions while although MFL converges, it does not recover the ground truth. In Figure 5(c), we show that the reconstructed velocity matches that of the ground truth in Figure 5(a).

## 5 Conclusion

We consider the problem of trajectory inference in latent space based on indirect observation, extending the theoretical analysis of the min-entropy estimator introduced in [] and the MFL dynamics algorithm introduced in []. Synthetic experiments were provided showing that the ability to include simple non-informative latent dynamics models, such as the “constant velocity” model, can dramatically improve the trajectory inference performance over the baseline MFL method, while also directly providing per-particle estimates of the hidden velocity state.

For future work, while we do here provide some flexibility for model misspecification via the unknown $\Psi$ potential, it would be interesting to further explore the stability of our method when the dynamics model $\Xi$ is misspecified. Further exploration of ensemble observability would also be a highly interesting fundamental direction to explore. Finally, we will seek to explore the various promising empirical use cases outlined in the introduction.

## Appendix A Ensemble Observability for Linear Systems

Recall that in classical observability [], the goal is to recover the dynamics of a single particle, while here we want to recover the dynamics of a probability distribution.
The notion of ensemble observability introduced in [] tackles this problem.
Consider the non-stochastic model with linear $\Xi(X)=A_{\Xi}X+B_{\Xi}$:

$$ $dX_{t}=-(A_{\Xi}X_{t}+B_{\Xi})\,dt$ (9) $$

with initial condition $\mathbf{P}_{0}$ and linear observations $Y_{t}=g(X_{t})=C_{g}X_{t}+D_{g}$.
For shorthand, we denote this system as $(A_{\Xi},B_{\Xi},C_{g},D_{g})$.

The following is the definition of ensemble observability as introduced in []: it does not consider stochasticity.

###### Definition 2 (Ensemble observability [ 56 , Def. 1] ) .

The linear system (9) is *ensemble observable* if given marginals $g_{\sharp}\mathbf{P}_{t}$ of $Y_{t}$ for all $t\in[0,1]$, the marginals $\mathbf{P}_{t}$ of $X_{t}$ are uniquely determined for all $t\in[0,1]$.

We can consider Definition 1 as an extension of ensemble observability. In particular, if we consider $\tau=0$ in Definition 1, we exactly recover ensemble observability. [] showed that classical observability is a necessary condition for ensemble observability, and provided several sufficient conditions as well. For a random variable $X$, we denote $\varphi_{X}$ to be its characteristic function. We assume the following on the initial distribution $X_{0}$.

###### Assumption A.1 .

Let $X_{0}$ be such that $s\mapsto\varphi_{X_{0}}(sv)$ is real-analytic for all non-zero $v\in\mathbb{R}^{n}$.

This assumption is not very strong and most “nice” distributions satisfy it, e.g. if they have a density. Recall that by [, Thm. 5.2], classical observability holds if and only if the observability matrix $O:=[C_{\Xi},C_{\Xi}A_{\Xi},\dots,C_{\Xi}A_{\Xi}^{n-1}]$ has rank $n$. [] provides two useful sufficient conditions(^9^99These are not the only concrete conditions provided therein. Furthermore, a more general sufficient condition is provided which is possible to check numerically.) for ensemble observability of such systems.

###### Proposition A.2 ( [ 56 , Thm. 8] ) .

Under Assumption A.1, if $(A_{\Xi},B_{\Xi},C_{g},D_{g})$ is observable and $\operatorname{rank}C_{g}=n-1$, then $(A_{\Xi},B_{\Xi},C_{g},D_{g})$ is ensemble observable.

###### Corollary A.3 ( [ 56 , Cor. 8] ) .

Under Assumption A.1, if $\mathcal{X}=\mathbb{R}^{2}$ and $(A_{\Xi},B_{\Xi},C_{g},D_{g})$ is observable, then $(A_{\Xi},B_{\Xi},C_{g},D_{g})$ is ensemble observable.

We now show that these two conditions can be carried over to stochastic systems, i.e., those with $\tau>0$. Consider adding stochasticity to (9) with the model

$$ $dX_{t}=-(A_{\Xi}X_{t}+B_{\Xi})\,dt+\sqrt{\tau}\,dB_{t}$ (10) $$

with initial condition $\mathbf{P}_{0}$, $\{B_{t}\}$ is an $\mathbb{R}^{n}$-valued Brownian motion, and observations $Y_{t}=C_{g}X_{t}+D_{g}$. We have the following result:

###### Corollary A.4 .

Suppose $(A_{\Xi},B_{\Xi},C_{g},D_{g})$ is ensemble observable (with $\tau=0$). Then the system (10) with known $\tau>0$ is ensemble observable.

###### Proof.

It is easy to see via direct calculation(^10^1010E.g. using an integrating factor.) that the solution to (10) is

$$ $X_{t}=e^{-A_{\Xi}t}X_{0}-\left(\int_{0}^{t}e^{-A_{\Xi}(t-s)}\,ds\right)B_{\Xi} +\sqrt{\tau}\int_{0}^{t}e^{-A_{\Xi}(t-s)}dB_{s},$ (11) $$

where we use matrix exponentials. Using arguments similar to those in [], we can characterize the covariance:

$$ $\Sigma_{t}:=\operatorname{Cov}\left[X_{t}\right]=\tau\int_{0}^{t}e^{-(A_{\Xi}+ A_{\Xi}^{\top})(t-s)}ds.$ $$

We also know that

$$ $\mu_{t}:=\mathbb{E}[X_{t}]=e^{-A_{\Xi}t}X_{0}-\left(\int_{0}^{t}e^{-A_{\Xi}(t- s)}\,ds\right)B_{\Xi}.$ $$

Then as the first two terms on the right-hand side of (11) have zero variance and the Itô integral of a deterministic integrand is normally distributed with mean zero, we know that (11) is distributed as

$$ $X_{t}\sim\mathcal{N}(\mu_{t},\Sigma_{t}).$ $$

As we know $\tau$, and the corresponding observability matrix to the system has full-rank, we see that the pushforward (to observation space) of every term in (11) is also fully recoverable as well. Note that it is possible to deconvolve(^11^1111E.g., using the fact that deconvolution is equivalent to division in the Fourier domain.) the known Gaussian noise from $X_{t}$, and hence the system is ensemble observable. This concludes the proof.
∎

Next, we extend Corollary A.3 to independent processes and apply Corollary A.4.

###### Proposition A.5 .

Let $\mathcal{X}=\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{n}$, where each $\mathcal{X}_{i}=\mathbb{R}^{2}$. Suppose $(A_{\Xi_{i}},B_{\Xi_{i}},C_{g_{i}},D_{g_{i}})$ is observable for each $i\in[n]$. Further, suppose the initial condition joint distribution for the $X_{0,i}$ satisfies Assumption A.1 with $[X_{0,i}]_{1},[X_{0,j}]_{2}$ conditionally independent conditioned on $[X_{0,i}]_{2},[X_{0,j}]_{1}$, for each $j\neq i$. If noise parameter $\tau>0$ is known and $\{B_{t}\}$ is an $\mathcal{X}$-valued Brownian motion, then the system

$$ $\left(\begin{pmatrix}A_{\Xi_{1}}\\ \vdots\\ A_{\Xi_{n}}\end{pmatrix},\begin{pmatrix}B_{\Xi_{1}}\\ \vdots\\ B_{\Xi_{n}}\end{pmatrix},\begin{pmatrix}C_{g_{1}}\\ \vdots\\ C_{g_{n}}\end{pmatrix},\begin{pmatrix}D_{g_{1}}\\ \vdots\\ D_{g_{n}}\end{pmatrix}\right)$ $$

is ensemble observable. Furthermore, the system remains ensemble observable under (known) permutations.

###### Proof.

This follows from a simple application of Corollaries A.3 and A.4.
∎

###### Definition 2 (Ensemble observability [ 56 , Def. 1] ) .

###### Assumption A.1 .

###### Proposition A.2 ( [ 56 , Thm. 8] ) .

###### Corollary A.3 ( [ 56 , Cor. 8] ) .

###### Corollary A.4 .

###### Proof.

###### Proposition A.5 .

###### Proof.

### A.1 Example: “Constant velocity” model

The two-dimensional “constant velocity” model, so named because the velocity would be constant if there were no process noise ($\tau=0$), uses a state vector

$$ $X=(x,y,\dot{x},\dot{y})\in\mathbb{R}^{4},$ $$

where here $(x,y)$ are two-dimensional positional coordinates, and $(\dot{x},\dot{y})$ is the current two-dimensional velocity. The “constant velocity” dynamics model uses $\Xi(X)=A_{\Xi}X$ where

$$ $A_{\Xi}=\left[\begin{array}[]{cccc}0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\end{array}\right],$ $$

which is a very simple matrix simply implying that the rate of change of $X_{1}=x$ is given by the current state vector $X_{3}=\dot{x}$, and similarly for $y$.

Having defined the dynamics, in this model, only the positions are observed. In other words, the observations $g(X)=[I_{2},0_{2\times 2}]X$, i.e. $C_{\Xi}=[I_{2},0_{2\times 2}]$ and $D_{\Xi}=0$.

Note that this experimental setting satisfies Proposition A.5 as the $x$ and $y$ dynamics are independent. Hence, it is sufficient to check if the system is observable. Here $C=[1,0]$ and $A=[0,1;0,0]$, so the observability matrix for each of these subsystems becomes

$$ $\left[\begin{array}[]{c}C\\ CA\end{array}\right]=\left[\begin{array}[]{cccc}1&0\\ 0&1\end{array}\right],$ $$

which is the identity and thus full rank. By observability theory, the system is classically observable, and by the results above, ensemble observable as well.

Note that ensemble observability can be extended to non-zero $\Psi$ in this setting. For instance, in the “constant velocity” model of the main text, $\nabla\Psi=[0;\psi]$ for constant but unknown $\psi$ will serve simply as a drift term on the mean of the hidden velocity state. Since without this drift the mean of the velocity is constant, this drift will be identifiable and the system will be ensemble observable.

Finally, as a brief remark, note that in Definition 1, we require $\Psi$ to be restricted to a class of functions $\mathcal{C}_{\Psi}$ as otherwise the SDE (1) may fail to satisfy classical observability. Further exploration of $\mathcal{C}_{\Psi}$ classes is left to future work.

## Appendix B Proofs for Consistency

We make this section as self-contained as possible, although we suppress some of the longer details when they are very similar to certain corresponding results in []. Whenever we do, we point to the fuller arguments in []. The main result in this section is the following:

###### Theorem B.1 (Consistency, (formal version of Thm. 3.1 )) .

Let $\mathbf{P}$ be the law of the SDE given in (1), restated below:

$$ $dX_{t}=-\Xi(t,X_{t})\,dt-\nabla\Psi(t,X_{t})\,dt+\sqrt{\tau}\,dB_{t},$ $$

with initial condition
$\mathbf{P}_{0}\in\mathcal{P}(\mathcal{X})$ such that $H(\mathbf{P}_{0}|\mathrm{vol})<+\infty$. Assume we have the following:

- (i)
$g:\mathcal{X}\to\mathcal{Y}$ is a smooth, measurable, bounded, time invariant function, and $(g,\Xi,\mathcal{C}_{\Psi})$ is $\mathcal{C}_{\Psi}$-ensemble observable.
- (ii)
For every $T\geq 1$, we have a sequence of ordered observation times $\{t_{i}^{T}\}_{i=1}^{T}$ between 0 0 and $1$, and $\{t_{i}^{T}\}_{i=1}^{T}$ becomes dense in $[0,1]$ as $T\to+\infty$.
- (iii)
For each $T$ and each $i\in[T]$, we have $N_{i}^{T}\geq 1$ random variables $\{Y_{i,j}^{T}\}_{j=1}^{N_{i}^{T}}$, which are i.i.d. and distributed according to $g_{\sharp}\mathbf{P}_{t_{i}^{T}}$.
- (iv)
The variables $Y_{i,j}^{T}$ and $Y_{i^{\prime},j^{\prime}}^{T^{\prime}}$ are sampled independently from their respective distributions except when $(T,i,j)=(T^{\prime},i^{\prime},j^{\prime})$.

Consider the functional (2), restated below:

$$ $\mathcal{F}(\mathbf{R}):=\mathrm{Fit}^{\lambda,\sigma}(g_{\sharp}\mathbf{R}_{t _{1}^{T}},\dots,g_{\sharp}\mathbf{R}_{t_{T}^{T}})+\tau H(\mathbf{R}|\mathbf{W} ^{\Xi,\tau}),$ $$

and let $\mathbf{R}^{T,\lambda,h}\in\mathcal{P}(\Omega)$ be its unique minimizer:

$$ $\mathbf{R}^{T,\lambda,h}:=\underset{{\mathbf{R}\in\mathcal{P}(\Omega)}}{\arg \min}\,\mathcal{F}(\mathbf{R}).$ $$

Then, we have the weak convergence

$$ $\lim_{h\to 0,\lambda\to 0}\left(\lim_{T\to\infty}\mathbf{R}^{T,\lambda,h} \right)=\mathbf{P},$ $$

almost surely.

###### Proof.

We use Theorem B.5 to take the limit $T\to+\infty$. By the law of large numbers and the weak convergence assumption, we have $\overline{\rho}_{t}=\Phi_{h}(\mathbf{P}_{t})$, almost surely. Define $\mathbf{R}^{\lambda,h}$ to be the limit of $\mathbf{R}^{T,\lambda,h}$ as $T\to+\infty$. By Theorem B.5, it is the unique minimizer of

$$ $\mathbf{R}\mapsto F_{\lambda,h}(\mathbf{R}):=\frac{1}{\lambda}\int_{0}^{1} \mathrm{DF}(\mathbf{R}_{t},\Phi_{h}\mathbf{P}_{t})\,dt+\tau H(\mathbf{R}| \mathbf{W}^{\Xi,\tau}).$ $$

By definition of the data-fitting term, the functional $G_{\lambda,h}$ in Theorem B.17 differs from $F_{\lambda,h}$ only by a constant. We see that

$$ $G_{\lambda,h}(\mathbf{R})=F_{\lambda,h}(\mathbf{R})-\int_{0}^{1}H(\Phi_{h} \mathbf{P}_{t})\,dt-C,$ $$

so $\mathbf{R}^{\lambda,h}$ must also be the unique minimizer for $G_{\lambda,h}$. Finally, we use Theorem B.17 to take the limit of $\mathbf{R}^{\lambda,h}$ as $h\to 0$ and $\lambda\to 0$. This concludes the proof.
∎

###### Theorem B.1 (Consistency, (formal version of Thm. 3.1 )) .

###### Proof.

### B.1 Variational characterization of the SDE

We recall some previously introduced preliminaries and notation. $\Omega=C([0,1]:\mathcal{X})$ is the set of $\mathcal{X}$-valued paths, $\{X_{t}\}_{t\in[0,1]}$ is our canonical process, and $\mathcal{F}$ is the Borel $\sigma$-algebra generated by the random variables $X_{s}$ for $s\leq t$ such that $\{\mathcal{F}_{t}\}_{t\in[0,1]}$ is a filtration. We use the notation $|\cdot|^{2}=\langle\cdot,\cdot\rangle\in\mathbb{R}$ for the quadratic variation of a process (similarly use this notation for cross-variation).

For the Girsanov transforms to be martingales, we have the following mild technical assumption.

###### Assumption B.2 (Novikov conditions on $\Xi$ ) .

Assume that the following Novikov conditions hold:

$$ $\mathbb{E}\left[\exp\left(\frac{1}{2}\int_{0}^{1}|\Xi|^{2}\,ds\right)\right]<+\infty$ $$

and

$$ $\mathbb{E}\left[\exp\left(\frac{1}{2}\int_{0}^{1}|\Xi+\nabla\Psi|^{2}\,ds \right)\right]<+\infty.$ $$

Also, assume that there exists $C<+\infty$ such that $\int_{0}^{1}|\Xi|^{2}\,ds\leq C$ and $\int_{0}^{1}|\Xi+\nabla\Psi|^{2}\,ds\leq C$.

This last condition is so that we can apply Girsanov’s on manifolds, e.g. [, Thm. 8.1.2].

###### Proposition B.3 (analogous to [ 33 , Prop. 2.11] ) .

Let $\mathbf{P}$ be the law of the SDE in (1). Then the Radon-Nikodym derivative of $\mathbf{P}$ with respect to $\mathbf{W}^{\Xi,\tau}$ is given $\mathbf{W}^{\Xi,\tau}$-a.e. by

$$ $\begin{split}&\frac{d\mathbf{P}}{d\mathbf{W}^{\Xi,\tau}}(X)=\frac{d\mathbf{P}_ {0}}{d\mathrm{vol}}(X_{0})\exp\left(\frac{\Psi(0,X_{0})-\Psi(1,X_{1})}{\tau} \right)\\ &\qquad\cdot\exp\left(\frac{1}{\tau}\left(\int_{0}^{1}\left(\partial_{s}\Psi- \frac{1}{2}|\nabla\Psi|^{2}-\langle\Xi,\nabla\Psi\rangle+\frac{\tau}{2}\Delta \Psi\right)(s,X_{s})\,ds\right)\right).\end{split}$ (12) $$

To prove this proposition, we do not use a martingale characterization as in [], but directly use the Girsanov theorem (which by our assumption on $\Xi$, can be applied on manifolds) and the Itô formula.

###### Proof.

By the chain rule, we have

$$ $\frac{d\mathbf{P}}{d\mathbf{W}^{\Xi,\tau}}(X)=\frac{d\mathbf{P}_{0}}{d\mathrm{ vol}}(X_{0})\cdot\frac{d\mathbf{P}}{d\mathbf{W}^{\tau}}\cdot\frac{d\mathbf{W}^ {\tau}}{d\mathbf{W}^{\Xi,\tau}}.$ $$

The first term follows from an averaging argument identical to that of [, Prop. 2.11]. For the second term, recall that $\mathbf{P}$ is the measure induced by the process $dX_{t}=-(\Xi+\nabla\Psi)\,dt+\sqrt{\tau}\,dB_{t}$ and $\mathbf{W}^{\tau}$ is the measure induced by the process $dY_{t}=\sqrt{\tau}\,dB_{t}$. We have

$$ $\displaystyle\frac{d\mathbf{P}}{d\mathbf{W}^{\tau}}$ $\displaystyle=\exp\left(\frac{1}{\sqrt{\tau}}\int_{0}^{t}(\Xi+\nabla\Psi)(s,X_ {s})\,dB_{s}-\frac{1}{2\tau}\int_{0}^{t}|\Xi+\nabla\Psi|^{2}(s,X_{s})\,ds\right)$ $\displaystyle=\exp\left(\frac{1}{\sqrt{\tau}}\int_{0}^{t}\nabla\Psi(s,X_{s})\, dB_{s}-\frac{1}{2\tau}\int_{0}^{t}|\nabla\Psi|^{2}(s,X_{s})\,ds\right)$ $\displaystyle\qquad\cdot\exp\left(\frac{1}{\sqrt{\tau}}\int_{0}^{t}\Xi(s,X_{s} )\,dB_{s}-\frac{1}{2\tau}\int_{0}^{t}(|\Xi|^{2}+2\langle\Xi,\nabla\Psi\rangle) (s,X_{s})\,ds\right)$ $\displaystyle=\exp\left(\frac{1}{\tau}\left(\Psi(0,X_{0})-\Psi(1,X_{1})+\int_{ 0}^{t}\left(\partial_{s}\Psi-\frac{1}{2}|\nabla\Psi|^{2}+\frac{\tau}{2}\Delta \Psi\right)(s,X_{s})\,ds\right)\right)$ $\displaystyle\qquad\cdot\exp\left(\frac{1}{\sqrt{\tau}}\int_{0}^{t}\Xi(s,X_{s} )\,dB_{s}-\frac{1}{2\tau}\int_{0}^{t}(|\Xi|^{2}+2\langle\Xi,\nabla\Psi\rangle) (s,X_{s})\,ds\right)$ $\displaystyle\begin{split}&=\exp\left(\frac{1}{\tau}\left(\Psi(0,X_{0})-\Psi(1 ,X_{1})+\sqrt{\tau}\int_{0}^{t}\Xi(s,X_{s})\,dB_{s}\right)\right),\\ &\qquad\cdot\exp\left(\frac{1}{\tau}\int_{0}^{t}\left(\partial_{s}\Psi-\frac{1 }{2}|\Xi+\nabla\Psi|^{2}+\frac{\tau}{2}\Delta\Psi\right)(s,X_{s})\,ds\right) \end{split}$ (13) $$

where the first line follows from the Girsanov theorem, [, Thm. 8.6.6] and the third line follows from Itô’s formula, [, Thm. 4.2.1]. Letting $\mathbf{W}^{\Xi,\tau}$ be the measure induced by the process $dZ_{t}=-\Xi(t,Z_{t})\,dt+\sqrt{\tau}\,dB_{t}$, we have

$$ $\frac{d\mathbf{W}^{\tau}}{d\mathbf{W}^{\Xi,\tau}}=\exp\left(\frac{1}{2\tau} \int_{0}^{t}|\Xi|^{2}\,ds-\frac{1}{\sqrt{\tau}}\int_{0}^{t}\Xi\,dB_{s}\right)$ (14) $$

by Girsanov. Combining (13) and (14) yields (12). The claim follows.
∎

The next result is the variational characteristic of the SDE.

###### Theorem B.4 (analogous to [ 33 , Thm. 2.1] ) .

Suppose $(g,\Xi,\mathcal{C}_{\Psi})$ is $\mathcal{C}_{\Psi}$-ensemble observable. Let $\Xi:[0,1]\times\mathcal{X}\to\mathcal{X}$ be a smooth function and $\Psi:[0,1]\times\mathcal{X}\to\mathbb{R}$ be a smooth potential. Let $\mathbf{P}$ be the law of the SDE

$$ $dX_{t}=-\Xi(t,X_{t})\,dt-\nabla\Psi(t,X_{t})\,dt+\sqrt{\tau}\,dB_{t}$ $$

with initial condition
$\mathbf{P}_{0}\in\mathcal{P}(\mathcal{X})$ such that $H(\mathbf{P}_{0}|\mathrm{vol})<+\infty$. If $\mathbf{R}\in\mathcal{P}(\Omega)$ is such that $g_{\sharp}\mathbf{R}_{t}=g_{\sharp}\mathbf{P}_{t}$ for all $t\in[0,1]$, we have

$$ $H(\mathbf{P}|\mathbf{W}^{\Xi,\tau})\leq H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})$ $$

with equality if and only if $\mathbf{P}=\mathbf{R}$.

The argument follows that of [] with our ensemble observable assumption and reference measure. Here, the proof is the same, but now we use the fact that our reference measure “cancels out” the stochastic integral, e.g. see Proposition B.3.

###### Proof.

Let $\mathbf{P}$ be the law of the solution of (1) and suppose $\mathbf{R}\in\mathcal{P}(\Omega)$ is another path measure such that $H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})<+\infty$. Let $p,r\in L^{1}(\Omega,\mathbf{W}^{\Xi,\tau})$ denote the Radon-Nikodym derivative of $\mathbf{P},\mathbf{R}$ with respect to $\mathbf{W}^{\Xi,\tau}$, respectively. By strict convexity of $x\mapsto x\log x$, we have

$$ $r\log r-p\log p\geq(1+\log p)(r-p),$ $$

$\mathbf{W}^{\Xi,\tau}$-almost everywhere, with equality if and only if $r=p$. Integrating with respect to $\mathbf{W}^{\Xi,\tau}$, we see

$$ $H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})-H(\mathbf{P}|\mathbf{W}^{\Xi,\tau})\geq \mathbb{E}_{\mathbf{R}}[1+\log p]-\mathbb{E}_{\mathbf{P}}[1+\log p].$ (15) $$

Using Proposition B.3, we have

$$ $\displaystyle\mathbb{E}_{\mathbf{R}}[1+\log p]=\mathbb{E}_{\mathbf{R}}\left[1+ \log\left(\frac{d\mathbf{P}_{0}}{d\mathrm{vol}}\right)(X_{0})+\frac{\Psi(0,X_{ 0})-\Psi(1,X_{1})}{\tau}\right.$ $\displaystyle\left.\qquad+\frac{1}{\tau}\int_{0}^{1}\left(\partial_{s}\Psi- \frac{1}{2}|\nabla\Psi|^{2}-\langle\Xi,\nabla\Psi\rangle+\frac{\tau}{2}\Delta \Psi\right)(s,X_{s})\,ds\right].$ $$

By definition of ensemble observability, if $g_{\sharp}\mathbf{R}_{t}=g_{\sharp}\mathbf{P}_{t}$ for all $t\in[0,1]$, this implies $\mathbf{R}_{t}=\mathbf{P}_{t}$ for all $t\in[0,1]$. Because this expression only depends on the temporal marginals of $\mathbf{R}$ as the Radon-Nikodym derivative of $\mathbf{P}$ with respect to $\mathbf{W}^{\Xi,\tau}$ does not contain a stochastic integral, the right-hand side of (15) vanishes if $g_{\sharp}\mathbf{R}_{t}=g_{\sharp}\mathbf{P}_{t}$ for all $t\in[0,1]$. This concludes the proof.
∎

###### Assumption B.2 (Novikov conditions on Ξ Ξ \Xi roman_Ξ ) .

###### Proposition B.3 (analogous to [ 33 , Prop. 2.11] ) .

###### Proof.

###### Theorem B.4 (analogous to [ 33 , Thm. 2.1] ) .

###### Proof.

### B.2 The main technical result: Theorem B.5

###### Theorem B.5 (analogous to [ 33 , Thm. 2.7] ) .

Fix $\lambda>0$ and assume we have the following:

- (i)
For every $T\in\mathbb{N}$, we have a sequence of ordered observation times $\{t_{i}^{T}\}_{i=1}^{T}$; a sequence of data smoothed by the heat-kernel $\hat{\rho}_{i}^{T}$ (a collection of $T$ probability measures on $\mathcal{Y}$); and a sequence of non-negative weights $\{\omega_{i}^{T}\}_{i=1}^{T}$.
- (ii)
There exists a $\mathcal{P}(\mathcal{Y})$-valued continuous curve $\overline{\rho}\in C([0,1]:\mathcal{P}(\mathcal{Y}))$ such that the following weak convergence holds: for all continuous functions $a:[0,1]\times\mathcal{Y}\to\mathbb{R}$,
$\lim_{T\to+\infty}\sum_{i=1}^{T}\omega_{i}^{T}\int_{\mathcal{X}}a(t_{i}^{T},x)
\hat{\rho}_{i}^{T}(dx)=\int_{0}^{1}\int_{\mathcal{X}}a(t,x)\overline{\rho}(dx)
\,dt.$

For each $T$, let $\mathbf{R}^{T}\in\mathcal{P}(\Omega)$ be the unique minimizer of

$$ $\mathbf{R}\mapsto F_{T}(\mathbf{R}):=\tau H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})+ \frac{1}{\lambda}\sum_{i=1}^{T}\omega_{i}^{T}\mathrm{DF}(g_{\sharp}\mathbf{R}_ {t_{i}^{T}},\hat{\rho}_{i}^{T}).$ (16) $$

Then as $T\to+\infty$, the sequence $\{\mathbf{R}^{T}\}$ converges weakly on $\mathcal{P}(\Omega)$ to the unique minimizer of

$$ $\mathbf{R}\mapsto F(\mathbf{R}):=\tau H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})+ \frac{1}{\lambda}\int_{0}^{1}\mathrm{DF}(g_{\sharp}\mathbf{R}_{t},\overline{ \rho}_{t}).$ $$

Before proving this theorem, we state the following result that is immediate from the non-negativity of our data-fitting term.

###### Fact B.6 (Non-negativity) .

With the assumptions of Theorem B.5, the functionals $F_{T}$ and $F$ are bounded from below by 0 0.

###### Proof of Theorem B.5 .

The argument follows that of []. Let $\mathbf{R}$ be a minimizer of $F$ and $\mathbf{R}^{T}$ be the minimizer of $F_{T}$. By optimality of the minimizers, we have $\mathcal{G}_{0}\mathbf{R}=\mathbf{R}$ and $\mathcal{G}_{0}\mathbf{R}^{T}=\mathbf{R}^{T}$. Using Proposition B.15, we can find a sequence $\tilde{\mathbf{R}}^{T}$ that converges weakly to $\mathbf{R}$ as $T\to+\infty$ such that

$$ $F(\mathbf{R})\geq\limsup_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^{T})\geq\limsup _{T\to+\infty}\min_{\mathcal{P}(\Omega)}F_{T}=\limsup_{T\to+\infty}F_{T}( \mathbf{R}^{T}).$ $$

In particular, the sequence is bounded, which by Fact B.6, implies the sequence $H(\mathbf{R}^{T}|\mathbf{W}^{\Xi,\tau})$ is bounded. Then from the compactness of the sublevel sets of the entropy, we have a limit point $\widehat{\mathbf{R}}$ of the sequence $\{\mathbf{R}^{T}\}$. Using the optimality of $\mathbf{R}$ and Proposition B.16, we see

$$ $\displaystyle F(\mathbf{R})\leq F(\mathcal{G}_{0}\widehat{\mathbf{R}})\leq \liminf_{T\to+\infty}F_{T}(\mathbf{R}^{T}).$ $$

Thus, we have equalities everywhere, so

$$ $F(\mathbf{R})=\limsup_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^{T})=\lim_{T\to+ \infty}F_{T}(\mathbf{R}^{T}).$ $$

Then we see

$$ $F_{T}(\tilde{\mathbf{R}})-F_{T}(\mathbf{R}^{T})=F_{T}(\tilde{\mathbf{R}}^{T})- \min_{\mathcal{P}(\Omega)}F_{T}$ $$

converges to 0 0 as $T\to+\infty$. By [, Lem. B.3], relative entropy is 1-convex with respect to the total variation, i.e. if $p,q,r$ are three probability measures,

$$ $H\left(\frac{p+q}{2}\Bigg{|}r\right)\leq\frac{1}{2}H(p|r)+\frac{1}{2}H(q|r)- \frac{1}{2}\|p-q\|_{\mathrm{TV}}^{2}.$ $$

Since the data-fitting term is also convex, the full objective $F_{T}(\cdot)$ is 1-convex with respect to the total variation. By a classic strong convexity argument, since $F_{T}(\tilde{\mathbf{R}})$ converges to the minimum value $\min_{\mathcal{P}(\Omega)}F_{T}$ achieved at $\mathbf{R}^{T}$,
$\|\tilde{\mathbf{R}}^{T}-\mathbf{R}^{T}\|_{\mathrm{TV}}$ must also converge to 0 0 as $T\to+\infty$. Recall that TV convergence is stronger than weak convergence. Then, using the weak convergence of $\tilde{\mathbf{R}}^{T}$ to $\mathbf{R}$, we see that $\mathbf{R}^{T}$ converges weakly to $\mathbf{R}$ as $T\to+\infty$. This concludes the proof.
∎

The remainder of Appendix B.2 is dedicated towards proving Theorem B.5.

###### Theorem B.5 (analogous to [ 33 , Thm. 2.7] ) .

###### Fact B.6 (Non-negativity) .

###### Proof of Theorem B.5 .

#### B.2.1 Heat flow and regularization of the marginals

Recall that we use $\Phi_{s}$ to denote the heat flow with width $s$. We use the heat flow to regularize the marginals. First, we have the following result showing that the density of $\Phi_{s}\mathbf{R}_{t}$ is continuous jointly in $t$ and $x$.

###### Proposition B.7 (analogous to [ 33 , Prop. 2.12] ) .

Let $s>0$. There exists a constant $C$ depending only on $\mathcal{X}$ and $\Xi$ for which the following hold:

- (i)
For each $\mathbf{R}\in\mathcal{P}(\Omega)$, its heat flow regularization $\Phi_{s}\mathbf{R}_{t}$ has density $\rho^{(s)}(t,\cdot)$ (with respect to the volume measure) that satisfies for all $t\in[0,1],x\in\mathcal{X}$, we have
$\rho^{(s)}(t,x)\geq\frac{1}{C_{s}}.$
- (ii)
For all $t_{1},t_{2}\in[0,1]$ and $x_{1},x_{2}\in\mathcal{X}$, we have
$|\rho^{(s)}(t_{1},x_{1})-\rho^{(s)}(t_{2},x_{2})|\leq C\left(\sqrt{\tau}\sqrt{
H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})+C+C\tau}\sqrt{|t_{1}-t_{2}|}+d_{\mathcal{X
}}(x_{1},x_{2})\right).$

###### Proof.

The first estimate is directly from []. The second estimate follows from [] and the following proposition.
∎

###### Proposition B.8 (analogous to [ 33 , Lem. 2.13] ) .

There exists a constant $C$ depending only on $\mathcal{X}$ and $\Xi$ such that for each $\mathbf{R}\in\mathcal{P}(\Omega)$,

$$ $\mathbb{E}_{\mathbf{R}}[d_{\mathcal{X}}(X_{t_{1}},X_{t_{2}})^{2}]\leq C\left(H (\mathbf{R}|\mathbf{W}^{\Xi,\tau})+C+C\sigma^{2}\right)\sigma^{2}|t_{1}-t_{2}|.$ $$

###### Proof.

The argument follows that of [].
For any $\eta>0$, using the dual representation of entropy with the function $X\mapsto\eta d_{\mathcal{X}}(X_{t_{1}},X_{t_{2}})$, we have

$$ $\eta\mathbb{E}_{\mathbf{R}}[d_{\mathcal{X}}(X_{t_{1}},X_{t_{2}})^{2}]\leq H( \mathbf{R}|\mathbf{W}^{\Xi,\tau})+\log\mathbb{E}_{\mathbf{W}^{\Xi,\tau}}[\exp( \eta d_{\mathcal{X}}(X_{t_{1}},X_{t_{2}})^{2}].$ $$

Using an upper bound on the heat kernel from [, Cor. 3.1] and that $\|\Xi\|_{L^{\infty}}<+\infty$, we have the following bound on the transition probability for $\mathbf{W}^{\Xi,\tau}$:

$$ $p_{\tau}(x,y,t)\leq\frac{Ce^{\|\Xi\|_{L^{\infty}}^{2}}}{(\tau t)^{d/2}}\exp \left(C\tau t-\frac{d_{\mathcal{X}}(x,y)^{2}}{C\tau t}\right).$ $$

Then the remainder of the argument of the proof of [, Prop. 2.13] yields the desired result.
∎

###### Proposition B.7 (analogous to [ 33 , Prop. 2.12] ) .

###### Proof.

###### Proposition B.8 (analogous to [ 33 , Lem. 2.13] ) .

###### Proof.

#### B.2.2 Heat flow and entropy on the space of paths

We introduce an auxiliary variational problem in which all the temporal marginals are fixed.

###### Definition 3 .

Let $\rho\in C([0,1]:\mathcal{P}(\mathcal{X}))$ be a $\mathcal{P}(\mathcal{X})$-valued continuous curve with respect to the weak topology. Define the problem $\mathcal{A}_{\tau}(\rho)$ to be

$$ $\mathcal{A}_{\tau}(\rho):=\underset{\mathbf{R}\in\mathcal{P}(\Omega)}{\inf}\{ \tau H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})\mid\forall t\in[0,1],g_{\sharp} \mathbf{R}_{t}=g_{\sharp}\rho_{t}\}.$ $$

We use the convention that $\mathcal{A}_{\tau}(\rho)=+\infty$ if the above problem has no admissible competitor.

Using a dual representation of $\mathcal{A}$, we can use PDE theory to solve this problem. First, we give a martingale characterization of a class of stochastic processes:

###### Proposition B.9 .

Suppose $\tilde{\mathbf{W}}^{\Xi,\tau}$ is the law of the SDE $dX_{t}=-\Xi\,dt+\sqrt{\tau}\,dB_{t}$ with arbitrary initial distribution. Let $\varphi:[0,1]\times\mathcal{X}\to\mathbb{R}$ be a smooth function. Then, the process whose value at $t\in[0,1]$ is given by

$$ $\exp\left(\frac{1}{\tau}\left(\varphi(t,X_{t})-\varphi(0,X_{0})-\int_{0}^{t} \left[\partial_{s}\varphi+\frac{1}{2}|\nabla\varphi|^{2}-\langle\Xi,\nabla \varphi\rangle+\frac{\tau}{2}\Delta\varphi\right](s,X_{s})\,ds\right)\right)$ (17) $$

is an $\mathcal{F}_{t}$-martingale under $\tilde{\mathbf{W}}^{\Xi,\tau}$.

###### Proof.

By Assumption B.2 on $\Xi$, the process

$$ $M_{t}^{\varphi}=\varphi(t,X_{t})-\varphi(0,X_{0})-\int_{0}^{t}\left[\partial_{ s}\varphi-\langle\Xi,\nabla\varphi\rangle+\frac{\tau}{2}\Delta\varphi\right](s ,X_{s})\,ds$ (18) $$

is an $\mathcal{F}_{t}$-martingale under $\tilde{\mathbf{W}}^{\Xi,\tau}$ by [, Thm 8.3.1]. We calculate the quadratic variation $\langle M^{\varphi}\rangle_{t}$ similar to [, Prop. 1.3.1]. First, we have

$$ $\displaystyle\varphi(t,X_{t})^{2}$ $\displaystyle=\varphi(0,X_{0})^{2}+M^{\varphi^{2}}_{t}+\frac{1}{2}\int_{0}^{t} [-\langle\Xi,\nabla\varphi^{2}\rangle+\Delta\varphi^{2}](s,X_{s})\,ds$ $\displaystyle=\varphi(0,X_{0})^{2}+M^{\varphi^{2}}_{t}+\frac{1}{2}\int_{0}^{t} [-2\varphi\langle\Xi,\nabla\varphi\rangle+\Delta\varphi^{2}](s,X_{s})\,ds.$ $$

Using Itô’s formula, we have

$$ $\displaystyle\varphi(t,X_{t})^{2}$ $\displaystyle=\varphi(0,X_{0})^{2}+2\int_{0}^{t}\varphi(s,X_{s})d\varphi^{2}(s ,X_{s})+\langle M^{\varphi}\rangle_{t}$ $\displaystyle=\varphi(0,X_{0})^{2}+2\int_{0}^{t}\varphi(s,X_{s})dM_{s}^{ \varphi}+\int_{0}^{t}\varphi(s,X_{s})[-\langle\Xi,\nabla\varphi\rangle+\Delta \varphi](s,X_{s})ds+\langle M^{\varphi}\rangle_{t}.$ $$

Equating the bounded variation parts, we see

$$ $\displaystyle\langle M^{\varphi}\rangle_{t}$ $\displaystyle=\frac{1}{2}\int_{0}^{t}\left[-2\varphi\langle\Xi,\nabla\varphi \rangle+\Delta\varphi^{2}+2\varphi\langle\Xi,\nabla\varphi\rangle-\varphi \Delta\varphi\right](s,X_{s})\,ds$ $\displaystyle=\frac{1}{2}\int_{0}^{t}\left[\Delta\varphi^{2}-\varphi\Delta \varphi\right](s,X_{s})\,ds$ $\displaystyle=\int_{0}^{t}|\nabla\varphi(s,X_{s})|^{2}\,ds.$ $$

Then, (17) is the exponential martingale of $M_{t}^{\varphi}$.
∎

Here, recall that Assumption B.2 ensures that (18) is a martingale. Otherwise, it is only a *local martingale* and we would need to check the $L^{1}$ convergence of the stopped process with an increasing sequence of stopping times that goes to $+\infty$. This is a standard argument in stochastic calculus, e.g. see [].

Now we give the dual representation mentioned above.

###### Proposition B.10 (analogous to [ 33 , Prop. 2.15] ) .

Let $\rho\in C([0,1]:\mathcal{P}(\mathcal{X}))$ be a $\mathcal{P}(\mathcal{X})$-valued continuous curve. We have

$$ $\displaystyle\mathcal{A}_{\tau}(\rho)=\tau H(\rho_{0}|\mathrm{vol})$ $\displaystyle+\sup_{\varphi}\left\{-\int_{\mathcal{X}}\varphi(0,x)\rho_{0}(dx) -\int_{0}^{1}\int_{\mathcal{X}}\left(\partial_{t}\varphi+\frac{1}{2}|\nabla \varphi|^{2}-\langle\Xi,\nabla\varphi\rangle+\frac{\tau}{2}\Delta\varphi\right )\rho_{t}(dx)\,dt\right\},$ $$

where the supremum is taken over all $\varphi\in C^{2}([0,1]\times\mathcal{X})$ such that $\varphi(1,\cdot)=0$.

###### Proof.

The argument follows that of []. From the duality result of [, Prop. 2.3], we have

$$ $\mathcal{A}_{\tau}(\rho)=\tau H(\rho_{0}|\mathrm{vol})+\tau\sup_{\psi}\left\{ \int_{0}^{1}\int_{\mathcal{X}}\psi\rho_{t}(dx)\,ds-\int_{\mathcal{X}}\left[ \log\mathbb{E}_{\mathbf{W}^{\Xi,\tau,x}}\exp\left(\int_{0}^{1}\psi\,dt\right) \rho_{0}(dx)\right]\right\},$ $$

where $\mathbf{W}^{\Xi,\tau,x}$ is the measure such that $\mathbf{W}^{\Xi,\tau}_{0}=\delta_{x}$ and the supremum is taken over $\psi\in C([0,1]\times\mathcal{X})$. Here, the characterization holds because the argument in [] only requires the reference measure to be uniform at all marginals.

Let $-\psi:=\frac{1}{\tau}(\partial_{t}\varphi+\frac{1}{2}|\nabla\varphi|^{2}-
\langle\Xi,\nabla\varphi\rangle+\frac{\tau}{2}\Delta\varphi)$ for some smooth $\varphi$ satisfying the terminal condition $\varphi(1,x)=0$. By Proposition B.9, we see

$$ $\mathbb{E}_{\mathbf{W}^{\Xi,\tau,x}}\left[\exp\left(-\frac{\varphi(0,X_{0})}{ \tau}+\int_{0}^{t}\psi(t,X_{t})\,dt\right)\right]=1.$ $$

As $X_{0}=0$ under $\mathbf{W}^{\Xi,\tau,x}$, we have

$$ $\displaystyle\int_{\mathcal{X}}\left[\log\mathbb{E}_{\mathbf{W}^{\Xi,\tau,x}} \exp\left(\int_{0}^{t}\psi(t,X_{t})\,dt\right)\right]$ $\displaystyle=\int_{\mathcal{X}}\left[\log e^{\tau^{-1}\varphi(0,x)}\right] \rho_{0}(dx)$ $\displaystyle=\frac{1}{\tau}\int_{\mathcal{X}}\varphi(0,x)\,\rho_{0}(dx).$ $$

The remaining argument of [] follows through.
∎

The main idea is that there is a contraction of $\mathcal{A}_{\tau}$ under the heat flow, which we can think of as a space-time counterpart of the contraction of entropy under the heat flow.

###### Proposition B.11 (analogous to [ 33 , Prop. 2.16] ) .

Let $\rho\in C([0,1]:\mathcal{P}(\mathcal{X}))$ be a $\mathcal{P}(\mathcal{X})$-valued continuous curve and for $s\geq 0$, define the new curve $\rho^{(s)}:t\mapsto\Phi_{s}\rho_{t}$. Let $K$ be a lower bound on the Ricci curvature of the manifold $\mathcal{X}$. Then, for any $s\geq 0$, we have

$$ $\mathcal{A}_{\tau}(\rho^{(s)})\leq e^{-2Ks}\mathcal{A}_{\tau}(\rho).$ $$

###### Proof.

Consider the dual formulation in Proposition B.10. If $\varphi:[0,1]\times\mathcal{X}\to\mathbb{R}$ is a $C^{2}$ function with boundary condition $\varphi(1,\cdot)=0$, then by the self-adjointness property of the heat semigroup, we have

$$ $\displaystyle\int_{\mathcal{X}}\varphi(0,\cdot)\rho_{0}^{(s)}+\int_{0}^{1}\int _{\mathcal{X}}\left(\partial_{t}\varphi+\frac{1}{2}|\nabla\varphi|^{2}-\langle \Xi,\nabla\varphi\rangle+\frac{\tau}{2}\Delta\varphi\right)\rho_{t}^{(s)}\,dt$ $\displaystyle\quad=\int_{\mathcal{X}}\{\Phi_{s}\varphi\}(0,\cdot)\rho_{0}+\int _{0}^{1}\int_{\mathcal{X}}\left(\partial_{t}\Phi_{s}\varphi+\frac{1}{2}\Phi_{s }|\nabla\varphi|^{2}-\Phi_{s}\langle\Xi,\nabla\varphi\rangle+\frac{\tau}{2} \Delta\Phi_{s}\varphi\right)\rho_{t}\,dt,$ $$

where $\Phi_{s}\partial_{t}\varphi=\partial_{t}\Phi_{s}\varphi$ by Schwarz’s theorem and $\Phi_{s}\Delta\varphi=\Delta\Phi_{s}\varphi$ by simple calculation. By properties of the *carré du champ* operator [, Cor. 3.3.19] and expanding out the inner product, we see that $\langle\Phi_{s}\Xi,\nabla\Phi_{s}\varphi\rangle\leq e^{-2Ks}\Phi_{s}\langle\Xi
,\nabla\varphi\rangle$. Thus, letting $\tilde{\varphi}=e^{2Ks}\Phi_{s}\varphi$ and $\tilde{\Xi}=e^{2Ks}\Phi_{s}\Xi$, we have

$$ $\displaystyle-\int_{\mathcal{X}}\varphi(0,\cdot)\rho_{0}^{(s)}-\int_{0}^{1} \int_{\mathcal{X}}\left(\partial_{t}\varphi+\frac{1}{2}|\nabla\varphi|^{2}- \langle\Xi,\nabla\varphi\rangle+\frac{\tau}{2}\Delta\varphi\right)\rho_{t}^{(s )}\,dt$ $\displaystyle\qquad\leq-e^{-2Ks}\left[\int_{\mathcal{X}}\tilde{\varphi}(0, \cdot)\rho_{0}-\int_{0}^{1}\int_{\mathcal{X}}\left(\partial_{t}\tilde{\varphi} +\frac{1}{2}|\nabla\tilde{\varphi}|^{2}-\langle\tilde{\Xi},\nabla\tilde{ \varphi}\rangle+\frac{\tau}{2}\Delta\tilde{\varphi}\right)\rho_{t}\,dt\right]$ $\displaystyle\qquad\leq e^{-2Ks}[\mathcal{A}_{\tau}(\rho)-\tau H(\rho_{0}| \mathrm{vol})],$ $$

where the last inequality is due to Proposition B.10. Taking a supremum over $\varphi$, we see that

$$ $\mathcal{A}_{\tau}(\rho^{(s)})\leq e^{-2Ks}\mathcal{A}_{\tau}(\rho)+\tau\left[ H(\Phi_{s}\rho_{0}|\mathrm{vol})-e^{-2Ks}H(\rho_{0}|\mathrm{vol})\right].$ $$

By [, Eq. B.3], the second term in the right-hand side is always non-positive, so the claim follows.
∎

Next, we define the regularizing operator $\mathcal{G}_{s}$ that acts at the level of laws on the space of paths.

###### Definition 4 .

For each $\mathbf{R}\in\mathcal{P}(\Omega)$ with $H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})<+\infty$ and for each $s\geq 0$, define

$$ $\mathcal{G}_{s}(\mathbf{R}):=\underset{\tilde{R}\in\mathcal{P}(\Omega)}{\arg \min}\{H(\tilde{\mathbf{R}}|\mathbf{W}^{\Xi,\tau})\mid\forall t\in[0,1],g_{ \sharp}\tilde{\mathbf{R}}_{t}=g_{\sharp}\Phi_{s}\mathbf{R}_{t}\}.$ $$

That is, among all probability distributions on the space of paths whose marginals in hidden space coincide with $t\mapsto g_{\sharp}\Phi_{s}\mathbf{R}_{t}$, the measure $\mathcal{G}_{s}(\mathbf{R})\in\mathcal{P}(\Omega)$ is the one with the smallest entropy.

Note that $\mathcal{G}_{s}(\mathbf{R})$ is well-defined because thanks to Proposition B.11, $\mathcal{A}_{\tau}((\Phi_{s}\mathbf{R}_{t})_{t})\leq e^{-2Ks}\mathcal{A}_{\tau
}((\mathbf{R}_{t})_{t})\leq e^{-2Ks}H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})<+\infty$, so the minimization problem has admissible solutions. Since sublevel sets of entropy are compact, there exists a minimizer, and from strict convexity of the entropy functional, it is unique. Now note that

$$ $\mathcal{A}_{\tau}((\Phi_{s}(\mathbf{R}_{t})_{t})=H(\mathcal{G}_{s}(\mathbf{R} |\mathbf{W}^{\Xi,\tau})).$ $$

This gives us the following result.

###### Proposition B.12 (analogous to [ 33 , Prop. 2.18] ) .

For each $\mathbf{R}\in\mathcal{P}(\Omega)$ such that $H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})<+\infty$, we have the following:

- (i)
For any $s\geq 0$, $H(\mathcal{G}_{s}(\mathbf{R})|\mathbf{W}^{\Xi,\tau})\leq e^{-2Ks}H(\mathcal{G}
_{0}(\mathbf{R})|\mathbf{W}^{\Xi,\tau})\leq e^{-2Ks}H(\mathbf{R}|\mathbf{W}^{
\Xi,\tau})$.
- (ii)
$\mathcal{G}_{s}(\mathbf{R})$ converges to $\mathcal{G}_{0}(\mathbf{R})$ weakly as $s\to 0^{+}$.

###### Proof.

The argument follows that of []. The first property is a rewriting of Proposition B.11 together with the definition of $\mathcal{G}_{s}$ and $\mathcal{A}_{\tau}$. The second property follows from our observability assumption and an analogous argument to that of the proof of [, Prop. 2.18]. Consider the following sequential characterization. Let $\{s_{n}\}_{n\in\mathbb{N}}$ be a sequence with $s_{n}\to 0$ as $n\to+\infty$. By the contraction estimate in (i) and that the Ricci curvature is bounded from below, we know that $H(\mathcal{G}_{s_{n}}\mathbf{R}|\mathbf{W}^{\Xi,\tau})$ is uniformly bounded in $n$. Let $\tilde{\mathbf{R}}$ be any limit point of $\mathcal{G}_{s_{n}}\mathbf{R}$. Notice that this limit point exists due to the compactness of the sublevel sets of $H(\cdot|\mathbf{W}^{\Xi,\tau})$.

We show that $\tilde{\mathbf{R}}=\mathcal{G}_{0}\mathbf{R}$ by a standard analytic argument. We consider a subsequence (which we do not relabel) $\mathcal{G}_{s_{n}}\mathbf{R}$ that converges to $\tilde{\mathbf{R}}$ as $n\to+\infty$. The marginals of $\tilde{\mathbf{R}}$ agree with those of $\mathbf{R}$ as we easily see that the marginals of $\mathcal{G}_{s_{n}}\mathbf{R}$ are the $\{\Phi_{s_{n}}\mathbf{R}_{t}\}_{t\in[0,1]}$, and $\Phi_{s_{n}}f\to f$ in $L^{1}(\mathcal{X},\mathrm{vol})$ as $s_{n}\to 0$. Then, using the lower semi continuity of entropy, the definition of $\mathcal{G}_{s_{n}}$, and the contraction estimate for $\mathcal{A}$, we have

$$ $\displaystyle H(\tilde{\mathbf{R}}|\mathbf{W}^{\Xi,\tau})$ $\displaystyle\leq\liminf_{n\to+\infty}H(\mathcal{G}_{s_{n}}\mathbf{R}|\mathbf{ W}^{\Xi,\tau})$ $\displaystyle=\liminf_{n\to+\infty}\mathcal{A}_{\tau}((\Phi_{s_{n}}\mathbf{R}_ {t})_{t})|\mathbf{W}^{\Xi,\tau})$ $\displaystyle\leq\liminf_{n\to+\infty}e^{-2Ks_{n}}\mathcal{A}_{\tau}((\mathbf{ R}_{t})_{t})=\mathcal{A}_{\tau}((\mathbf{R}_{t})_{t}).$ $$

This shows that $\tilde{\mathbf{R}}=\mathcal{G}_{0}\mathbf{R}$, which concludes the proof.
∎

###### Definition 3 .

###### Proposition B.9 .

###### Proof.

###### Proposition B.10 (analogous to [ 33 , Prop. 2.15] ) .

###### Proof.

###### Proposition B.11 (analogous to [ 33 , Prop. 2.16] ) .

###### Proof.

###### Definition 4 .

###### Proposition B.12 (analogous to [ 33 , Prop. 2.18] ) .

###### Proof.

#### B.2.3 The data-fitting term

We recall the definition of the data-fitting term here:

$$ $\displaystyle\mathrm{DF}^{\sigma}(g_{\sharp}\mathbf{R}_{t^{T}_{i}},\hat{\rho}_ {i}^{T,h})$ $\displaystyle:=\int_{\mathcal{Y}}-\log\left[\int_{\mathcal{X}}\exp\left(-\frac {\|g(x)-y\|^{2}}{2\sigma^{2}}\right)d\mathbf{R}_{t^{T}_{i}}(x)\right]d\hat{ \rho}_{i}^{T,h}(y)$ $\displaystyle:=H(\hat{\rho}_{i}^{T,h}|g_{\sharp}\mathbf{R}_{t_{i}^{T}}* \mathcal{N}_{\sigma})+H(\hat{\rho}_{t_{i}^{T}})+C,$ $$

where $\mathcal{N}_{\sigma}$ is the Gaussian kernel. First, we have the following result, which is immediate from properties of entropy.

###### Proposition B.13 .

The function $r\mapsto\mathrm{DF}(r,p)$ is convex and lower semi continuous on $\mathcal{P}(\mathcal{X})$.

We will require a quantitative control on the effect of the heat flow on the data-fitting term.

###### Proposition B.14 (analogous to [ 33 , Prop. 2.22] ) .

Assume $g:\mathcal{X}\to\mathcal{Y}$ is measure preserving.(^12^1212Suppose that $(\mathcal{X},\lambda_{\mathcal{X}}),(\mathcal{Y},\lambda_{\mathcal{Y}})$ are measure spaces with Lebesgue measure. $g$ is measure preserving if for every Borel set $B\in\mathcal{X}$, $\lambda_{\mathcal{X}}(A)=\lambda_{\mathcal{Y}}(g_{\sharp}A)$.) Let $p,r\in\mathcal{P}(\mathcal{X})$. There exists a constant $C>0$ depending only on $\mathcal{X}$, $g$, and $\sigma$ such that for every $s>0$,

$$ $\mathrm{DF}(g_{\sharp}\Phi_{s}r,g_{\sharp}p)\leq\mathrm{DF}(g_{\sharp}r,g_{ \sharp}p)+s\cdot C.$ $$

Above, for simplification of the notation, we pushforward both parameters of the data-fitting term by $g$. This makes the argument below much cleaner.

###### Proof.

The argument follows that of [], but it is much simpler due to our different data-fitting term. In particular, we do not need a bound on the Fisher information. By an abuse of notation, denote $r\in L^{1}(\mathcal{X},\mathrm{vol})$ the density of $r$ with respect to $\mathrm{vol}$. Denote $r(s,\cdot)$ to be the density of $\Phi_{s}r$ with respect to $r$. It satisfies the heat equation

$$ $\frac{\partial r}{\partial s}=\Delta r.$ $$

Then, we have

$$ $\displaystyle\frac{d}{ds}\mathrm{DF}(g_{\sharp}\Phi_{s}r,g_{\sharp}p)$ $\displaystyle=\frac{d}{ds}\int_{\mathcal{Y}}-\log\left[\int_{\mathcal{X}}\exp \left(-\frac{\|g(x)-g(y)\|^{2}}{2\sigma^{2}}\right)r(s,x)\mathrm{vol}(dx) \right]p(y)\mathrm{vol}(dy)$ $\displaystyle=-\int_{\mathcal{Y}}\log\left[\int\exp\left(-\frac{\|g(x)-g(y)\|^ {2}}{2\sigma^{2}}\right)\frac{\partial}{\partial s}r(s,x)\mathrm{vol}(dx) \right]p(y)\mathrm{vol}(dy)$ $\displaystyle=-\int_{\mathcal{Y}}\log\left[\int_{\mathcal{X}}\exp\left(-\frac{ \|g(x)-g(y)\|^{2}}{2\sigma^{2}}\right)\Delta r(s,x)\mathrm{vol}(dx)\right]p(y) \mathrm{vol}(dy)$ $\displaystyle\leq C\int_{\mathcal{Y}}p(y)\mathrm{vol}(dy)\leq C,$ $$

where the inequality follows from properties of the Gaussian integral and the fact that $\int\Delta r(s,x)\mathrm{vol}(dx)=1$. Integrating yields the desired result.
∎

###### Proposition B.13 .

###### Proposition B.14 (analogous to [ 33 , Prop. 2.22] ) .

###### Proof.

#### B.2.4 Two results on limits of functionals

We require two results of the functional $F_{T}$ defined in (16). We use these for the $\Gamma$-convergence theory required in the proof of Theorem B.5.

###### Proposition B.15 (analogous to [ 33 , Prop. 2.24] ) .

Use the notation and assumptions of Theorem B.5. Suppose $\mathbf{R}\in\mathcal{P}(\Omega)$ with $F(\mathbf{R})<+\infty$ and $\mathcal{G}_{0}\mathbf{R}=\mathbf{R}$. Then there exists a sequence $\tilde{\mathbf{R}}^{T}$ which converges weakly to $\mathbf{R}$ as $T\to+\infty$ and

$$ $\limsup_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^{T})\leq F(\mathbf{R}).$ $$

###### Proof.

The argument follows that of []. Let $s>0$. Combining Proposition B.14 for the data-fitting term and Proposition B.12 for the relative entropy on the space of paths, we see that

$$ $\displaystyle F(\mathcal{G}_{s}(\mathbf{R})$ $\displaystyle=\tau H(\mathcal{G}_{s}\mathbf{R}|\mathbf{W}^{\Xi,\tau})+\frac{1} {\lambda}\int_{0}^{1}\mathrm{DF}(\Phi_{s}\mathbf{R}_{t},\overline{\rho_{t}})\,dt$ $\displaystyle\leq\tau e^{-2Ks}H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})+\frac{1}{ \lambda}\int_{0}^{1}\mathrm{DF}(\mathbf{R}_{t},\overline{\rho_{t}})\,dt+s\cdot C,$ $$

so we have

$$ $\limsup_{s\to 0}F(\mathcal{G}_{s}\mathbf{R})\leq F(\mathbf{R}).$ $$

Now as $-\log[\mathcal{G}_{s}\mathbf{R}]_{t}$ is a continuous function of $t$ and $x$ by Proposition B.7, we can use the weak convergence of $\hat{\rho}^{T}$ to $\overline{\rho}$ to write, for $s>0$,

$$ $\lim_{T\to+\infty}\sum_{i=1}^{T}\omega_{i}^{T}\mathrm{DF}\left([\mathcal{G}_{s }\mathbf{R}]_{t_{i}^{T}},\hat{\rho}_{i}^{T}\right)=\int_{0}^{1}\mathrm{DF}([ \mathcal{G}_{s}\mathbf{R}]_{t},\overline{\rho}_{t})\,dt.$ $$

This implies for all $s>0$, we have $\lim_{T\to+\infty}F_{T}(\mathcal{G}_{s}\mathbf{R})=F(\mathcal{G}_{s}\mathbf{R})$, so it is sufficient to let $\tilde{\mathbf{R}}:=\mathcal{G}_{s_{T}}\mathbf{R}$ for a sequence $\{s_{T}\}_{T\geq 1}$ that decays to 0 0 sufficiently slowly as $T\to+\infty$. This concludes the proof.
∎

###### Proposition B.16 (analogous to [ 33 , Prop. 2.25] ) .

Use the notation and assumptions of Theorem B.5. For each $T\geq 1$, let
$\tilde{\mathbf{R}}^{T}\in\mathcal{P}(\Omega)$ and assume that it converges weakly to some $\mathbf{R}\in\mathcal{P}(\Omega)$ as $T\to\infty$. Then

$$ $F(\mathcal{G}_{0}\mathbf{R})\leq\liminf_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^ {T}).$ $$

###### Proof.

The argument follows that of []. Assume that $\liminf_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^{T})<+\infty$ otherwise we are done. Then, up to a subsequence (that we do not relabel), we have $\sup_{T}H(\tilde{\mathbf{R}}^{T}|\mathbf{W}^{\Xi,\tau})<+\infty$. Combining Proposition B.14 for the data-fitting term and Proposition B.12 for the relative entropy on the space of paths, we have

$$ $\displaystyle F_{T}(\mathcal{G}_{s}\tilde{\mathbf{R}}^{T})$ $\displaystyle=\tau H(\mathcal{G}_{s}\tilde{\mathbf{R}}^{T}|\mathbf{W}^{\Xi, \tau})+\frac{1}{\lambda}\sum_{i=1}^{T}\omega_{i}^{T}\mathrm{DF}\left(g_{\sharp }\Phi_{s}\tilde{\mathbf{R}}_{t_{i}^{T}}^{T},\hat{\rho}_{i}^{T}\right)$ $\displaystyle\leq\tau e^{-2Ks}H(\tilde{\mathbf{R}}^{T}|\mathbf{W}^{\Xi,\tau})+ \frac{1}{\lambda}\mathrm{DF}\left(\tilde{\mathbf{R}}_{t_{i}^{T}}^{T},\hat{\rho }_{i}^{T}\right)+\frac{sc}{\lambda}.$ $$

Now we rewrite the above as

$$ $F_{T}(\tilde{\mathbf{R}}^{T})\geq F_{T}(\mathcal{G}_{s}\tilde{\mathbf{R}}^{T}) -C(s),$ $$

where

$$ $C(s)=\tau|e^{-2Ks}-1|\sup_{T}H(\tilde{\mathbf{R}}^{T}|\mathbf{W}^{\Xi,\tau})+ \frac{sc}{\lambda}$ $$

is upper bounded by a quantity independent of $T$ and $\lim_{s\to 0^{+}}C(s)=0$. For the data-fitting term, define the sequence of functions $a_{s}^{T}(t,x)$ to be

$$ $a_{s}^{T}(t,x):=-\log\left[\int\exp\left(-\frac{\|g(z)-x\|^{2}}{2\sigma^{2}} \right)d\Phi_{s}\tilde{\mathbf{R}}_{t}^{T}(z)\right],$ $$

which is parametrized by $T$. Notice that from the definition of the data-fitting term, we have

$$ $\sum_{i=1}^{T}\omega_{i}^{T}\mathrm{DF}(g_{\sharp}\Phi_{s}\tilde{\mathbf{R}}_{ t_{i}^{T}}^{T},\hat{\rho}_{i}^{T})=\sum_{i=1}^{T}\omega_{i}^{T}\int_{\mathcal{ X}}a_{s}^{T}(t_{i}^{T},x)\hat{\rho}_{i}^{T}(dx).$ $$

For a fixed $s>0$, the family of functions $a_{s}^{T}(t,x)$ indexed by $T$ is uniformly equicontinuous due to $g$ being continuous and Proposition B.7. Then there exists a subsequence (that we do not relabel) that converges uniformly on $[0,1]\times\mathcal{Y}$ as $T\to\infty$ to the function

$$ $\displaystyle a_{s}^{T}(t,x)$ $\displaystyle=-\log\left[\int\exp\left(-\frac{\|g(z)-x\|^{2}}{2\sigma^{2}} \right)d\Phi_{s}\mathbf{R}_{t}(z)\right]$ $\displaystyle=-\log\left[\int\exp\left(-\frac{\|g(z)-x\|^{2}}{2\sigma^{2}} \right)d[\mathcal{G}_{s}\mathbf{R}]_{t}(z)\right].$ $$

Using this uniform convergence with the weak convergence of $\hat{\rho}_{i}^{T}$ to $\overline{\rho_{t}}$, we see

$$ $\displaystyle\lim_{T\to+\infty}\sum_{i=1}^{T}\omega_{i}^{T}\mathrm{DF}\left( \Phi_{s}\mathbf{R}_{t_{i}^{T}}^{T},\hat{\rho}_{i}^{T}\right)$ $\displaystyle=\lim_{T\to+\infty}\sum_{i=1}^{T}\omega_{i}^{T}\int_{\mathcal{X}} a_{s}^{T}(t_{i}^{T},x)\hat{\rho}_{i}^{T}(dx)$ $\displaystyle=\int_{0}^{1}\int_{\mathcal{X}}a_{s}(t,x)\overline{\rho}_{t}(dx) \,dt$ $\displaystyle=\int_{0}^{1}\mathrm{DF}(\mathcal{G}_{s}\mathbf{R}_{t},\overline{ \rho}_{t})\,dt.$ $$

Using lower semi continuity of entropy, we have $F(\mathcal{G}_{s}\mathbf{R})\leq\liminf_{T\to\infty}F_{T}(\mathcal{G}_{s}
\tilde{\mathbf{R}}^{T})$. Thus, for each $s>0$, we have

$$ $\liminf_{T\to+\infty}F_{T}(\tilde{\mathbf{R}}^{T})\geq F(\mathcal{G}_{s} \mathbf{R})-C(s).$ $$

Finally, we use Proposition B.12 to take $s\to 0^{+}$ using the lower semi continuity of $F$ and the convergence of $\mathcal{G}_{s}\mathbf{R}$ to $\mathcal{G}_{0}\mathbf{R}$ when $s\to 0^{+}$.
∎

###### Proposition B.15 (analogous to [ 33 , Prop. 2.24] ) .

###### Proof.

###### Proposition B.16 (analogous to [ 33 , Prop. 2.25] ) .

###### Proof.

### B.3 Γ Γ \Gamma roman_Γ -convergence: taking h → 0 , λ → 0 formulae-sequence → ℎ 0 → 𝜆 0 h\to 0,\lambda\to 0 italic_h → 0 , italic_λ → 0

###### Theorem B.17 (analogous to [ 33 , Thm. 2.9] ) .

Let $\mathbf{P}\in\mathcal{P}(\Omega)$ with $H(\mathbf{P}|\mathbf{W}^{\Xi,\tau})<+\infty$. For each $\lambda>0$ and $h>0$, let $\mathbf{R}^{\lambda,h}$ be the minimizer of the functional

$$ $\mathbf{R}\mapsto G_{\lambda,h}(\mathbf{R}):=\tau H(\mathbf{R}|\mathbf{W}^{\Xi ,\tau})+\frac{1}{\lambda}\int_{0}^{1}H(\Phi_{h}\mathbf{P}_{t}|\mathbf{R}_{t}* \mathcal{N}_{\sigma})\,dt.$ $$

Then, as $h\to 0,\lambda\to 0$, the measure $\mathbf{R}^{\lambda,h}$ converges to the minimizer of $\mathbf{R}\mapsto H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})$ among all measures such that $g_{\sharp}\mathbf{R}_{t}=g_{\sharp}\mathbf{P}_{t}$ for all $t\in[0,1]$. Furthermore, if $\mathbf{P}$ is the law of the SDE in (1), then $\mathbf{R}^{\lambda,h}$ converges to $\mathbf{P}$.

###### Proof.

The argument follows that of []. First, consider $\mathbf{R}:=\mathcal{G}_{h}\mathbf{P}\in\mathcal{P}(\Omega)$ as a competitor in $G_{\lambda,h}$. Using the contraction estimate given by Proposition B.12, we have

$$ $\min_{\mathcal{P}(\Omega)}G_{\lambda,h}=G_{\lambda,h}(\mathbf{R}^{\lambda,h}) \leq\tau H(\mathcal{G}_{h}\mathbf{P}|\mathbf{W}^{\Xi,\tau})\leq\tau e^{-Kh}H( \mathcal{G}_{0}\mathbf{P}|\mathbf{W}^{\Xi,\tau}).$ $$

As $K>-\infty$ and $H(\mathbf{P}|\mathbf{W}^{\Xi,\tau})$ by assumption, we see that $G_{\lambda,h}(\mathbf{R}^{\lambda,h})$ is uniformly bounded in $\lambda$ and $h$. Thus, $H(\mathbf{R}^{\lambda,h}|\mathbf{W}^{\Xi,\tau})$ is uniformly bounded as well. Due to [, Prop. B.2], this implies that the family $\mathbf{R}^{\lambda,h}$ belongs to a compact set in the weak topology. Let $\tilde{\mathbf{R}}$ be any limit point in the limit as $\lambda\to 0,h\to 0$. We only need to show that $\tilde{\mathbf{R}}=\mathcal{G}_{0}\mathbf{P}$. Note that

$$ $\tau H(\mathbf{R}^{\lambda,h}|\mathbf{W}^{\Xi,\tau})\leq G_{\lambda,h}(\mathbf {R}^{\lambda,h})\leq\tau e^{2Kh}H(\mathcal{G}_{0}\mathbf{P}|\mathbf{W}^{\Xi, \tau}).$ $$

By taking $h\to 0$ and using the lower semi continuity of entropy, we see

$$ $H(\tilde{\mathbf{R}}|\mathbf{W}^{\Xi,\tau})\leq H(\mathcal{G}_{0}\mathbf{P}| \mathbf{W}^{\Xi,\tau}).$ $$

Now using Fatou’s lemma, the chain rule for relative entropy, and joint lower semi continuity of the entropy, we have

$$ $\displaystyle\int_{0}^{1}H(\mathbf{P}_{t}|\tilde{\mathbf{R}}_{t}*\mathcal{N}_{ \sigma})\,dt$ $\displaystyle\leq\liminf_{\lambda\to 0,h\to 0}\int_{0}^{1}H(\Phi_{h}\mathbf{P} _{t}|\mathbf{R}_{t}^{\lambda,h}*\mathcal{N}_{\sigma})\,dt$ $\displaystyle\leq\liminf_{\lambda\to 0,h\to 0}\int_{0}^{1}H(\Phi_{h}\mathbf{P} _{t}|\mathbf{R}_{t}^{\lambda,h})\,dt$ $\displaystyle\leq\liminf_{\lambda\to 0,h\to 0}\left(\lambda\sup_{\lambda,h}G_{ \lambda,h}(\mathbf{R}^{\lambda,h})\right)=0.$ $$

Thus, it follows that $g_{\sharp}\tilde{\mathbf{R}}_{t}=g_{\sharp}\mathbf{P}_{t}$ for almost every $t$. Therefore, by definition of $\mathcal{G}_{0}$, we have $\tilde{\mathbf{R}}=\mathcal{G}_{0}\mathbf{P}$. This concludes the proof.
∎

###### Theorem B.17 (analogous to [ 33 , Thm. 2.9] ) .

###### Proof.

## Appendix C Reduced Formulation

### C.1 Proof of Theorem 3.2

We use the following result to prove Theorem 3.2. Here, the statement is identical to that of [], but we consider a different reference measure.

###### Lemma C.1 (analogous to [ 14 , Prop. B.2] ) .

There exists a constant $C>0$ such that, for any $\mathbf{R}\in\mathcal{P}(\Omega)$ and $t_{1}^{T},\dots,t_{T}^{T}$ a collection of time instants, it holds

$$ $\displaystyle H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})$ $\displaystyle{\geq}H(\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}|\mathbf{W}^{\Xi, \tau}_{t_{1}^{T},\dots,t_{T}^{T}})$ $\displaystyle{\geq}\sum_{i=1}^{T-1}H(\mathbf{R}_{t_{i}^{T},t_{i+1}^{T}}|p_{ \tau_{i}}^{\Xi}(\mathbf{R}_{t_{i}^{T}}\otimes\mathbf{R}_{t_{i+1}^{T}}))-\sum_{ i=1}^{T-1}H(\mathbf{R}_{t_{i}^{T}}|\mathbf{W}_{t_{i}^{T}}^{\Xi,\tau})+C.$ $$

The first inequality $({\dagger})$ becomes an equality if and only if

$$ $\mathbf{R}(\cdot)=\int_{\mathcal{X}^{T}}\mathbf{W}^{\Xi,\tau}(\cdot|x_{1}, \dots,x_{T})\,d\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}(x_{1},\dots,x_{T}),$ $$

where $\mathbf{W}^{\Xi,\tau}(\cdot|x_{1},\dots,x_{T})$ is the law of $\mathbf{W}^{\Xi,\tau}$ conditioned on passing through $x_{1},\dots,x_{T}$ at times $t_{1}^{T},\dots,t_{T}^{T}$, respectively. In addition, the second inequality $(*)$ becomes an equality if and only if $\mathbf{R}$ is Markovian.

###### Proof.

Using the fact that $\Xi$ is divergence-free and that $\mathbf{W}^{\Xi,\tau}$ has the Markov property, the proof from [] holds. We provide the full proof for completeness.
The first inequality $({\dagger})$ and the equality case follows from the behavior of entropy with respect to a Markov measure under conditioning, e.g. [, Eq. 11]. In particular, we have

$$ $\displaystyle H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})=H(\mathbf{R}_{t_{1}^{T}, \dots,t_{T}^{T}}|\mathbf{W}_{t_{1}^{T},\dots,t_{T}^{T}}^{\Xi,\tau})$ $\displaystyle\quad+\int H\left(\mathbf{R}(\cdot|x_{1},\dots,x_{T})|\mathbf{W}^ {\Xi,\tau}(\cdot|x_{1},\dots,x_{T})\right)d\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{ T}}(x_{1},\dots,x_{T}),$ $$

where the second term vanishes if and only if the conditional distributions $\mathbf{R}(\cdot|x_{1},\dots,x_{T})$ follow the law of $\mathbf{W}^{\Xi}$, for $\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}$ almost every $(x_{1},\dots,x_{T})$. The second inequality $(*)$ follows from [, Lem. 3.4], which states

$$ $H(\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}|\mathbf{W}_{t_{1}^{T},\dots,t_{T}^{T} }^{\Xi,\tau})\geq\sum_{i=1}^{T-1}H(\mathbf{R}_{t_{i}^{T},t_{i+1}^{T}}|\mathbf{ W}_{t_{i}^{T},t_{i+1}^{T}}^{\Xi,\tau})-\sum_{i=2}^{T-1}H(\mathbf{R}_{t_{i}^{T} }|\mathbf{W}_{t_{i}^{T}}^{\Xi,\tau})=:E,$ $$

with equality if and only if $\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}$ is Markovian. As in [], we reorganize the terms in $E$.

Without loss of generality, assume that $\mathbf{R}_{t_{i}^{T}}$ are absolutely continuous with density $d\mathbf{R}_{t_{i}^{T}}(x)/dx:=r_{i}(x)$ and let $V_{\mathcal{X}}$ be the Lebesgue volume of $\mathcal{X}$. Since $\mathbf{W}_{t_{i}^{T}}^{\Xi,\tau}$ is the uniform measure on $\mathcal{X}$ for every $t_{i}^{T}$, we have

$$ $H(\mathbf{R}_{t_{i}^{T}}|\mathbf{W}_{t_{i}^{T}}^{\Xi,\tau})=H(\mathbf{R}_{t_{i }^{T}})+\log V_{\mathcal{X}}.$ $$

Letting $\tau_{i}:=\tau(t_{i+1}^{T}-t_{i}^{T})$, we also have

$$ $\mathbf{W}_{t_{i}^{T},t_{i+1}^{T}}^{\Xi,\tau}(dx,dy)=\frac{1}{V_{\mathcal{X}}} p_{\tau_{i}}^{\Xi}(x,y)\,dx\,dy.$ $$

Thus, we see that for any $\mu,\nu\in\mathcal{P}(\mathcal{X})$ with finite differential entropy and $\gamma\in\Pi(\mu,\nu)$, we have

$$ $\displaystyle H(\gamma|\mathbf{W}_{t_{i}^{T},t_{i+1}^{T}}^{\Xi,\tau})$ $\displaystyle=\int\log\left(\frac{d\gamma}{dx\otimes dy}\frac{V_{\mathcal{X}}} {p_{\tau_{i}}^{\Xi}}\right)d\gamma(x,y)$ $\displaystyle=\log V_{\mathcal{X}}+\int\log\left(\frac{d\gamma}{p_{\tau_{i}}^{ \Xi}d(\mu\otimes\nu)}\frac{d\mu}{dx}\frac{d\nu}{dy}\right)d\gamma$ $\displaystyle=\log V_{\mathcal{X}}+H(\gamma|p_{\tau_{i}}^{\Xi}(\mu\otimes\nu)) +H(\mu)+H(\nu),$ $$

where the last line follows from [, Lem. 1.6]. Now using the fact that $\mathbf{R}_{t_{i}^{T},t^{T}_{i+1}}\in\Pi(\mathbf{R}_{t^{T}_{i}},\mathbf{R}_{t^
{T}_{i+1}})$, we have

$$ $\displaystyle E$ $\displaystyle=\log V_{\mathcal{X}}+\sum_{i=1}^{T}H(\mathbf{R}_{t_{i}^{T},t_{i+ 1}^{T}}|p_{\tau_{i}}^{\Xi}(\mathbf{R}_{t_{i}^{T}}\otimes\mathbf{R}_{t_{i+1}^{T }}))+\sum_{i=1}^{T-1}H(\mathbf{R}_{t_{i}^{T}}),$ $$

which proves the formula.
∎

###### Theorem C.2 (Thm. 3.2 , restated) .

Let $\mathrm{Fit}:\mathcal{P}(\mathcal{Y})^{T}\to\mathbb{R}$ be *any* function and let $\Xi$ be bounded and divergence-free.

- (i)
If $\mathcal{F}$ admits a minimizer $\mathbf{R}^{*}$ then $(\mathbf{R}_{t_{1}^{T}}^{*},\dots,\mathbf{R}_{t_{T}^{T}}^{*})$ is a minimizer for $F$.
- (ii)
If $F$ admits a minimizer $\boldsymbol{\mu}^{*}\in\mathcal{P}(\mathcal{X})^{T}$, then a minimizer $\mathbf{R}^{*}$ for $\mathcal{F}$ is built as
$\mathbf{R}^{*}(\cdot)=\int_{\mathcal{X}^{T}}\mathbf{W}^{\Xi,\tau}(\cdot|x_{1},
\dots,x_{T})\,d\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}(x_{1},\dots,x_{T}),$
where $\mathbf{W}^{\Xi,\tau}(\cdot|x_{1},\dots,x_{T})$ is the law of $\mathbf{W}^{\Xi,\tau}$ conditioned on passing through $x_{1},\dots,x_{T}$ at times $t_{1}^{T},\dots,t_{T}^{T}$, respectively and $\mathbf{R}_{t_{1}^{T},\dots,t_{T}^{T}}$ is the composition of the optimal transport plans $\gamma_{i}$ that minimize $T_{\tau_{i},\Xi}(\boldsymbol{\mu}^{*(i)},\boldsymbol{\mu}^{*(i+1)})$, for $i\in[T-1]$.

###### Proof.

The proof from [] holds using our transition probability densities and OT plans. We provide it for completeness. First, note that a minimizer $\mathbf{R}^{*}\in\mathcal{P}(\Omega)$ of $\mathcal{F}(\mathbf{R})=\mathrm{Fit}(\mathbf{Q}_{t_{1}^{T}},\dots,\mathbf{Q}_{
t_{T}^{T}})+\tau H(\mathbf{R}|\mathbf{W}^{\Xi,\tau})$ is of the form in Lemma C.1. Let $\boldsymbol{\mu}^{(i)}:=\mathbf{R}_{t_{i}^{T}}^{*}$ be its marginals and $\gamma^{(i)}:=\mathbf{R}_{t_{i}^{T},t_{i+1}^{T}}^{*}$, which clearly satisfies $\gamma^{(i)}\in\Pi(\boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(i+1)})$. Using $C:=\log V_{\mathcal{X}}$, we see that

$$ $\displaystyle\mathcal{F}(\mathbf{R}^{*})$ $\displaystyle=\mathrm{Fit}(g_{\sharp}\boldsymbol{\mu}^{(1)},\dots,g_{\sharp} \boldsymbol{\mu}^{(T)})+\tau\sum_{i=1}^{T-1}H(\gamma^{(i)}|p_{\tau_{i}}^{\Xi}( \boldsymbol{\mu}^{(i)}\otimes\boldsymbol{\mu}^{(i+1)}))+\tau H(\boldsymbol{\mu })+C$ $\displaystyle\geq\mathrm{Fit}(g_{\sharp}\boldsymbol{\mu}^{(1)},\dots,g_{\sharp }\boldsymbol{\mu}^{(T)})+\frac{\tau}{\tau_{i}}\sum_{i=1}^{T-1}T_{\tau_{i},\Xi} (\boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(i+1)})+\tau H(\boldsymbol{\mu})+C,$ $$

where the inequality becomes an equality if and only if $\mathbf{R}^{*}_{t_{i}^{T},t_{i+1}^{T}}=\gamma^{(i)}$ is optimal in the definition of $T_{\tau_{i},\Xi}(\boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(i+1)})$. The claim follows.
∎

###### Lemma C.1 (analogous to [ 14 , Prop. B.2] ) .

###### Proof.

###### Theorem C.2 (Thm. 3.2 , restated) .

###### Proof.

## Appendix D Mean-Field Langevin Dynamics

Recall that the MFL dynamics is defined as the solution of (8), which we restate below:

$$ $\begin{cases}dX_{s}^{(i)}=-\nabla V^{(i)}[\boldsymbol{\mu}_{s}](X_{s}^{(i)})\, ds+\sqrt{2(\tau+\epsilon)}\,dB_{s}^{(i)}+d\Phi_{s}^{(i)},&\mathrm{Law}(X_{0}^{ (i)})=\boldsymbol{\mu}_{0}^{(i)}\\ \boldsymbol{\mu}_{s}^{(i)}=\mathrm{Law}(X_{s}^{(i)}),&i\in[T],\end{cases}$ $$

where $d\Phi_{s}^{(i)}$ is the boundary reflection in the sense of the Skorokhod problem. The family of laws $\{\boldsymbol{\mu}_{s}\}_{s\geq 0}$ of this stochastic process are characterized by the following system of PDEs:

$$ $\partial_{s}\boldsymbol{\mu}_{s}^{(i)}=\nabla\cdot(\nabla V^{(i)}[\boldsymbol{ \mu}_{s}]\boldsymbol{\mu}_{s}^{(i)})+(\tau+\epsilon)\Delta\boldsymbol{\mu}_{s} ^{(i)},$ (19) $$

which are coupled via the quantity $\nabla V^{(i)}[\boldsymbol{\mu}_{s}]$. The link between (8) and (19) follows from the Itô-Tanaka formula, see e.g. [, Lem. C.3]. This is a multi-species PDE where each of the species $\boldsymbol{\mu}^{(i)}$ attempts to minimize $\frac{\Delta t_{i}}{\lambda}\mathrm{Fit}^{\lambda,\sigma}(\cdot,\hat{\rho}_{i}
^{T})+(\tau+\epsilon)H$ via a drift-diffusion dynamics, and it is connected to $\boldsymbol{\mu}^{(i-1)}$ and $\boldsymbol{\mu}^{(i+1)}$ via Schrödinger bridges.

### D.1 Properties of G 𝐺 G italic_G and F 𝐹 F italic_F

We describe some properties of functions $G$ (4) and $F$ (5).

Recall that the *first-variation* of $G:\mathcal{P}(\mathcal{X})^{T}\to\mathbb{R}$ at $\boldsymbol{\mu}$ is the unique (up to an additive constant) function $V[\boldsymbol{\mu}]\in C(\mathcal{X})^{T}$ such that for all $\boldsymbol{\nu}\in\mathcal{P}(\mathcal{X})^{T}$,

$$ $\lim_{\epsilon\to 0}\frac{1}{\epsilon}[G(1-\epsilon)\boldsymbol{\mu}+\epsilon \boldsymbol{\nu})-G(\boldsymbol{\mu})]=\sum_{i=1}^{T}V^{(i)}[\boldsymbol{\mu}] (x)\,d(\boldsymbol{\nu}-\boldsymbol{\mu})^{(i)}(x).$ $$

###### Proposition D.1 (analogous to [ 14 , Prop. 3.2] ) .

The function $G$ is convex separately in each of its inputs (but not jointly), weakly continuous and its first-variation is given for $\boldsymbol{\mu}\in\mathcal{P}(\mathcal{X})^{T}$ and $i\in[T]$ by

$$ $V^{(i)}[\boldsymbol{\mu}]=\frac{\delta\mathrm{Fit}}{\delta\boldsymbol{\mu}^{(i )}}[\boldsymbol{\mu}]+\frac{\varphi_{i,i+1}}{t_{i+1}^{T}-t_{i}^{T}}+\frac{\psi _{i,i-1}}{t_{i}^{T}-t_{i-1}^{T}},$ $$

and

$$ $\frac{\delta\mathrm{Fit}}{\delta\boldsymbol{\mu}^{(i)}}[\boldsymbol{\mu}]:x \mapsto-\frac{\Delta t_{i}}{\lambda}\int\frac{\mathcal{N}_{\sigma}(g(x)-y)}{( \mathcal{N}_{\sigma}*g_{\sharp}\boldsymbol{\mu}^{(i)})(y)}\,d\hat{\rho}(y),$ $$

where $(\varphi_{i,j},\psi_{i,j})\in C^{\infty}(\mathcal{X})$ are the Schrödinger potentials for $T_{\tau_{i},\Xi}(\boldsymbol{\mu}^{(i)},\boldsymbol{\mu}^{(j)})$, with the convention that the corresponding term vanishes when it involves $\psi_{1,0}$ or $\varphi_{T,T+1}$. The function $F$ is jointly convex and admits a unique minimizer $\boldsymbol{\mu}^{*}$, which has an absolutely continuous density (again denoted by $\boldsymbol{\mu}^{*}$) characterized by

$$ $(\boldsymbol{\mu}^{*})^{(i)}\propto e^{-V^{(i)}[\boldsymbol{\mu}^{*}]/\tau}, \quad\text{for }i\in[T].$ $$

Here, the Schrödinger potentials are classically $L^{1}$ by standard (entropic) OT theory, but we can extend them to $C^{\infty}$ functions, as discussed in [].

###### Proof.

The argument is similar to that of []. The properties of $G$ and its first-variation are clear. In particular, the first-variation of $T_{\tau_{i},\Xi}$ follows from the fact that $g$ is smooth and [, Prop. 7.17], and the first-variation of $\mathrm{Fit}$ follows by direct calculation. The convexity of $G$ follows from the convexity of $T_{\tau_{i},\Xi}$ and the fact that the pushforward of $g$ is linear. The joint convexity of $F$, its unique minimizer, and the characterization of the minimizer
follow directly from the argument in the proof of [, Prop. 3.2].
∎

###### Proposition D.1 (analogous to [ 14 , Prop. 3.2] ) .

###### Proof.

### D.2 Noisy particle gradient descent

Let $m\in\mathbb{N}$ be the number of particles used in the discretization for each of the time marginals $\boldsymbol{\mu}^{(i)}$. For computation, we approximate the MFL dynamics by running noisy gradient descent on the function $G_{m}:(\mathcal{X}^{m})^{T}\to\mathbb{R}$ defined as $G_{m}(\hat{X}):=G(\hat{\boldsymbol{\mu}}_{\hat{X}})$, where

$$ $\hat{\boldsymbol{\mu}}_{\hat{X}}^{(i)}:=\frac{1}{m}\sum_{j=1}^{m}\delta_{\hat{ X}_{j}^{(i)}}.$ $$

From [, Prop. 2.4], we see that $m\nabla_{X_{j}^{(i)}}G_{m}(\hat{X})=\nabla V^{(i)}[\hat{\boldsymbol{\mu}}_{
\hat{X}}](\hat{X}_{j}^{(i)})$. Thus, this yields the discretization of (8):

$$ $\begin{cases}\hat{X}_{j}^{(i)}[k+1]=\hat{X}_{j}^{(i)}[k]-\eta\nabla V^{(i)}[ \hat{\boldsymbol{\mu}}[k]](\hat{X}_{j}^{(i)}[k])+\sqrt{2\eta(\tau+\epsilon)}Z_ {j,k}^{(i)},&\hat{X}_{j}^{(i)}[0]\overset{i.i.d.}{\sim}\boldsymbol{\mu}_{0}^{( i)}\\ \hat{\boldsymbol{\mu}}^{(i)}[k]=\frac{1}{m}\sum_{j=1}^{m}\delta_{\hat{X}_{j}^{ (i)}[k]},&i\in[T],\end{cases}$ (20) $$

where $\eta>0$ is a step-size, the $Z_{j,k}^{(i)}$ are i.i.d. standard Gaussian variables, and all the particles should be projected onto $\mathcal{X}$ at each step if $\mathcal{X}$ has boundaries. The MFL dynamics are recovered in the limit as $m\to\infty$ and $\eta\to 0$, e.g. see
[, , ].

### D.3 Exponential convergence

###### Theorem D.2 (Thm. 3.4 , restated) .

Assume $\mathcal{X}$ is the $d$-torus.
Let $\boldsymbol{\mu}_{0}\in\mathcal{P}(\mathcal{X})^{T}$ be such that $F(\boldsymbol{\mu}_{0})<+\infty$. Then for $\epsilon\geq 0$, there exists a unique solution $(\boldsymbol{\mu}_{s})_{s\geq 0}$ to the MFL dynamics (8). Let $\epsilon>0$ and assume that $\boldsymbol{\mu}_{0}$ has a bounded absolute log-density, it holds

$$ $F_{\epsilon}(\boldsymbol{\mu}_{s})-\min F_{\epsilon}\leq e^{-Cs}(F_{\epsilon}( \boldsymbol{\mu}_{0})-\min F_{\epsilon}),$ $$

where $C=\beta e^{-\alpha/\epsilon}$ for some $\alpha,\beta>0$ independently of $\boldsymbol{\mu}$ and $\epsilon$. Moreover, taking a smooth time-dependent $\epsilon_{s}$ that decays asymptotically as $\tilde{\alpha}/\log s$ for some $\tilde{\alpha}>\alpha$, it holds $F_{0}(\boldsymbol{\mu}_{s})-F_{0}(\boldsymbol{\mu}^{*})\lesssim\log\log s/\log
s\to
0$ and $\boldsymbol{\mu}_{s}$ converges weakly to the min-entropy estimator $\boldsymbol{\mu}^{*}$.

###### Proof.

As in [], we simply need to verify the assumptions in [, Thm. 3.2]. Recall that the objective function is of the form $F_{\epsilon}=G+(\tau+\epsilon)H$. The stability and regularity of the first-variation $V$, [, Assumption 1], is immediate from [, Prop. C.2] and that $g$ is bounded. The convexity of $F_{0}$ and existence of a minimizer for $F_{\epsilon}$, [, Assumption 2], follows from Proposition D.1.

For the uniform log-Sobolev inequality (LSI), [, Assumption 3], first note that the $i$th component of the first-variation of $F_{0}$ is given by $V^{(i)}[\boldsymbol{\mu}]+\tau\log\boldsymbol{\mu}^{(i)}$. Define $D:=\operatorname{diam}{\mathcal{X}}$ and $E:=\operatorname{diam}g_{\sharp}\mathcal{X}$, where $D<+\infty$ by assumption and $E<+\infty$ as $g$ is bounded. Note that $\operatorname{osc}V^{(i)}[\boldsymbol{\mu}]<+\infty$ as the gradient formula for $\delta\mathrm{Fit}/\delta\boldsymbol{\mu}^{(i)}$ is non-negative and is bounded by $Ee^{E^{2}/(2\sigma^{2})}$ and by [, App. A, Eq. 17], the Schrödinger potential $\varphi_{i,i+1}$ has an oscillation bounded by

$$ $\sup_{x,y\in\mathcal{X}}c_{t_{i}^{T},t_{i+1}^{T}}(x,y)-\inf_{x,y\in\mathcal{X} }c_{t_{i}^{T},t_{i+1}^{T}}(x,y)\leq D^{2}/2.$ $$

Following the argument in the proof of [, Thm. 3.3], the probability measure proportional to $e^{-(V^{(i)}[\boldsymbol{\mu}]+\tau\log\boldsymbol{\mu}^{(i)})/\epsilon}$ satisfies a LSI with constant $\rho\geq\alpha e^{-\beta/\epsilon}$ for some $\alpha,\beta$ independent of $s,\epsilon,\boldsymbol{\mu}_{0}$.

Then, [, Thm. 3.2] guarantees the exponential convergence with rate $e^{-Cs}$ with $C=2\epsilon\rho$. Furthermore, the convergence result with simulated annealing follows from [, Thm. 4.1].
∎

###### Theorem D.2 (Thm. 3.4 , restated) .

###### Proof.

## Appendix E Additional Experiments

Figure: Figure 6: Varying velocity. We see that as the ground truth initial velocity increases, MFL breaks while PO-MFL remains robust.
Refer to caption: extracted/5644114/figures/velocity/velocity0.png

### E.1 Settings for experiments in main text

##### “Constant velocity” model

In this experiment, the diffusivity parameter is set at $\tau=0.05$. Particles are initialized from $X_{0}\sim\mathcal{N}(0,0.1^{2}\cdot I)$ and simulated over the time interval $t\in[0,5]$ with marginals sampled at 5 evenly spaced intervals. Both PO-MFL and MFL are applied using $m=100$ particles, we observe $32$ particles at each time point, and we use a kernel width of $\sigma=1.0$ for the data-fitting term. The optimization procedure is initialized with $\eta=0.5$ and continues for 2,000 iterations. The number of Sinkhorn iterations for entropic OT is capped at 500 iterations.

For the crossing paths experiment, the diffusivity parameter is set at $\tau=0.0005$, and the time interval is $[0,2.25]$, and marginals are sampled at $10$ evenly spaced intervals we use $m=50$ particles.

##### Circular motion model

In the circular motion experiment, the diffusivity parameter is set at $\tau=0.0002$. Particles are initialized from $X_{0}\sim\mathcal{N}(0,0.1^{2}\cdot I)$ and simulated over the time interval $t\in[0,3.14]$ with marginals sampled at 15 evenly spaced intervals. Both PO-MFL and MFL are applied using $m=100$ particles, we observe $32$ particles at each time point, and we use a kernel width of $\sigma=1.0$ for the data-fitting term. The optimization procedure is initialized with $\eta=0.5$ and continues for 4,000 iterations. The number of Sinkhorn iterations for entropic OT is capped at 500 iterations.

Figure: Figure 7: Varying number of observations at the intermediate times. Increasing number of observations improves the optimization.
Refer to caption: extracted/5644114/figures/observations/observations1.png

Figure: Figure 8: A high observation regime at the intermediate time points.
Refer to caption: extracted/5644114/figures/observations/observations128.png

In the following sections, if a parameter is not stated, we assume the same setting of parameters as in the main text.

### E.2 Varying velocity

Experiments varying the mean of the ground truth initial velocity distribution are shown in Figure 6. At the endpoints, we observe $32$ particles, and in the intermediate stages, we observe just $2$ particles per time point. Note that in the small velocity regime, although MFL converges, it converges to the wrong distribution.

### E.3 Varying number of observed particles

Figures 7 and 8 show results when the number of observed samples at the intermediate time points are varied (the number of observations at the endpoints is held constant at $32$). Here, we try the same settings as above, but now we consider velocity $(\dot{x},\dot{y})=(2,4)$. We try the number of observations $1,2,4,8,16,32,128,256$. Even in a large number of observation regime, the MFL algorithm is not capable of reconstructing the full trajectory, instead clustering around the center of the overall trajectory.

### E.4 Varying temporal sampling density Δ ⁢ t Δ 𝑡 \Delta t roman_Δ italic_t

In Figure 9, we show results for increasing the density of temporal sampling. At the endpoints, we observe $32$ particles, and in the intermediate stages, we observe $2$ particles. MFL was sensitive to hyperparameter values as we needed to try different parameters to get semi-reasonable results for the figure. We used $\sigma=0.1,m=25$.

Figure: Figure 9: Varying the number of observed time points for fixed time window, i.e. varying $\Delta t$. Note that PO-MFL is always robust. MFL does better with more observations, but the method still tends to collapse inwards because its model suggests that, in expectation, particles should not be moving (as the Brownian motion reference measure has 0 expectation).
Refer to caption: extracted/5644114/figures/timepoints/timepoints3.png

## Peer Reviews

confidence: 2
contribution: 2
initial_rating: 3
presentation: 1
questions: - I don't understand precisely how optimal transport fits into the framework -- eq. 11 is a Wasserstein gradient flow w.r.t. a convexified functional in eq. 8. Is it possible to simplify the framework via Wasserstein gradient flow?
- The paper mentions an unknown potential function $\Psi$ in eq. 1, but does not mention outside the appendix afterwards. Could you explain if inferring $\Psi$ is left for future work, or if it is omitted somewhere in the later notations?
- I don't understand the explanation at the beginning of Section 3.4. Why do we need to increase the entropy by $\epsilon$ when $G + \tau H$ is already convex?
soundness: 3
strengths: While I can't seem to understand part of the paper, I think the overall approach of applying latent variable models to trajectory inference is is a sound and novel approach. I think trajectory inference for latent variables are also a key open problem in this field, and this paper proposes an approach to solving this problem. I also think that the paper's main idea is technically sound for the parts I am able to understand. 

The empirical illustration looks convincing and showcases the validity of the algorithm. While I have some reservations about whether or not it is a fair comparison with MFL, I think the paper's approach of inferring latent trajectories + a push-forward function g is overall more preferable than trajectory inference in the observation space.
summary: This paper proposes a latent variable model for trajectory inference, i.e., inferring a time-indexed collection of particles. We assume a latent state vector $(X_t)$ follows some trajectory driven by an SDE, and project the state vector into the observation space via a function $g$, and the objective is to recover the latent trajectories that give rise to the observations. 

To infer the latent trajectories, we formulate a minimum entropy functional (an approximation of the KL-divergence with Gaussian kernel smoothing) that can be optimized by an entropy-regularized gradient flow that can be simulated via a McKean-Vlasov process, i.e., the PO-MFL algorithm. 

The paper evaluates PO-MFL empirically via toy examples and Wikipedia traffic data, showing improvements compared to the original MFL.
weaknesses: While I think the paper presents an interesting approach to trajectory inference, I find the paper's presentation overall unsatisfactory. The explanations are hard to follow, and the notations are perplexing. I think the lack of readability makes it unsuitable to grant an acceptance for the current version, and it makes it difficult to assess the contribution.

- The paper is hard to follow overall. While there is a "notations" section, new notations appear throughout the rest of the paper. Many notation can be inferred from context, but that seems to defeat the purpose of a notation section.  
- Some of the confusing notation include 2 different ways to denote convolving variables with a Gaussian density in eq. 4, e.g., both $\mathcal{N}_\sigma$ and $\hat{\rho}_i^{T, h}$. $\gamma$ is both a multiplicative constant in eq. 3 and part of the OT formulation in eq. 6. 
- Some of the theoretical explanations in the paper, such as $C_\Psi$-ensemble observability and the general assumptions on $\mathcal{X}$ and $\mathcal{Y}$ are mentioned in the paper, but they are not really included in the discussion.
- There are a number of "filler" type of sentences about "recall that", "note that" which often are re-statements or clarifications. I think it is helpful to reformulate the explanation in a more cogent way. Perhaps it is a good idea to dwell on the most important concepts of the paper, and relegate some less relevant but still beneficial explanation to the appendix. 
- Some notations seem unnecessarily complex. For example, I don't understand why the time variables need to be formulated as $t_T^T$. 
- The paper makes many references to previous works in the abstract, which makes it look more like a background overview. I think the abstract needs to be restructured in a more self-contained fashion. 
- PO-MFL seems to empirically outperform other trajectory inference methods, and latent variable models are ostensibly harder to infer. But this experimental setting (e.g., Figure 4) seems more beneficial, as the latent velocity variables are easier to infer.

---

confidence: 3
contribution: 2
initial_rating: 3
presentation: 3
questions: 1. What can you say about how this method scales with the dimension of $X_t$?

2. What do you do if $\Xi$ and $\tau$ are unknown?

3. Having a the same diffusion parameter $\tau$ for all components of $X_t$ (e.g., for both the evolution of the positions and of the velocities) seems very limiting. Is there any hope of relaxing this assumption?

4. I didn't understand the "crossing paths" experiment discussed in Lines 477--500 and Figure 4. Can you explain this more clearly?

5. What is the real-world significance of the "Wikipedia traffic data" experiment. In this case, all the traffic data has already been observed. What is the real-world utility of the estimated trajectories?

6. In the caption of Table 1, what does "MCMC" refer to here? I didn't see MCMC methods being mentioned anywhere else.
soundness: 2
strengths: 1. **Clarity.** I believe that this work is overall fairly well written (though see some of the comments/questions below, especially with regards to the explanation of some of the experiments).

2. **Novelty/Originality.** The proposed methodology seems new. The main innovation is the extension of a previous approach to allow for a non-zero drift term $\Xi$. This seems fairly moderate, but IMHO sufficient given that it requires non-trivial extensions to the theoretical guarantees which the authors provide.

3. **Quality/Correctness.** This paper comes with a number of rigorously derived theoretical guarantees (though I did not have time to verify all details nor check the derivations in the appendix).
summary: This work proposes methodology for sampling trajectories of a diffusion which is only partially (but repeatedly) observed at a finite number of time points. At each iteration, the proposed methodology uses optimal-transport ideas to construct a joint distribution over the states of the diffusion at observation times from their marginal distributions (the latter are approximated via samples/"particles"). At the end of each iteration, a mean-field Langevin algorithm is then used to optimise the particle locations.
weaknesses: **Main issues:**


1. **Lack of comparison with simple Monte Carlo alternatives.** I believe the method needs to be compared against a simple Monte Carlo alternative like a particle filter (with backward sampling to obtain trajectories) or a simple MCMC method. The method is currently only compared against an alternative, denoted "MFL", which seems to be so heavily biased that it makes almost anything look good by comparison.

### EDIT: 2024-11-22: As explained in my comment below, Monte Carlo alternatives may likely need to be modified in a not-completely-trivial way to be applicable here. Therefore, I agree that it is fine not to compare against these. 


2. **Seemingly high computational cost.** The actual computation time and total number of iterations used in the numerical experiments should be mentioned in the main text. The only mention I found is in Appendix F.1 where it says "[s]ynthetic experiments take a few minutes to run, and Wikipedia experiments take a few hours to run." However, the phrase "a few minutes" is too vague. Further, the synthetic experiments are fairly small (the dimension of $X_t$ seems to be between 2 and 4). I would expect that a simple Monte Carlo alternative (like those mentioned above) would run in mere seconds rather than minutes.


3. **Variance of target distribution seems poorly captured.** As shown in Figure 1c, and commented on in Lines 467--469, the *average* of the sampled velocities seems to match the ground truth. However, the variance seems to be heavily overestimated (or is this just a visual artifact of using scatter plots with different numbers of points?). Additionally, Figure 1a seems to indicate that the variance of the ground truth increases over time but this does not seem to be reflected in the trajectories produced by the proposed method shown in Figure 1b.


4. **Lack of clarity about the real-data experiment.** I find the "Wikipedia traffic data" experiment impossible to understand. In Line 511, the dynamics(?) $x_{t+1} = \theta_1 x_t + \theta_2 x_{t-6}$ are stated. But it is not clear to me how this discrete-time autoregression relates to the SDE fro (1)? That is, what are $\Xi$ and $\tau$ here? Furthermore, are the three websites modelled/estimated separately or somehow jointly (e.g., do the trajectories shown in Figure 5b represent a single website or all three)?


**Minor issues:**

* In Line 32, it should be explained what "unpaired observations" means.

* The term "MFL" appears multiple times throughout the manuscript but sometimes refers to the minimisation algorithm (also used within the proposed methods) and sometimes to the algorithm used as a benchmark in the numerical experiments. I found this confusing, especially because it makes it difficult to find the reference for the benchmark algorithm.

* Figure 2: The tick labels on the 1st axis in the left panel are confusing. Later in the text it is explained that these the final 500 iterations rather than Iterations 1 to 500.

* All axes need to be labelled in all figures. And the 1st-axis labels in Figure 5a are all overlapping and thus unreadable.

* Figures should not appear in the text before they are referenced. This is particularly confusing with Figure 1 which appears on Page 2 but is only referenced/used on Page 9. But there are similar problems with Figures 2 and 3.

* The number of footnotes seems excessive. Many of these, e.g., Footnote 2 an 3, should be included in the text.

* Bibliography: Most entries in the bibliography are missing the journal/conference name. There are also many typos (e.g., missing capitalisation such as "wasserstein", "schroedinger", "brownian", etc).



**Typos:**

* L135: $H(\mu|\nu)$ is defined for probability measures $\mu$ and $\nu$ here but is later used with non-probability measures.

* L144: The last line in the "notation" paragraph seems incomplete since neither $g$ nor $\xi$ have been previously defined and it should probably be stated explicitly that $g_{\sharp\mu}$ denotes the pushforward measure.

* L231: "measurement" -> "measurements"

* Equation 4: Define the "$*$" operator.

* L251: What is $\hat{\mu}_i^{T, h}$?

* L253: "computationally more effective" -> "computationally more efficient"

* L280: Equation 7 is referenced before it appears in the text.

* L289: Shouldn't the first term in the cost function be $-\tau_i$ rather than $-\Delta t_i$?

* L335: Shouldn't "approximate (10)" be replaced by "approximate the first two terms in (10)"?

* L338: There seems to be a space missing at the end of the sentence.

* L362: What is "$\xi$" (i.e., without a superscript) here?

* L367: Use natbib's \citet rather than \citep

* Algorithm 1, Line 8: Is there a "hat" accent missing on $\mathbf{m}$ in the comment?

* Algorithm 1, Lines 8 and 10 (and in the "Require" line): Shouldn't this be "$T$" rather than "$t$"?

---

confidence: 2
contribution: 4
initial_rating: 8
presentation: 3
questions: 1. How does the computational complexity scale against other methods such as regular MFL?

2. It would improve the strength of the experiments to compare to other methods. Would comparing to something like rejection sampling based methods work? Even with a high computational expense, it could be worth it to compare to as a baseline?

3. Are there any scenarios (within the Partially Observed Trajectory Inference setting) in which PO-MFL will break down? What restrictions does the method have, if you know of any? Are there any limiting factors?

4. Are there any limitations or concerns related to relying on Assumption 2.1? Does the restriction for $\Xi$ to be known, divergence-free and Lipschitz continuous restrict the applicability to some real world systems that have more complex dynamics?
soundness: 4
strengths: 1. The paper introduces a well developed framework, extending mean-field Langevin dynamics with latent dynamics modelling for partially observed trajectory inference, making substantial advancements in the field. 

2. The paper is theoretically robust - providing a wealth of theory that seeks to capture the applicability and performance of the method, such as the consistency and convergence results.

3. The experiments are varied on synthetic and real-world data, provide insights into the method, and empirically back up theoretical claims.

4. The paper is extremely well written, providing a good background into the subject and gradually introducing concepts to build up to a final goal.

5. The work is framed well within the global context of the problem, highlighting relevant works and providing comparisons and gaps in the literature when needed. Applications of the problem are presented in many cases, showcasing the general applicability of the need for solving the Partially Observed Trajectory Inference problem.
summary: The paper proposes an extension of existing methods for the trajectory inference problem, focusing on where states are partially observed only through "snapshots" at a given timepoint. These snapshots, for example, capture position but not velocity. They are also unlabelled, for example, you cannot know whether a particle at timepoint t_1 corresponds to a particle at timepoint t_2. The objective is to infer the trajectory distribution of the particles over time. 

The authors introduce the PO-MFL (Partially observed mean field Langevin) algorithm to address this challenging problem, developing a deep theoretical contribution that robustly explores their algorithm, including a weak consistency result (section 3.1) showing that their proposed objective function minimises to the true law of the SDE, under some assumptions and in the limit, in ideal conditions. As well as a convergence result (section 3.4), indicating that the PO-MFL algorithm’s approximation of the true trajectory distribution improves over time and eventually stabilises, showing the robustness of PO-MFL.

Finally, the paper grounds their theory in robust experiments - for example, showing empirically that the convergence from Theorem 3.4 holds in Figure 2. Additional experiments show the usefulness of the method, testing robustness across various settings on synthetic and real-world data.

I congratulate the authors on a well constructed paper.
weaknesses: 1. The computational complexity of the algorithm is $O(NTm^2)$, which is not explored further than it is stated, nor is it compared against other methods in the experiments section. Some details on real-world runtime could be useful, due to its (presumably) large scaling cost.

2. Experiments show comparisons to regular MFL, which is not suited for the unobserved case, and therefore there are no comparisons to methods that are suited for this domain. However, I am unfamiliar with the literature, so not sure if there are other methods that can be used in this case.

3. Whilst there is some discussion on the limitation towards model misspecification in the summary section, limitations of the methodology are not discussed in great detail.

Meta-review: This work extends trajectory inference to latent state-space models using partially observed data and specified dynamics (e.g., constant velocity/acceleration). It introduces the PO-MFL algorithm with theoretical guarantees, building on prior stochastic differential equation frameworks. Experiments show robust performance, exponential convergence, and significant improvement over latent-free baselines.

Two reviewers recommended acceptance, noting that the paper makes substantial contributions. One reviewer highlighted the soundness and novelty of applying latent variable models to trajectory inference, stating that the empirical illustrations convincingly demonstrate the algorithm's validity. Another reviewer praised the paper for its clarity and originality.

However, the third reviewer recommended rejection, citing a lack of clear presentation in the paper.

The AC sides with the majority vote, acknowledging that the paper's strengths significantly outweigh its weaknesses, such as poor presentation. The AC strongly encourages the authors to improve the paper's presentation, as they committed to doing in their rebuttal.

Decision: Accept (Poster)