Multi-Marginal Flow Matching with Adversarially Learnt Interpolants
Authors: Oskar Kviman
Kirill Tamogashev
Nicola Branchini
Víctor Elvira
Jens Lagergren
Nikolay Malkin
Venue: ICLR 2026 Poster (2026)
Keywords: flow matching
stochastic interpolants
adversarial learning
scRNA-seq
trajectory inference
Primary area: generative models
TL;DR: We learn neurally parametrised interpolants in multi-marginal flow matching using a GAN-inspired adversarial loss.

Abstract: Learning the dynamics of a process given sampled observations at
  several time points is an important but difficult task in many
  scientific applications.
  When no ground-truth trajectories are available, but one has only
  snapshots of data taken at discrete time steps, the problem of
  modelling the dynamics, and thus inferring the underlying
  trajectories, can be solved by multi-marginal generalisations of
  flow matching algorithms.
  This paper introduces a novel flow matching method that overcomes
  the limitations of existing multi-marginal trajectory inference algorithms.
  Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss
  to fit neurally parameterised interpolant curves between source and
  target points such that the marginal distributions at intermediate
  time points are close to the observed distributions.
  The resulting interpolants are smooth trajectories that, as we
  show, are unique under mild assumptions.
  These interpolants are subsequently marginalised by a flow matching
  algorithm, yielding a trained vector field for the underlying dynamics.
  ALI-CFM outperforms existing baselines on spatial transcriptomics and
  cell tracking problems, while performing on par with them on
  single-cell trajectory prediction, which showcases its versatility and scalability.

Published as a conference paper at ICLR 2026
MULTI-MARGINAL FLOW MATCHING WITH ADVERSARIALLY LEARNT INTERPOLANTS
Oskar Kviman' KTH
Kirill Tamogasher'
Nícola Branchini
University of Edinburgh
University of Edinburgh
Victor Elvira
Jens Lagergren KTH
Nikolay Malkin
University of Edinburgh
University of Edinburgh
CIFAR Fellow
ABSTRACT
Learning the dynamics of process given sampled observations at several time points is an important but difficult task many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring generalisations of
the underlying trajectories, can be solved by flow matching algorithms. This paper introduces a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss fit neurally parameterised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. ALI-CFM outperforms existing baselines spatial transcriptomics and cell tracking problems, while performing on par with them single-cell trajectory prediction, which showcases its versatility and scalability.
INTRODUCTION
Modelling the time-dependent dynamics of system given experimental observa- tions is central task in many scientific problems in biology (see Schiebinger et al. (2019); Bunne et al. (2023)), medicine (see Oeppen and Vaupel (2002): Hay et al. (2021)), and other areas. The problem involves a collection of data snapshots taken at various time steps that together provide an empirical account of some process. Examples of such processes include recordings of health measurements, evolution of disease (Waddington, 1942; Hay et al., 2021), and time-series single-cell RNA sequencing data (scRNA-seq: Macosko al. (2015); Klein et al. (2015)).
Go
du
41
Formally, (deterministic) system in R" can be de- scribed by an ordinary differential equation (ODE) dx, vy(x;) dr, where is a time-dependent vector field. In a bimarginal case where samples are given from pair of marginal distributions 4o. 91. we aim to find v, such that the ODE's integration map from f 0 to = pushes qo to 91 In the multi-marginal case, the marginal distribu- tions p, induced by the dynamics with initial conditions Po = 4o should also satisfy intermediate
Adversarially learnt inter- polants (red eurves) follow pushforward distributions (red densities) that approx- imate the intermediate-time marginal dis- tributions 4,, (blue) and, by construction, have the correct end-marginals 4o and 41-
Figure 1:
Equal contribution. Correspondence tattaaaaaeeesssmms.... ed uk

---

Published as a conference paper at ICLR 2026
Data
Prior interpolant-based methods
Ours
Inte seerpolant
Subset of data
Piecewise linear interp.
Cubic spline interp
LAND metric interp.
Adversarial interp.
Marginal flow
0,
marginals
OFECCPM
OT-MMFM
OT-MPM
OT-LLLCPM
Figure 2: Comparison of CFM (Tong et al., 2024), MFM (Kapušniak et al., 2024), MMFM (Rohbeck al., 2025) and our ALI-CFM method on synthetic 2D 'knot' distribution. See $4 for details.
conditions, namely, Рt, q4, for a set of times 11 12 < lg 1. The intermediate distributions 94, are provided by datasets of samples.
While this problem can be tackled by applying flow matching (Lipman et al., 2023; Albergo et al., 2025; Liu et al., 2023; Tong et al., 2024) to learn flows for every pair of consecutive marginals 4n 9: (Tong et al., 2024), we show that such approaches can lead to non-smooth interpolation curves. Recently, several methods specialising in multi-marginal problems have been proposed. Rohbeck et al. (2025) proposes to use cubic splines to build trajectories that pass through samples from intermediate distributions. However, spline interpolation methods do not seem scale well to high dimensions (Lee et al., 2025). Another line of work (Neklyudov et al., 2024; Kapuániak et al., 2024) assumes that the interpolants in each interval [t, 4++]] follow certain geometry that can be learnt. These methods fit an interpolant and use it for learning the vector field of the underlying dynamics. However, the interpolants still piecewise and are difficult marginalise with precision, as we show in $4.1 and $4.2 (see, e.g., Fig. 2).
To overcome the limitations of previous methods. we propose to learn the interpolations between distributions using GAN-like adversarial objective (Goodfellow et al., 2014; Huang et al., 2024) obtain what we call adversarially learnt interpolants (ALIs). ALI directly matches the target intermediate marginals with those of the learnt interpolants (Fig. 1). The interpolants can then be marginalised by conditional flow matching (CFM) method, allowing us model complex time-dependent behaviour while using available intermediate-time information. The full algorithm, called ALI-CFM, approximates the data distribution at each time step, as opposed to prior methods that force interpolants pass through samples explicitly, which makes method especially useful in cases where the provided dataset is noisy.
Our core contributions in this paper are the following:
(1) We propose novel technique for learning interpolants using a GAN-like adversarial objective. We show that the introduced algorithm be easily used for multi-marginal problems with snapshots in either discrete or continuous time. We show the versatility of ALI-CFM on both synthetic and real-data examples.
(2) We successfully apply our method to a tumour coordinate inference problem using spatial transcriptomics (ST) data, where method significantly outperforms all the existing baselines. (3) We demonstrate the scalability of ALI-CFM on scRNA-seq trajectory inference problem.
METHOD
Background. We summarise CFM, roughly following the setting and exposition of Lipman et al. (2023); Tong et al. (2024); Pooladian et al. (2023). All statements below hold under regularity conditions whose details not relevant in this paper.
2

---

Published as a conference paper at ICLR 2026
We assume dynamical system in R= given by ordinary differential equation (ODE) dx, vr(x,) dr. Its integration map ,, RR" - R" from time to time r satisfies
d
(x)) (xo)). dt
vx xo-
(1)
The integration map, together with stochastic initial conditions No ~ po. defines a probability path Pr (W/))Po. where Pr is the marginal distribution of xf,
In the bimarginal FM setting, one observes samples from two marginal distributions, xo ~ qo and x1 41, and fixes a vector field v, such that its integration map 1o, satisfies (1/1)a90 = 41. (This vector field is not tractably computable, but is described by interpolants, as we discuss below.) Having fixed v, target, the goal is approximate it with neural net uf with weights 0. This could be done via the FM objective
LFM E1- Il (x) (x)
(2)
However, since v, and Pt are intractable to compute and sample from, the loss (2) is intractable.
Instead, one assumes family of interpolant curves, one for every pair Xi in the support of qo 3 q1. These curves are denoted G (x. and should satisfy ((x..x1. = for f = 0, 1. For example, a linear interpolant is described by G (xp,, X1. fx + (1 r)xp. We define v, (,, xo. := d G (xx..1,1) f). For any joint distribution π over R" R" whose marginals are qo and qı, q1.respectivg di it can then be shown that the marginal vector field
respectively,
(x,,) == E (x, x1)
X; = (), (xo. ~ n,
pushes qo to qı and can thus be used as the learning target for u²- The marginals pr of the resulting dynamies tractably sampled by drawing (xx,x1) ~ æ and setting x, G X1, While v, itself is still not tractable, one can replace (2) by the following conditional flow matching (CFM) objective:
LCFM = [o- Ilu, (x,) (xy]Ao.xx1)1 where x, Gx.
(4)
It is easy to show that the gradients of (4) and (2) coincide, and (4) thus provides a tractable way to learn the target vector field (Lipman et al., 2023).
The above setting leaves two choices open: the coupling π and the interpolants G. Past work has proposed an independent coupling π = qo @ q1, giving objectives equivalent to those in Liu et al. (2023); Albergo et al. (2025), couplings computed by (possibly minibatch entropic) optimal transport (Tong et al., 2024; Pooladian et al., 2023). The latter has been shown to result in straighter integration curves and solve the dynamic optimal transport problem. For the interpolants, linear and trigonometric (Albergo et al., 2025) curves have been proposed, as well as those trained to pass through areas of high data distribution density (Kapušniak et al., 2024). In the multi-margin marrinal setting, piecewise linear (Tong et al., 2024) and cubic spline (Rohbeck et al., 2025) interpolants have been used. See $3 for a discussion.
2.1 ADVERSARIAL LEARNING OF INTERPOLANTS
We move from the bimarginal to the multi-marginal setting, where data has been collected from sequence of K marginal distributions, Xr, 9r, with corresponding time stamps f1 12 < < fx= 1. Given a coupling π between go and qı and neural network fa, we consider neural interpolants (Neklyudov et al., 2024; Kapušniak et al., 2024)
sequuee
G (xo. x|. r) (1 - (1-r)xo + xxi + +(1 - 1) (Xo, x).
(5)
Our aim is match the intermediate distributions at I; of the interpolants when (xn, x the given marginals 4t, that is, to enforce
(Go(s:54i)arz = 4t,-
(6)
(The parametrisation (5) guarantees (XO. X1, = X2 for r = 0, 1, so (6) holds automatically for i E (1,K).) In order to approximately enforce (6), we use an adversarial learning scheme. Let (x,,x, 1) be a second neural network, tasked to discriminate between marginal samples, X, qr, and the learnable interpolants in (5). Optimising the min-max GAN objective (Goodfellow et al., 2014) for each Ij
min max B(xm.xx)-m [log(1 I - Dy (x0. X1, x1,1), + By,, [log (x,(x,, [i)]
(7)
Go D,
LaaN(Go.D,.t)
3

---

Published as a conference paper at ICLR 2026
is then equivalent, under the assumption of an optimal discriminator, minimising the Jensen- Shannon divergence (Goodfellow et al., 2014) between On. and, this case, C Notably, "generators' the interpolants the targeted gt, While the noise GANs typically comes from a fixed distribution, in (7) the pair (xo. plays the role of "noise".
conditioned scalar-valued time imputs, associated with
The solutions to the min-max problem in (7) are not unique and can induce arbitrarily curved interpolants. To this end, we propose regularising terms, Leg (G guarantee unique interpolants. See $2.3 for explicit formulations of the proposed regularisers. In Algorithm we provide pseudocode for training ALIs.
fo. in the learning objective that
Algorithm Training adversarially learnt interpolants
Require: coupling æ, trainable correction network fø. trainable discriminator Dy regulariser Lreg.
regularising weight A
while Training do
2:
Draw (No, x1) ~ n, i
1)), qu, Go(xo, x1, + (1 4;) Xo + 11 + 41(1 - 11) fo (Xo, x1,1) LGAN - estimate GAN loss with X₁, and G (x0..)111)
3:
Equation (5)
4:
Equation (7)
5:
Treg estimate regularising term
Regularisers suggested in $2.3
6:
LALI LGAN + ALLcg
+ Equation (8)
7:
Update o and y using gradient V.yy TALL
8:
return adversarially learnt interpolants G
2.2
COMPLETE METHOD: ALI-CFM
The loss (7) and the above result motivate the full adversarially learnt interpolants (ALI) objective:
LA &. Dy) (2....r((......--I [LGAN(G DyjAt)
Dyjjt) + ALog
(8)
where À > O is regularisation weight.
Once the interpolants G have been trained using ALI, they can be marginalised using the CFM objective (4), using the same coupling æ." At convergence, this yields dynamical system, defined by vector field u, whose marginals P, at each f match the interpolants' marginals (G s( , f)),. further (6) has been satisfied, then the resulting flow solves the multi-marginal transport problem: Pu 4r, for all i.
We refer the complete method as ALI-CFM: consists of (i) learning interpolants G using ALI (Algorithm 1), then (ii) marginalising them to yield the time-dependent vector field u. (Algorithm 2). The prefixes I- and OT- in the algorithm names specify if the coupling used is independent or (minibatch) optimal transport plan, respectively.
Algorithm 2 Training ALI-CFM
Require: coupling trained ALI Go, trainable CFM net uº
1: while Training
2: (xo.x1)
U(0. 1]
3:
(x(Xo.X1,1) + (1 1) xx ((=
Equation (5)
4:
di G.((x.MM11) - - x xo +10+111-1 x1) +1, 21) f x1,
5:
LALLCEM - (xx (x..
Estimate CFM loss
6: Update 0 using gradient V LALI-CFM
7:
return learned vector field n
'The objective requires differentiating the learnt interpolants with respect which is done with little overhead using autograd. Note that the interpolants' parameters fixed this stage.
4

---

Published as a conference paper at ICLR 2026
2.3 REGULARISERS FOR INTERPOLANTS
In this section we propose three regularising terms to use in (8) with provable uniqueness guarantees for the first two options. See $4.3 for ablation study of the different regularisers.
Linear reference regulariser. Here we introduce a regularising term in the ALJ learning objective which penalises deviations from the linear interpolant between coupled samples from the end- marginals. Letting ((xo. X1, f) (1 f)xo (x), we define
Greg(G (i) ()), | xi) ((xo, x11
The problem of matching marginals (6) while minimising the regulariser (9) enjoys unique solutions:
Theorem 2.1. Fix E (0, qn, and a coupling π between qo and q1 such that the distribution (G.r, DuuT is absolutely continuous (a.c.) w.r. the Lebesgue measure. Then the interpolant G(...))
Rd x Rd - Rd minimising
(oox)==llG(xooX) - x1,
(10)
subject to G(.,,,)a7 = qr, exists and is unique the support of up to almost-everywhere equality.
We provide the proof in Appendix A. Note that the assumption of an a.c. target interpolant is satisfied under number of conditions, e.g., if π is the solution entropic OT with squared-euclidean cost (or either of its limiting cases: (nonentropic) OT or the independent coupling) and either qo or qı is a.c.
Piecewise linear reference regulariser. In settings where the supports of the intermediate marginal distributions differ from those of the end marginals, the linear reference from the previous section may restrict G from accurate modelling of the target distributions, especially in high dimensions. Also, when K is small, the learning of fø may benefit from objectives evaluated on the pseudo-time unit interval, opposed to on a discrete set of time stamps. As such, we propose a second regulariser that regresses Go to a piecewise linear interpolant, marginalised over / E [O. 1]
x1, (20. o)]² where we let ne, be a Markov-chained OT coupling (Tong et al., 2024) n., ( (x with
Leeg (o: f,) = ( Mi, [[|G
(11)
(oxo))
(1x 41 (x)((11))Ng
1 ft.
E(x, (No, X1, N),
(12)
1-6
li.
and which comes with the following uniqueness guarantee.
Theorem 2.2. Fix E (0, 1), qt, and a Markov-chiined OT coupling nt, = x(xI xx) T(x,
xo) (xo) between qo and qı. Let ((x, x1,x), be the piecewise linear reference map in (12), and assume that the distribution E( "..,:,)))), is a.c. with respect to the Lebesgue measure. Then the interpolant G(.,,,)) Rd x Bd - Rd minimising
B (x x1)~ [G C(x, Xx,,
(13)
subject to ((: = where π is the (No, x1 maarrinaa of Rt, exists and is unique on the support of up almost-everywheree equality. Moreover, is is averaged according to U(O. 1] in (11),
Leeg( (G: E,-U10,11 Eix0
[[(xx,,),,) B - E(x, XI,
(14)
the same G()).t) minimises Leg for U-e.e.
See Appendix A for a proof.
Norm of the second derivative regulariser. An alternative way to introduce regularisation is to add a component that would enforce a smoothness constraint on the entire interpolant. One way to do that is use the integral of the norm of the second derivative of the interpolant. A similar idea motivated the use of cubic splines in Rohbeck et al. (2025). In contrast, we directly incorporate the integral of the norm of the second derivative as regularisation term:
0² 0+
Leeg(0) (b) = (xxx0xx)
G (x00x001100 dl.
(15)
5

---

Published as a conference paper at ICLR 2026
Computing the second derivative of the neural network exactly would be prohibitively costly. Instead, we use a numerical approximation of the second derivative:
8² 0r2
G a (x) (1, h) G (No. x1.1 - -
(16)
h2
We found a Monte Carlo estimate of the integral using three samples of ~ U[O, 1] to be sufficient.
3 RELATED WORKS
Here, we describe previously proposed multi- marginal flow matching methods and relate them to our novel ALIs. We begin by stressing a distinction between our approach and all existing approaches: ALIS are interpolants that follow distributions which match the intermediate-tine marginals, 9t, in distribution (see Fig. 1), while all existing multi- multi-marginal interpolation methods match the intermediate marginals pointwise.
The work of Kapušniak et al. (2024); Neklyudov et al. (2024) is closest to our method, as they also learn nonlinear interpolants parameterised by time-dependent neural network. However, there are differences in both the algorithms and the underlying motivations. Notably, neither of these approaches explicitly approximates divergence between the intermediate time marginals and the interpolants, nor do they use adversarial training. Kapušniak et al. (2024) motivate their approach, metric flow matching (MFM), by addressing distributions that supported on a manifold with metric learning approach (Cox and Cox, 2008; Xing et al., 2002). The metric used in MFM can be chosen as either time-independent or time-dependent. The former setting trivially contrasts to ALIS as our generator and discriminator networks are time-dependent, while the latter case results in piecewise interpolants (the interpolants are conditioned to pass through the marginal samples), approach. Furthermore, our method does not rely on
which again differs from our distribution-fitie specifying a particular underlying metric, and, importantly, we show experimentally in $4.1 and $4.2 how dime-independent metrics prevent MFM from learning valuable geodesics when the geometry of the underlying dynamics changes with time.
Rohbeck et al. (2025); Lee et al. (2025) propose explicit parametrisation of interpolants using splines. However, these methods- - along with the ones discussed before (Tong et al., 2024; Neklyudov et al., 2024; Kapušniak et al., 2024) assume that the obtained interpolants should pass through the available samples marginal distributions. In addition to that, spline- interpolants (Rohbeck al., 2025; Lee al., 2025) may scale poorly in high dimensions.
EXPERIMENTS
We conduct four different experiments. To showcase the flexibility of our ALIS to solve multi- marginal problems for large K, with noisy data, and where the data geometry changes with time, consider a toy experiment $4.1 and an experiment with real cell tracking data in $4.2. In $4.3 and $4.4, experiment with scRNA-seq and ST data, respectively, where K [4. In all experiments, we compare our method with the baselines in terms of earth mover's distances (EMDs) and/or visual verifications. Finally, note that all methods here are trained with the OT coupling (see $2), but that comparison of I-ALI-CFM and OT-ALI-CFM is provided in Appendix D.2.
4.1 SYNTHETIC DATA
First, we showcase the flexibility of our ALI-CFM method on synthetic data: a sequence of 1,200 marginal distributions centred along a knot (Fig. 2). We obtain 10 samples from each marginal distribution. The experiment shows that ALI-CFM is the only method capable of accurately capturing the time-dependent geometry of the 'knot' distribution (formalised in Appendix B). For OT-MFM, considered the time-dependent LAND metric, where sequence of LAND metrics is constructed from the data at f; and f++1 and the interpolants are learnt using all the data. More experimental details are provided in Appendix D.I.
Notably, the piecewise linear interpolant (Tong et al., 2024), the cubic spline interpolants (Rohbeck al., 2025) and OT-MFM with time-dependent LAND metrics are non-smooth, which results in the

---

Published as a conference paper at ICLR 2026
4.5
5.30
(a) ALIs
(b) Piecewise linear interp.
(c) All data
(f) Subset of data
36
450
(c) OT-ALI-CFM
(d) OT-CFMM
(g) OT-ALI-CFM
(h) OT-CFM
Figure 3: (a) and (b) show ten ALIs and piecewise linear interpolants, respectively, based on the subsampled cell tracking data, while (c) and (d) depict the resulting OT-ALI-CFM and OT-CFM vector fields. In the 2D figures we plot the centroids (one per frame) of (e) all of the segmentation data, (f) the subset training data, (g) the OT-ALI-CFM trajectories and (h) the OT-CFM trajectories. Note that the OT-CFM trajectories diverge in (d) and (h).
corresponding CFM objective having high gradient variance. This was not the case when training OT-ALI-CFM the smooth ALI interpolants. In Fig. 8 visually compare the smoothness of ALIS and OT-MFM interpolants for different choices of K, demonstrating that MFM scales poorly with K, while our ALIs consistently produce smooth and accurate interpolants.
4.2 LEARNING SPATIO-TEMPORAL CELLULAR DYNAMICS WITH CELL TRACKING DATA
Encouraged by our results in Fig. 2, we aim to learn the mattiolemmporl dynamics of the positions of cell. That is, given segmentation masks of glioblastoma-astrreee times steps//onsecutive frames, provided by The Cell Tracking Challenge (Maška al., 2023), we consider learning vector fields that, when integrated, produce trajectories that follow the tracked cell movements. This is challenging as the shape of the cell contracts and expands while moves along a loopy path (Fig. 3e).
U373 cell collected in 115
To demonstrate ALIs' capacity to handle noisy datasets, we independently subsample ten segmenta- tion coordinates per time stamp. Based on the subset, we learn both interpolants and CFMs. Then, we use all Xo samples when integrating the vector field, i.c., we push all Xo samples in the dataset through the learnt vector fields with 115 integration steps (same the number of frames). Given the similar performances of OT-CFM and OT-MMFM in Fig. 2, we compare OT-ALI-CFM to OT-CFM and OT-MFM this experiment. Note that, here, we implemented two versions of OT-MFM using the time-independent and time-dependent LAND metric. The latter setting is the same as in $4.1, while the former mimics the LiDAR experiment setup in Kapušniak al. (2024), where the metric is inferred from all the data, while the interpolants are learnt using only samples from qo and q1-
Inspecting the resulting interpolants in Fig. ALIS provide a smooth map between (1) pairs while the piecewise linear interpolants, again, introduce instability reflected as non-smoothness in the marginalised trajectories. Therefore, we could not train an OT-CFM without the integrated vector fields diverging, whereas fitting an OT-ALI-CFM was straightforward. In Fig. 4, we overlay the training data and the inferred CFM trajectories on the microscopy images, visualising also the inability of time-independent OT-MFM to learn accurate dynamics when the data geometry changes with time. The plots in Fig. 10 support the latter observation. Meanwhile, the time-dependent OT-MEM produces similar interpolants OT-CFM which are not smooth due their piecewise nature, causing the vector field integration to diverge (Fig. 11). We constrained the analysis here to visual evaluations the results in the mentioned figures conclusive.
7

---

Published as a conference paper at ICLR 2026
Training data
OT-ALI-CFM
OT-CFM
OT-MFM
Figure 4: From left to right we visualize frame 1, 25, 50, 75, 100 and 115 in the sequence of microscopy images. From top to bottom we overlay the images with the subsampled training data (10 samples per frame), and all segmentation samples No pushed to the frame-specife times via the marginalised vector fields of OT-ALI-CFM, OT-CFM and OT-MFM. In concordance with the centroid plot in Fig. 3h, the trajectories from OT-CFM eventually diverge and become inaccurate. Meanwhile the fime-independent metric in OT-MFM results trajectories that do not capture the cell positions" temporal dependence.
Table I: Trajectory inference on 5D PCA scRNA-seq data. Accuracy measured in EMD (smaller better) w.r.t. the
Algorithm | Dataset +
Cite
EB
Multi
I-CFM
1.236+0.000
1.156x00.2
1.150=00091
OT-CFM
1.142:0.085
0.809
0.975
OT-MFM
0.793+00019
0.71 +0.050
0.890=0.123
left-out marginal distributions, averaged over five indepen- dent runs.
I-MMEM (Cubic splines)
2.068
2..40.0.650
1.528200000
OT-MMFM (Cubic splines)
1.099
3.530
1..07
OT-ALI-CFM (ours)
0.910s0.024
0.742
0.925
To produce Fig. 3, we trained the CFM nets for 50,000 iterations with a learning rate of 10-3 and minibatches of size 128. The CFM nets were 3-hidden-layy layer MLPs with 256 units in each layer and SELU activations. ALI was regularised with 2 and (9). More details on data and hyperparameters are provided in D.2.
4.3 SINGLE-CELL TRAJECTORY INFERENCE
As another real- world example, we experiment with time-series single-cell data, predicting the gene expression levels of cell populations that evolve over time. The samples from the K consecutive populations are naturally unpaired, and the dynamics of the gene expressions are unknown. We follow preceding works (Tong et al., 2024) on predicting the marginal distributions of held-out intermediate- time data, measuring accuracy via the carth-mover distance (EMD) between the CFM trajectories and the held-out data the corresponding time stamp. That is, using the remaining data, we train ALIs by optimising the objective in (8) and then learn ALI-CFMs by minimising (4). In OT-ALI-CFM to other CFM methods on Embryoid
Tables and 2, we compare the performance body (EB; Moon al. (2019)) data, and on Cite-seq (Cite) and Multiome (Multi) data from Lance et al. (2021). The preprocessing pipeline single-cell data closely follows that of Kapušniak et al. (2024). The only difference is that normalise the data to ensure stability during training and denormalise only for metric computations. More details are given in Appendix D.3, where we also share ablation study of the effect of regularisers the trajectory curvature and the EMDs.
Although our algorithm on par with existing baselines, we believe that the nature of adversarial training makes difficult completely outperform OT-MFM. Since our adversarially learnt inter- polant matches the points at each time step in a distribution distribution-mmtchingg matching sense, might lose in pointwise metrics to methods that trained to overfit the given data.
8

---

Published as a conference paper at ICLR 2026
Table 2:
Trajectory infer-
Dim.
50
100
ence on 50D and 100D PCA scRNA-seq data. The accura- cies here are measured in the same way in Table 1.
Algoridhm | Datasct
Cite
Multi
Cite
Multi
I-CFM
42.478a00530
51.098e0,300
49.929
57. =9.30
OT-CEM
38.367.4029
47,205+0.16
45.148-0.ar
54,630a0405
I-MFM
41.172
48.415:0.70
46.339.0.15
53.667.0.66
OT-MEM
36.471 40.40
45.87940488
42/232-0.249
41.169.a55
OT-ALI-CFM (ours)
41,44910942
46.45410.58
48.496.0.814
54.554.0812
4.4 TUMOUR COORDINATE INFERENCE WITH SPATIAL TRANSCRIPTOMICS DATA
Understanding cancer evolution is a key problem to solve for accurate tumor analysis and effective treatment development (Greaves and Maley, 2012). ST (Stâhl al., 2016) has recently enabled studies of spatial tumor dynamics (Shafighi et al., 2024). In an ST platform, thin section of tissue is placed with coarsely distributed features, where cells in the tissue are sequenced, while maintaining the spatial locations of the cells. As such, given a single tissue section, any ST platform returns 2D distribution of gene expression data, often accompanied by an hematoxylin and eosin (H&E) stained image of the sec- tion. Multiple sections are typically processed, allowing for informative analyses of the spatial organisation of cancer cells.
Table 3: Average EMD on ST data using learnt vector fields.
Method
EMD (!)
slide
OT-CFM
109.76+9.98
OT-MMFM
109.17
OT-MFM
183.88:53.92
OT-ALI-CFM
98.91+2.03
We consider the breast cancer dataset published in Mo et al. (2024), using = 4 sections, each processed with the Visium 10x Genomics platform. There are more than 1,000 samples in each section, each X, IR? and the number of samples vary with r. We seek to model the distribution of tumour regions in unobserved tissue section. As far as been attempted, however it is well-suited to being tackled using FM as the samples (coordinates) in each section are naturally umpaired.
aware, this task has not previously
The four sections are spatially ordered along the z-axis, and the authors of the dataset have assigned the coordinates in each section with tumour purity score based on the measured gene expressions. We classified all coordinates with a score above 0.8 as belonging to a tumour region' and aligned the sections (details in Appendix C), which enabled to consider common coordinate system across sections. We visualise the preprocessed data in Fig. 5 and the raw images in Fig. 7.
Following the notational convention, we let pseudo-times E [0, 1] correspond normalised z- coordinates, such that go is the distribution of coordinate samples in the bottom section, and so on. We assume equal spacing between the sections, which is reasonable assumption based on the documentation in Mo et al. (2024). We experiment with leave-one-out interpolation, omitting either section two (t = 1/3) or three (t = 2/3) from the training data. We then compare the EMD between the held-out data and the integrated vector fields. The EMD results are shown in Table 3, and visualisation provided in Fig. 6.
This dataset is challenging as the marginal distributions are highly multimodal, with modes that are not present all sections. Also, tissues can shrink when stored before processing which makes alignment challenging (Appendix C; Zeira et al. (2022)). These challenges explain the OT-CFM and OT-MMFM results, while we believe that this makes the time-dependent LAND metric in OT-MFM inaccurate.
Interestingly, although GAN optimisation is considered sensitive to multimodal data distributions (Arjovsky et al., 2017; Huang al., 2024), we found that our regulariser in (11) along with the Markov-chained OT coupling made training stable, with clearly impressive results.
All CFMs were trained using MLPs with three hidden layers, 128 units per layer and SELU activation functions. The CFMs were trained for 10,000 epochs with a batch size of 128. D, and G s were
2The threshold was chosen based on visual agreement between the tumour-classifee classified coordinates and the tumour regions in the corresponding H&F-stained images (darker regions typically imply higher density of tumour cells).

---

Published as a conference paper at ICLR 2026
Figure 5: The aligned breast cancer H-E-stiieed images with overlaid scatter plots of tumor annotated coordinates (in light red).
(h)
(c)
(d)
(e)
Figure H&E-stained histology image of the third breast cancer tissue section overlayed with (a) true tumour annotated coordinates, and CFM trajectories integrated 2/3 using (b) OT-ALI-CFM, (c) OT-CFM, (d) OT-MMFM and (e) OT-MPM. OT-ALI-CFM achieves the lowest EMD score w.r.t. held-out coordinates in (a) (Table 3).
small MIP nets with two hidden layers, 64 units per layer and ELU activations. In the ALI objective (8) we used the piecewise-linear interpolant regulariser in (11). See Appendix D.4 for more details.
DISCUSSION
We propose ALI-CFM, novel approach to learning interpolants in multi-marginal flow matching. ALI is a conceptually novel way to learn interpolations by matching the distributions of intermediate marginals, as opposed to the strict pointwise assignment used in all existing methods. We have demonstrated the superior capabilities of our method to fit complex dynamics with noisy samples from hundreds ($4.2) or thousands ($4.1) )) marginal distributions no other FM algorithm managed demonstrate comparable performance this task. Moreover, across datasets and dimensions, our ALI-CFM performs on par with the state-of-the-ar methods on single-cell trajectory inference.
Our interpolants are learnt using adversarial learning objective, which fits well the density-free problem setting in flow matching. Although GANs are challenging to train, we have proposed regularising terms the objective that, apart from guaranteeing unique interpolants, empirically stabilise training.
In fact, compared to baselines (see the final paragraph in Appendix D.I), our adversarial training can reduce the complexity of training accurate interpolants in flow matching. Additionally, there is rich literature on GAN optimisation, e.g., regarding alternative objectives (Arjovsky al., 2017; Jolicoeur- Martineau, 2019; Sun et al., 2020; Huang et al., 2024).
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
The authors are grateful to Alex Tong for helpful discussions about single-cell data preprocessing and methods from prior work. The authors also thank Hosein Toosi for providing valuable guidance regarding the preprocessing of the ST data.
The work of VE is supported by the Advanced Research and Invention Agency (ARIA). JL acknowl- edges support from the Swedish Research Council (project ID 2022-03516_ NM acknowledges support from the CIFAR Learning in Machines and Brains programme.
This work was enabled by the computational resources of the Edinburgh International Data Facility (EIDF) with funding from the Edinburgh Generative Al Laboratory (GAIL).
REFERENCES
Michael Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions. Journal of Machine Learning Research, 26(209):1-80, 2025.
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks.
International Conference on Machine Learning (ICML), 2017.
John A Bogovic, Philipp Hanslovsky, Allan Wong, and Stephan Saalfeld. Robust registration of calcium images by learned contrast synthesis. 2016 IEEE 13th International Symposiun on Biomedical Imaging (ISBI), pages 1123-1126. IEEE, 2016.
Charlotte Bunne, Stefan G Stark, Gabriele Gut, Jacobo Sarabia Del Castillo, Mitch Levesque, Kjong-Van Van Lehmann, Lucas Pelkmans, Andreas Krause, and Gunnar Rätsch. Learning single-cell perturbation responses using neural optimal transport. Nature Methods, 20(1 10111):159-1770 2023.
Michael AA Cox and Trevor Cox. Multidimenssonal scaling. In Handbook of data visualization, pages 315-347. Springer, 2008.
Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Neural Information Processing Systems (NeurIPS), 2014.
Mel Greaves and Carlo C Maley. Clonal evolution cancer. Nature, 481 (7381):300633,, 2012.
James A Hay, Lee Gennedy-Shaffee Sanjat Kanjilal, Niall J Lennon, Stacey B Gabriel, Marc Lipsitch, and Michael J Mina. Estimating epidemiologic dynamics from cross-sectional viral load distributions. Science, 37366552)eeab06635 2021.
Nick Huang, Aaron Gokaslan, Volodymyr Kuleshov, and James Tompkin. The GAN is dead; long live the GAN! modern GAN baseline. Neural Information Processing Systems (NeurIPS), 2024.
Alexia Jolicoeur-Marineau. The relativistic discriminator: A key clement missing from standard GAN. International Conference Learning Representations (ICLR), 2019.
Kacper Kapuániak, Peter Potaptchik, Teodora Reu, Leo Zhang. Alexander Tong. Michael Bronstein, Avishek Joey Bose, and Francesco Di Giovanni. Metric flow matching for smooth interpolations on the data manifold. Neural Information Processing Systems (NeurIPS), 2024.
Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. International Conference on Learning Representations (ICLR), 2015.
Allon M Klein, Linas Mazutis, llke Akartuna, Naren Tallapragada, Adrian Veres, Victor Li, Leonid Peshkin, David A Weitz, and Mare W Kirschner. Droplet barcoding for single-cell transcriptomics applied to embryonic stem cells. Cell, 161(5):11877 117-):::18-110111 2015.
Christopher Lance, Malte D Luecken, Daniel B Burkhardt, Robrecht Cannoodt, Pia Rautenstrauch, Anna Laddach, Aidyn Ubingazhibov, Zhi-Jie Cao, Kaiwen Deng, Sumeer Khan, et al. Multimodal single cell data integration challenge: results and lessons learned. Neural Information Processing Systems (NeurIPS) Competitions and Demonstrations, 2021.

---

Published as a conference paper at ICLR 2026
Justin Lee, Behnaz Moradijamei, and Heman Shakeri. Multi-marginal stochastic flow matching for high- nigh-dimensional snapshot data at irregular time points. International Conference on Machine Learning (ICML), 2025.
Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling. International Conference on Learning Representations (ICLR), 2023.
Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. International Conference on Learning Representations (ICIR), 2023.
Evan Z Macosko, Anindita Basu, Rahul Satija, James Nemesh, Karthik Shekhar, Melissa Goldman, Itay Tirosh, Allison R Bialas, Nolan Kamitaki, Emily M Martersteck, et al. Highly parallel genome- wide expression profiling of individual cells using nanoliter droplets. Cell, (61(5):1202-1244 2015.
Martin Maška, Vladimír Ulman, Pablo Delgado- -odriige, Estibaliz Gómez-de Mariscal, Tereza Nečasová, Fidel Guerrero Peña, Tsang Ing Ren, Elliot M Meyerowitz, Tim Scherr, Katharina Löffler, et al. The cell tracking challenge: 10 years of objective benchmarking. Nature Methods, 20(7):100-1020, 2023.
Chia-Kuei Mo, Jingxian Liu, Siqi Chen, Erik Storrs, Andre Luiz Targino da Costa, Andrew Houston, Michael C Wendl, Reyka G Jayasinghe, Michael D Iglesia, Cong Ma, et al. Tumour evolution and microenvironment interactions in 2D and 3D space. Nature, 1 178-1116, 2024.
Kevin R Moon, David Van Dijk, Zheng Wang. Scott Gigante, Daniel B Burkhardt, William S Chen, Kristina Yim, Antonia van den Elzen, Matthew Hir, Ronald Coifman, et al. Visualizing structure and transitions in high-dimensional biological data. Nature Biotechnology. 37(12):
1482-1492, 2019.
Kirill Neklyudov, Rob Brekelmans, Alexander Tong, Lazar Atanackovic, Qiang Liu, and Alireza Makhzani. A computational framework for solving Wasserstein lagrangian flows. International Conference on Machine Learning (ICML), 2024.
Jim Oeppen and James W Vaupel. Broken limits to life expectancy. Science, 29665570):102--1031. 2002.
Pooladian, Heli Ben-Hamu, Carles Domingo-Enrich, Brandon Amos, Yaron Lipman, and Ricky TQ Chen. Multisample flow matching: Straightening flows with minibatch couplings. International Conference on Machine Learning (ICML), 2023.
Martin Rohbeck, Edward De Brouwer, Charlotte Bunne, Jan- Christian Huetter, Anne Biton, Kelvin Chen, Aviv Regev, and Romain Lopez. Modeling complex system dynamics with flow matching across time and conditions. International Conference on Learning Representations (ICLR), 2025.
Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould, Siyan Liu, Stacie Lin, Peter Berube, et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell, 176(4):928-9))) 943, 2019.
Johannes Schindelin, Ignacio Arganda-Carreras, Carreras, Erwin Frise, Verena Kaynig, Mark Longair, Tobias Pietzsch, Stephan Preibisch, Curtis Rueden, Stephan Saalfeld, Benjamin Schmid, et al. Fiji: an open-source platform for biological-image analysis. Nature Methods, 9(7):676-66 6622,, 2012.
Shadi Shafighi, Agnieszka Geras, Barbara Jurzysta, Alireza Sahaf Nacini, Igor Filipiuk, Alicja Raczkowska, Hosein Toosi, Lukasz Koperski, Kim Thrane, Camilla Engblom, et al. Integrative spatial and genomic analysis of tumor heterogeneity with tumoroscope. Nature Communications, 15(1))9343, 2024.
Patrik L Ståhl, Fredrik Salmén, Sanja Vickovic, Anna Lundmark, José Fernández Navarro, Jens Mag- nusson, Stefania Giacomello, Michaela Asp, Jakub O Westholm, Mikael Huss, al. Visualization and analysis of gene expression tissue sections by spatial transcriptomics. Science, 353(4294): 78-82, 2016.
12

---

Published as a conference paper at ICLR 2026
Ruoyu Sun, Tiantian Fang, and Alexander Schwing. Towards better global loss landscape of GANs. Neural Information Processing Systems (NeurIPS), 2020.
Alexander Tong, Jessie Huang, Guy Wolf, David Van Dijk, and Smita Krishnaswamy. TrajectoryNet: A dynamic optimal transport network for modeling cellular dynamics. International Conference on Machine Learning (ICML), 2020.
Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector- Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions on Machine Learning Research, 2024.
Conrad H Waddington. The epigenotype. Endeavour, 1:18-20, 1942.
Eric Xing, Michael Jordan, Stuart Russell, and Andrew Ng. Distance metric learning with application to clustering with side-information. Neural Information Processing Systems (NeurIPS), 2002.
Ron Zeira, Max Land, Alexander Strzalkowski, and Benjamin J Raphael. Alignment and integration of spatial transcriptomicc data. Nature Methods, 19655567-575, 2022.
13

---

Published as a conference paper at ICLR 2026
A PROOF OF UNIQUENESS THEOREMS
Here we restate the theorems from $2.3 and provide their proofs.
Theorem 2.1. Fix E (0, qi, and a coupling x between qo and q1 such that the distribution EC. ..TT is absolutely continuous (a.c.) w.r.t the Lebesgue measure. Then the interpolant G(::.t)=
Rd x Rdd - Rd minimising
(x0x))==llG(Xo... - x(,0,x1,1)1
(10)
subject to G(.,.,002 = 4n exists and is unique on the support of π up to almost-everywhere equality.
Proof of Theorem 2.1. Let G(:,,t) be any function satisfying the constraint. Consider the following joint distribution X xX,
[(xx, dx() = (f(s)))), G(: (()..))a...
(17)
The marginals of II over the first and second components are the a.c. distributions ((.,,))*T and ar- respectively. The distribution II is a Kantorovich plan between these two marginals whose cost is given by (10). This cost has unique minimiser over all transport maps (up to a.e. equality on the supports) because the first marginal is a.c., and the minimiser has II deterministic over the second component given the first, i.e., is given by (Id, T)x((-.,,)) for some function T X + X.
This minimum is indeed uniquely achieved by G T E f. Conversely, the optimal transport plan from ((, ((,,,))u Duu to qr yields an interpolant in the obvious manner, showing existence.
Theorem 2.2. Fix E (0. qt, and a Markov-chained OT coupling ne, = π(xi xx) (X xo) go(xx) between qo and q1. Let E(x, x,, be the piecewise linear reference map in (12), and assume that the distribution E( "'.""''" is a.c. with respect to the Lebesgue measure. Then the interpolant G(.,.,t) Rd x Rd Rd minimising
B x1) [[0(xxx)) [(x, XI. rIl?
(13)
subject to G(.,.,)) = qr. where π is the (No, maaamrialll Rt, exists and is unique on the support of up elmost-everywhere equality. Moreover, if is averaged according to U(O, 1] in (11),
Leeg (;) E,-U[o.1] B(xo, x1) mi [[G(xox1,,)) - [(x, 1 X10.X1. x),
(14)
the same G(:)()) minimises Legg for U-e..
Proof of Theorem 2.2. The proof of the first part of the theorem follows similar arguments in Theorem 2.1. Let G(:,.() be any function satisfying the constraint. Consider the following joint distribution on X x X:
Idxx, dx,) (e(- (5),), G(::1))
(18)
The marginals of II over the first and second components are the a.c. distributions E(- 1.. DuT, and qı, respectively. As in Theorem 2.1, the distribution a Kantorovich plan between these two marginals whose cost is
[ xTT=xx)) = (xx)xxx (lt.aa))),,/GGAAaaaaa (G,
E(x, xo. x1, olf
(19)
i.e. the regulariser in (11). This cost has unique minimiser over all transport maps II (up a.e. equality on the supports) because the first marginal is a.c., and the minimiser has II deterministic over the second component given the first, i.e., II (Id, T)*H4 for some function T X - X. where 14 ((- | 1, (), This minimum is indeed uniquely achieved by
GIx 1) = T o E(x, ()),
(x ~ Ri,
(20)
which proves existence and uniqueness on the support of л. Since the argument is pointwise in r, integrating with respect to any U on [0, (as in (11)) preserves optimality for U-a.e. f.
14

---

Published as a conference paper at ICLR 2026
B KNOT DISTRIBUTION
We let / E [O, / ensure that the number of marginals, K, a multiple of three, and set
i= Imax
1.5.
(21)
We then partition 7 into three equally sized intervals,
I 1,x/3, I2 = ===1//31:2233
I3 [2K/341KK.
and sample X and Y coordinates as follows:
X()) Y(r)
N (110() 0012
(22)
with 0.1 and
3(5+
ifr Ii
px(1)= = (
i I 3(1-0.5) i E I3
(23a)
tanb 1)) 0.5
if E I if E I
(1 = sin(2x(1 - -0.75))
(23b)
0.5 tanh((((1 1)) 0.5
if Is
Finally, divide by three and collect ten samples from the bivariate distribution of (X(1), (r)) each the time stamps.
C PREPROCESSING THE SPATIAL TRANSCRIPTOMICS DATA
sequence of spatial transcriptomics tissue sections (e.g., in Fig. 7) is not naturally aligned since each slice is cut separately, for instance, at slightly different orientations. Small physical distortions occur during sectioning, staining, storing (the tissues are typically frozen post-sectioning), and imaging, causing stretching, folding, rotation relative to neighbouring sections. In fact, practitioners manually place the tissues on slide. As a result, the spatial coordinates reported for each section exist in their own local coordinate system and cannot be directly mapped the spots of other sections without an alignment preprocessing step. We are particularly interested aligning the spatial coordinates of the spots (the spots, or features, contain the RNA expression).
We aligned the spots by first visually aligning the raw H&E-stained images shown in Fig. 7 using the Warp plugin (Bogovic al., 2016) in Fiji (Schindelin al., 2012), an open-source biological- image analysis software. After downscaling the H&E-stained images by a factor of 1/10, we pairwise aligned all images the fourth section (Fig. 7d). To align pair of images in BigWarp one manually chooses landmarks which the software then utilises to nonlinearly warp the source image to the target image (again, Fig. is the target image all image pairs). The above accurately aligned the pixel coordinates of the source image to the target, which was clear from visual inspection.
Finally, we transformed the spot coordinates in the source section to the coordinate system of the target section. In particular, the nonlinear transformation used above (thin plate spline transformation) provided a mapping from the source pixel coordinates to the target pixel coordinate system. We found that feeding these maps to scipy. interpol ate..ate....................ee.e Reg.Reggurrae.ee iintrrriIngg iIIiteeInee epolaat object to transform the source spot coordinates accurately aligns the spot coordinates with the warped source images. The result of our alignment preprocessing is shown in Fig. 5.
D ADDITIONAL EXPERIMENTAL DETAILS AND RESULTS
D.1 KNOT EXPERIMENT
Our proposed method is trained using two-layer MLPs with 128 hidden units and ELU activation function for both fo and Dy- We d in (8), the linear reference regulariser from (9), and
15

---

Published as a conference paper at ICLR 2026
(a)
(d)
Figure 7: The raw H&E-stained images of the four tissue sections provided by Mo et al. (2024).
K 30
K 60
K 120
K 240
K 480
K 1200
LAND-ME
Figure 8: Comparison between learnt denendennndnt LAND-based interpolants used in OT-MFM, and the ALI interpolants for different number K of marginal distributions the knot dataset ($4.1).
train the interpolant for 50, 000 epochs with a batch size of 128. The nets fo and D, are optimised using separate Adam optimisers with learning rate of 10-3. The flow matching networks are parameterised as two-layer MLPs with 32 hidden units and SELU activations in every experiment, and the Adam optimiser (Kingma and Ba, 2015) is used for training. OT-ALI-CFM, OT-CFM and OT-MMFF were trained for 30000 epochs with a learning rate of 10-4, See below for OT-MFM details.
OT-MFM hyperparameters with time-independent LAND metric We train OT-MFM for 10, 000, and 40, 000 epochs with Adam optimiser and learning rate 10-4 for both the interpolant and flow matching networks. We experimented with other learning rates; however, we observed no difference degraded performance. We kept the default hyperparameters from the MFM codebase (Kapušniak et al., 2024) with the LAND metric. Following the net parameter choices for ALI on this data, the interpolant network has two hidden layers with 128 units, and the vector field net has two layers and 64 hidden units. All networks have ELU activations. The batch size is set to 128.
OT-MFM with time-dependent LAND metric We additionally ablate the use of LAND metire in MFM (Kapušniak et al., 2024) algorithm. Fig. 8 shows how the learnt interpolants for MFM and ALI change as the number of marginal distributions increases. When OT-MFM equipped with time-dependent LAND metric, the interpolants are constrained to pass through the marginal samples pointwise. As such, is unsurprising that the resulting geopath interpolants resemble the piecewise linear and cubic spline interpolants in Fig. 2. As we see, the interpolants, trained using LAND MFM become highly non-smoon non-smooth, whereas ALI interpolants stay smooth regardless the number of marginals.
As for the other methods in Fig. 2, we trained the vector field for 30,000 iterations with a learning rate of 14 and two-layered MLP with 32 hidden units per layer and SELU activations.
OT-MFM with a time-dependent RBF metric Beyond the setting above, we also attempted to run MFM on the knot data with the tinedeependent RBF metrie, it is implemented in the publications' associated code (see the single-cell trajectory inference section Kapušniak et al. (2024) when the dimensions are 50 or 100D).
16

---

Published as a conference paper at ICLR 2026
IALION OT-AL-CPM
140000
120000
100000
80000
60300
20000
10k
20k
30k
40k
50k
(a) I-ALI-CFM loss vs. OT-ALI-CFM loss
(b) T-ALI
(c) OT-ALI
Figure 9: Comparison of I-ALI-CFM and OT-ALI-CFM on the cell tracking data. As expected, the CFM losses in (a) show the increased variance in the CFM objective when using the independent coupling (red) compared to the OT coupling (blue). The effect of the increased variance is shown in (b) and (c), where the I-ALI trajectories overlap to a greater extent than the OT-ALI trajectories.
As outlined in 1, there are K 1200 marginal distributions with ten samples from each distribution. Before we can train the MFM interpolant (geopath) network, we therefore need to train I 199 RBF networks via gradient descent.
We optimised the hyperparameters in the RBF training in order to jointly optimise training time and metric loss, without getting NaN losses when training the 1199 networks. Each network is, by the construction of the experiment and the time-dependent RBF implementation, fitted to 20 samples.
With five clusters, hundreds of RBF had NaN losses independent of learning rates. This happened due to numerical issues when computing the empirical variance of the cluster-assigned points (line 76 in attpps:/giitie mfm/ wwt.geeericceeeeeeeeeeeeeee sepeerics/rbbb
com/ kkapusniak aaaaa//eete htt//reerccceeeememmmeo foc-fleee flow mat ching/ / bblb//////// ob/maiiiii. main//
We found that two clusters did not result in NaN losses, and then used 50 training epochs per RBF net with the default learning rate, ensuring converging loss curves. The training time to fit the metrics, no training of the interpolant or flow nets was around 400 minutes, i.e., 6.5 hours, on an NVIDIA GeForee RTX 3080 GPU. Recall that this is 2D data.
D.2 CELL TRACKING EXPERIMENT
The data can be found on this website, or explicitly via this link: https: data. Belltrackingchallengg ieg.eeggggneeeeeeeeeeeneiiii.n htw.rraiiinnnctttttte//--------a
zip. We use the frames available in folder 01 and the corresponding segmentation masks from folder 01_ST/SEG. We used the segmentation masks for the cell with label four.
ALI hyperparamters. Both the interpolant and the discriminator are parameterised as 2-layered MLPs with ELU activations and 256 hidden units. We train ALI for 70, 000 epochs. During training. we smooth the time-input to the generator by adding Gaussian noise with standard deviation 10-3. Both networks are trained with learning rates of 10- and separate Adam optimisers.
OT-MFM with a time-independent LAND metric. For the cell tracking experiments OT-MFM is trained for 40, 000 epochs with learning rate 10 for both interpolant and flow matching networks. The weights of both networks are optimised with Adam. We use 0.4 and 0.9 performing similarly, while leaving other hyperparameters of the LAND metric default. Training for longer also does not
17

---

Published as a conference paper at ICLR 2026
(a)
(b)
(d)
Figure 10: Different 3D views of all time-independent OT-MFM interpolants are shown in (a) and (b), while in (c) and (d) we show the 3D visualisations of the marginalised OT-MFM trajectories. In (b) we observe that the interpolants start to bifurcate around r = 0.5 and then abruptly change direction in order to satisfy the end marginal constraint. We believe the bifurcation happens due to the guidance of the metric that was fitted on all training samples. Crucially, the bifurcation in the interpolation paths causes part of the vector field to diverge, which seen (c) and (d). Note the extreme range of x-values in the 3D plots in (c) and (d) when mapping them to the scatter plots in Fig.
yield better results with these settings. Batch size set to 128. The neural networks have 128 hidden units and 3 hidden layers.
OT-MFM with a time-dependent LAND metric. For these experiments we trained both networks (same architecture as for other OT-MFM experiments) for 75000 epochs and learning rate 10-4 (Adam); best performing hyperparameters were y = 0.2 and other LAND hyperparameters set default. While it would be possible to OT every possible batch of (x,,xx,)) during training, instead precompute OT maps between the full successive pairs of marginals before training once and reuse them, as the former option was too computationally expensive.
I-ALI-CEM vs. OT-ALI-CMM In Fig. 9 we compare I-ALI-CFM and OT-ALI-CFM. As expected, the CFM loss curves exhibit higher variance for I-ALI-CFM, while the resulting I-ALI trajectories overlap to a greater extent than the OT-ALJ trajectories.
D.3 SCRNA-SEQ EXPERIMENTS
For single-cell experiments, we use the dataset available on htttps://date mende ey. hmmm/ddtaastt//hnyffffyy
the EB data used in Tong et al. (2020) can be found on https://githu/ com/ Kri haaaaaswwwaaa//aaaetttrry// eb_velocity__5.1 npz. Following Tong et al. (2024), we additionally whiten the data in the 5D experiments.
wwt.tmaatee///e//eaammmaaaaaeeet /aterr/daaaa. /
Hyperparameters for ALI-CFM. We parameterise learnable correction fo and Dy two-layered MLPs with 64 (for the 5D experiments) or 1024 (for the 50 or 100D experiments) hidden units in each layer with ELU activations. In accordance with Kapuśniak et al. (2024), we three-layered MLP with SELU activations to model u²- The number of hidden units in the layers of u, is chosen be either 64 or 1024, depending on the data dimensions, as for fo and D, above. We find that normalising the data and adding small noise (0.01 -e. E ~ NO, 1)) to the time input of G helps when training the GAN, as it makes the interpolant smoother with respect time input. At inference, we push all the samples from marginal i-1 1 to the time associated with the held-out marginal f, as in Tong et al. (2024).
For all of the single-cell datasets we pretrain the interpolant for 2. 000 steps and then train it using adversarial losses for 70, 000 steps. We empirically found that the best results on single-cell data are obtained by using R3GAN Huang et al. (2024) with the decond-derivative regulariser. We adjust the scale of the regulariser term in such way that it has the same value range as the generator loss. We use learning rate 10-3 for pretraining, 5 10-5 for both discriminator and generator for adversarial training and 10 -3 for the OT-CFM training. We find that OT coupling is important to achieve better results. We use multi-marginal OT coupling in all our experiments. We also investigate the alternatives for the Lo-norm. Following Kapušniak et al. (2024), we investigate substituting the La-norm with the "LANDD" metric in all of the previously discussed regularisers. Given the dataset
18

---

Published as a conference paper at ICLR 2026
Figure 11: Top: The time-dependent OT-MFM marginalised vector field in 3D on the cell tracking data. Bottom: the trajectories overlayed on the subset of the microscopy image frames (see Fig. 4).
D = ((x 1 and c > O we let (x,,1) - (LAND(X,, Ge (x,, = where
+ +cl)
||x; - xx,l?
||r - s|1
ha 1x)
Σ (x, - xxx))x)ppppp (xx.))E2)
exp
isas d,
(24)
Y/
with d being the dimension of x and y1 and y2 being the kernel sizes. Using "LAND" improved results. We conduct our final experiments on single-cell data using the regulariser detailed in Equation (15) with the "LAND" metric.
Reproduced results. In order to fairly compare our method to the existing ones, we rerun all the baselines hardware. The reported scores 1/OT-MFM and 1/TT-CFM are obtained by running the code provided by Kapušniak et al. (2024) without any changes made to the code. For I-MMFM and OT-MMFM (Rohbeck et al., 2025), Gaussian probability path, to align with the other considered methods. We only ran 1/OTMMFMM on the 5D experiments, because this method poorly scales to the high dimensions.
implemented the cubic splines interpolants without the
D.3.1 ABLATION STUDY FOR THE DIFFERENT REGULARISATION TERMS
We conduct ablation study different dimensions of the CITE-seq data, ablating regularisers ($2.3), regulariser coefficients, and the effect of these on EMD, computed on the left out time steps, and path energy, which quantifies the straightness of the trajectories (higher numbers, less straight paths) and is computed in the following way:
L/ ||u (x,) x go
path energy = E
(25)
The results are shared Tables 4, 5, 6.
Interestingly, we see that increasing the regularising coefficient, combined with the piecewise linear regulariser, makes the trajectories less smooth. We argue that this because the piecewise linearity forces interpolants to pass through the marginal samples pointwise, akin to OT-CFM, MFMM and MFM with a time-dependent metric, and so the trajectories become less straight.
Meanwhile, a stronger regularising weight combined with the other regularisers in general straightens interpolants (decreases the path energy). This expected as they directly penalise curvature.
D.4 SPATIAL TRANSCRIPTOMICS EXPERIMENT
In Fig. 6, we visualise some of the resulting tumour coordinate inference results.
When marginalising the vector fields during inference of tumour coordinates at the held-out time f; push the samples from the observed 44-, to using 101 integration steps and the dopri5
19

---

Published as a conference paper at ICLR 2026
Table 4:
EDM (L) for the learnt interpolant (Interpolant) and the marginalised vector field (CFM), as well as path energy (L) for the marginalised vector field (CFM path energy), computed on the left out time steps for Cite 50D and piecewise linear regulariser
Left out time step
Left out time step 2
A
Interpolant
CFM
CFM path energy
Interpolant
CFM
CFM path energy
0.01
46.96
47.29
300.01
38.12
42.17
358.98
0,05
41.11
41.07
697.19
37.70
41.17
385.26
0.10
40.76
41.19
1243.33
37.28
40.51
551.10
0.20
41.21
42.20
2979.58
38.28
41.75
238.59
0.50
41.82
40.93
5666.17
37.00
39.48
268.40
5
42,23
42.59
36264.89
35.93
41.70
6297.56
10
41.38
43.81
87052.14
35.57
43.06
8484.26
Table EDM (L) for the learnt interpolant (Interpolant) and the marginalised vector field (CFM), as well as path energy (L) for the marginalised vector field (CFM path energy), computed on the left out time steps for Cite 50D and 'the norm the second derivative' regulariser
Left out time step
Left time step
A
Interpolant
CFM
CFM path energy
Interpolant
CFM
CFM path energy
1 1.00E-06
41.50
42.17
12.30
39.63
45.66
7.58
5.00E-07
41.48
42.14
12.00
39.55
45.51
7.55
1 1.00E-07
41.44
42.06
12.56
39.35
45.23
7.68
5,00E-08
41.38
42.23
13.91
39.31
44.99
8.10
1.00E-08
40.75
41.59
26.00
39.06
41.61
13.21
5,00E-09
41.39
42.24
41.98
39.04
40.14
18.51
1.00E-09
42.89
43.81
56.39
41.24
40.04
39.65
ODE solver. For all methods, the dataset is normalised to make the coordinates be in [0, 1]. After training the CFMs, the marginalised CFM trajectories are denormalised and compared with the unnormalised true data.
ALI hyperparameters. We train Go and D, for 70, 000 epochs with the piecewise linear reference in (11), and = 10. We use batch size of 256. Additionally, pretrain G for 1000 epochs using a simple L2 regression objective between G and samples from the observed qu, During training, we add a small Gaussian noise variable with standard deviation parameter 0.001 to the time input of the generator. The learning rates in the adversarial training were 10 for both nets, and we used the Adam optimiser with default hyperparameters.
CFM hyperparameters. ALI-CFM trained using a learning rate of 10-3, Although we have tested multiple learning rates for OT-CFM and OT-MMFM, the best performance is attained using 10-3 for OT-CFM and 10-4, All nets are optimised using the Adam optimiser.
OT-MFM hyperparameters. We train OT-MFM for 40, 000 epochs with learning rate 10-4, 2- 10-4 and Adam for the interpolant and flow matching network respectively. We note that we do not report results with the multi-marginn version of OT-MFM using Eq. (20) Kapušniak et (2024); in our preliminary tests, this version exhibited worse results than the standard OT-MFM.
Batch size is set 128. We use the LAND metric this is the recommended one for low-dimensional data in Kapuśniak et al. (2024). We set the hyperparameters of the metric to 0.2 for ST data and 0.4 for cell tracking data (other values perform worse, respectively); p 10 for ST data seems to perform better than the default; the remaining hyperparameters are set to MFM defaults. The neural networks have 128 hidden units and 3 hidden layers.
20

---

Published as a conference paper at ICLR 2026
Table 6: EDM (1) for the learnt interpolant (Interpolant) and the marginalised vector field (CFM), as well as path energy (L) for the marginalised vector field (CFM path energy), computed on the left out time steps for Cite 100D and 'the norm of the second derivative' regulariser
Left out time step
Left out time step = 2
A
Interpolant
CFM
CFM path energy
Interpolant
CFM
CFM path energy
1,00E-06
47.00
49.89
6.90
44.78
51.12
6.90
5,00E-07
47.03
49.37
6.66
44.88
50.92
6.60
1,00E-07
47.03
49.65
10.02
44.80
50.99
7.09
5,00E-08
46.87
49.04
11.25
44.77
51.19
11.86
1.00E-08
46.66
48.36
25.43
44.65
49.98
31.18
5.00E-09
46.52
47.84
30.93
44.74
50.00
37.22
1.00E-09
46.97
47.92
55.48
45.97
49.60
28.22
21


Reviews:
[{"review_id": "K60rHOjyet", "reviewer": "Reviewer_1Zg7", "summary": "This paper tackles multi-marginal flow matching from  intermediate time marginal data. The paper proposed a 2-stage approach, first learn a non-linear interpolant to match intermediate marginals via adversarial learning and then fit a vector field with the CFM loss on those interpolants. Experiments on synthetic, cell-tracking, scRNA-seq, and spatial transcriptomics are provided to demostrate the marginal matching property of the proposed method.", "questions": "1. Could the authors explain the reason behind weaker performance in Section 4.3 ? \n2. Can the method work well with independent coupling ? How does it compare to OT-based? What would be the challenges? \n3. Could the authors provide ablation study for different regularizations? \n4. Could you discuss the possible challenges for high dimensional applications ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper is clearly written and easy to understand.\n2. Theoretical analysis showing existence of the interpolants strengthens the paper.\n3. Practical benefits are demonstrated in the experiments.", "weaknesses": "1. Compared to MFM, the method differs in use of GAN based distribution matching for intermediate marginals. The added complexity of adversarial training makes the idea less appealing compared to existing methods that do not require adversarial training.\n2. The performance on Section 4.3 single-cell RNA-seq seems to be weak compared to MFM that does not require adversarial training. \n3. Only OT coupling based implementation is used in the experiments. OT adds additional complexity and is challenging to compute in high dimension. \n4. Ablations are not provided for OT/Non-OT, different regularizations, etc.\n5. The empirical validation is limited to low dimensional applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ci2QvjeX0r", "reviewer": "Reviewer_xv91", "summary": "This paper proposes a new approach to learn probability distributions dynamics with multiple marginals, based on flow matching. Recognizing that previous approaches to define the interpolants across multiple marginals (piecewise linear or splines) may exhibit high curvature, thereby leading to suboptimal training, the authors propose to *learn* the interpolants. \n\nTo learnt the interpolants, the authors enforce that the marginal distributions match at each time steps, using a adversarial loss in practice. Using the learnt interpolants, the authors then plug them directly into the flow matching machinery.\n\nThe authors propose multiple trajectories regularization for learning the interpolants, such as linearity, or curvature minimization.\n\nThe authors then show qualitatively that this results in smoother trajectories on synthetic and cell tracking data, leading to more accurate flow. Quantitatively, they evaluate their approach on cellular trajectories and tumor proliferation and demonstrate that their approach is competitive with state of the art methods.", "questions": "- I’m surprised by how non-smooth the trajectories of the splines are in Figure 2. Did you compute splines between points that are matched with OT  between successive time steps ? Or abitrary points ? I think MMFM uses OT coupling between distributions.\n- The authors state `Although our algorithm is on par with existing baselines, we believe that the nature of adversarial\ntraining makes it difficult to completely outperform OT-MFM. Since our adversarially learned\ninterpolant matches the points at each time point in a distribution-matching sense, it might lose in\npointwise metrics to methods that are trained to overfit to the given points.` That is an interesting point, although the metric here is EMD so why do the authors refer to pointwise metrics ? Also, by the same argument as used by the authors, do we expect that this method may generate non-“realistic” samples ? That is, because the interpolants don't exactly align with the real data points, the model never learns to exactly generate \"realistic\" samples (it generates something slightly off). I would like the authors to comment on that as it can be a significant limitation of the method.\n- Echoing my comment above: could you please comment on having a discriminator at each time step - whether you did this or not - and potential limitations in terms of sample size.\n- Also echoing my other comment above, could the authors add the metrics I pointed to above - or  if not, could they argue why it's not relevant ? ` An important motivation for this work is to have interpolant with less curvature, hopefully leading to improved training. As such, one experimental result that is lacking from this work is the cumulative curvature of the learnt interpolants, as well as some notion of divergence between the pushforward distribution with the GAN interpolants and the observed marginal distributions (for all different interpolant strategies). Ideally, depending on the hyperparameters and the strength of the regularization used, there would be some trade-off between the cumulative curvature and the matching at each time step. In my opinion, that would really nail down the contribution of the paper.`\n- I understand that this is very challenging give the time constraints, but I think it would be great to have an established benchmark where your method outperforms previous baselines. If not, please consolidate why the ST task is actually relevant biologically.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- This paper proposes a creative approach to learning more meaningful interpolants in flow matching, which is of significant interest when dealing with multiple marginal distributions.\n- I appreciate the design of the new experiment using spatial data, although the biological relevance of this experiment is questionable as it only takes into account the spatial distribution of the tumor, discarding all the single cell RNA seq resolution of the data.\n- The paper is easy to read and the authors expose their ideas and contributions clearly.\n- The authors demonstrate the performance of their method graphically, which makes the evaluation of performance of the method easier for the reader.", "weaknesses": "- An important motivation for this work is to have interpolant with less curvature, hopefully leading to improved training. As such, one experimental result that is lacking from this work is the cumulative curvature of the learnt interpolants, as well as some notion of divergence between the pushforward distribution with the GAN interpolants and the observed marginal distributions (for all different interpolant strategies). Ideally, depending on the hyperparameters and the strength of the regularization used, there would be some trade-off between the cumulative curvature and the matching at each time step. In my opinion, that would really nail down the contribution of the paper.\n\n- The discriminator in the GAN seems to be common across all time steps. This seems suboptimal, especially for trajectories that intersect, like the synthetic knot experiment. In my opinion, you need a different discriminator for each time step, for the argument to hold, which may be challenging to train correctly when you don't have many samples per time step.\n\n- The most established benchmark in this paper is the scRNAseq trajectory inference. However, this model does not improve on that benchmark. That undermines the main motivation for this paper. The spatial dataset is, from my understanding of biology, somewhat toy-ish too, as I expect there may much better suited methods for that problem that flow matching. If this model does not improve performance on tasks that are established to be meaningful, it raises the question of \"what problem this paper is really solving?\".", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HssiXRMq23", "reviewer": "Reviewer_1S64", "summary": "This paper tackles *multi-marginal trajectory inference* from unpaired snapshots observed at discrete times. The proposed method, ALI-CFM (Adversarially Learnt Interpolants + Conditional Flow Matching), departs from hand-crafted paths (linear, spline, piecewise-linear) by *learning* a time-conditioned interpolant $G_\\phi$ whose intermediate-time pushforwards match the observed marginals.", "questions": "1. **Stability of adversarial training (W1).** Did you observe mode collapse or time-wise imbalance across $\\{t_i\\}$? How sensitive are results to $\\lambda$ and to the GAN loss (vanilla vs. R3GAN in Appx. D.3)? Any training heuristics that were essential (e.g., critic:generator update ratios, spectral norm, gradient penalties)?\n\n2. **Compute and $\\partial_t f_\\phi$ (W2).**\n   * Authors stated \"little overhead\" but would be great to report wall-clock and scaling with $(d,K,N)$ for ALI, CFM, and end-to-end ALI-CFM, compared to OT-MFM/OT-CFM.\n   * Algorithm 2 requires $\\partial_t f_\\phi(x_0,x_1,t)$. Do you autograd through $t$ every CFM step? What is the measured overhead relative to settings with analytic $\\dot{G}$?\n\n3. **Regularizer selection.** You mix linear reference, piecewise-linear reference, and $\\|\\partial_{tt} G\\|^2$ across datasets. Can you provide practical guidance (e.g., stationary vs. evolving geometry, large $K$, noise levels) and an ablation isolating each regularizer per dataset?\n\n4. **High-dimensional performance (W3).** Beyond the pointwise-vs-distribution-matching explanation, what limits ALI-CFM in 50D/100D (critic capacity, coupling $\\pi$, time-conditioning)? Have you tried stronger critics or multi-scale time features?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "* **(S1) Interpolant learning via distribution matching.** Adversarial learning of $G_\\phi$ avoids brittle pointwise constraints and yields smoother, noise-robust paths.\n\n* **(S2) Non-stationary geometry.** Clear wins on datasets with evolving topology/geometry (e.g., Knot, Cell Tracking), where time-independent metrics (e.g., MFM) underperform.\n\n* **(S3) High $K$ stability.** Handles very large numbers of time points (e.g., $K=1200$) without the \"kinks\" typical of piecewise methods, easing CFM training.\n\n* **(S4) Theory.** Uniqueness results for linear / piecewise-linear reference regularizers (Thms. 2.1–2.2) provide identifiability-style guarantees.\n\n* **(S5) Spatial transcriptomics.** Strong results on a challenging multimodal ST task and competitive performance on cell tracking benchmarks.", "weaknesses": "* **(W1) Adversarial stability.** No analysis of GAN stability, failure modes (e.g., mode collapse), or sensitivity to $\\lambda$, critic class, or choice of GAN loss.\n\n* **(W2) Compute overhead.** Two-stage training plus the need to evaluate $\\partial_t f_\\phi$ during CFM adds cost; the paper lacks wall-clock and complexity breakdowns versus single-stage baselines.\n\n* **(W3) High-dimensional scRNA-seq.** In 50D/100D, ALI-CFM is only on par with (or slightly below) OT-MFM; limited discussion of causes or remedies.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b0wNipdYAP", "reviewer": "Reviewer_YnZ4", "summary": "The paper proposes Adversarially Learned Interpolants: a GAN-style objective that directly matches intermediate-time marginals to a learnable interpolant. The full ALI-CFM pipeline first learns adversarially (with novel regularisers for uniqueness/smoothness), then regresses a time-dependent vector field via a standard CFM loss. The generator and discriminator are time-dependent, and no explicit metric is specified.\n\nThe method is evaluated against existing flow matching baselines on several tasks: a synthetic 2D knot dataset , a real-world cell tracking dataset, a standard single-cell trajectory inference benchmark , and a novel task of inferring tumour coordinates from spatial transcriptomics data.", "questions": "1. **Baseline fairness and time-localization of LAND.**  \n   Did your baselines using the LAND metric employ the *adjacent-time* constraint as in the original MFM implementation?  \n   If not, could you please report results using the time-step–localized (i.e., nearest-time) implementation of LAND as discussed in the Weaknesses section, and reflect these results in Figures 2, 4, 7, and Table 3?\n\n2. **Validity of the “geometry varies with time” claim.**  \n   If the above concern holds, the statement that LAND/RBF-based and prior methods are *“not suitable when the data geometry varies with time”* may not be accurate. Is there any additional theoretical or empirical reason that supports this claim?\n\n3. **Novelty.**  \n   If the above concern is valid and a fair, time-localized MFM performs well, what do the authors consider to be the primary novelty of ALI-CFM beyond offering an alternative formulation of a time-conditioned interpolant?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 4, "contribution": 2, "strengths": "1.  **Novel Regularisers** - the linear-reference and curvature penalties are well-motivated for stabilising adversarial training and encouraging uniqueness of interpolants\n2.  **New Benchmark Task** - Tumour coordinate inference with spatial transcriptomics is a relevant multi-marginal benchmark; the paper positions ALI-CFM credibly there and reports reproducible settings \n3.  **Problem Importance**  - The paper proposes an elegant approach to well motivated problem", "weaknesses": "1. **MFM Baseline Comparison**  \n   The paper’s central argument is that MFM is *“not suitable when the data geometry varies with time”* because their metric is *“time-independent.”* However, the statement *“their metric is time-independent, which contrasts with our generator and discriminator networks which are both time-dependent”* is somewhat misleading.  The authors explicitly state that their MFM implementation *“follows the setup … where the metric is inferred from all the data.”* This is a **specific, time-independent implementation choice**, not an inherent limitation of the MFM framework.  In fact, the LAND metric can be defined **locally in time** (e.g., using only data from the nearest time points, as done in MFM’s single-cell setup). Such a *time-localized* definition would make the metric effectively time-dependent and would likely yield more accurate MFM interpolants, especially in Figure 2.  This would basically be equivalent to time dependance learnt through Adversarial training as its the only granularity of information that we posses. \n\n2. **Novelty Claim**  \n   The paper frames its novelty as introducing a time-dependent adversarial approach that overcomes previous work time-independence. However, a time-conditioned discriminator (as used in ALI) and a time-localized LAND metric (as in a correctly implemented MFM) are conceptually similar—both use data from neighboring time points to define geometry along the interpolant.   If a fair, time-localized MFM implementation performs competitively, the core premise of the paper weakens: the proposed method may represent an alternative, but not fundamentally new, way of learning time-dependent interpolants—albeit through a more complex, GAN-based training procedure.\n\n3. **Minor: Attribution of the Interpolant Form**  \n   The interpolant parameterization in Equation (5), $G_{\\phi}(x_{0},x_{1},t)=(1-t)x_{0}+tx_{1}+t(1-t)f_{\\phi}(\\cdot)$,  is a standard form introduced in prior works (e.g., Neklyudov et al., 2024; Kapuśniak et al., 2024).  While these papers are cited in the related work section, the equation itself appears without a direct attribution, which could unintentionally suggest it is novel.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)