Carré du champ flow matching: better quality-generalisation tradeoff in generative models
Authors: Jacob Bamberger
Iolo Jones
Dennis Duncan
Michael M. Bronstein
Pierre Vandergheynst
Adam Gosztolai
Venue: ICLR 2026 Poster (2026)
Keywords: flow matching
diffusion geometry
manifold learning
regularisation
Primary area: generative models
TL;DR: We improve the quality-generalisation tradeoff in flow matching by adding geometric noise to probability paths.

Abstract: Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carré du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff, even when used as a latent space generation model. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.

Carré du champ FLOW MATCHING: BETTER QUALITY- GENERALISATION TRADEOFF IN GENERATIVE MODELS
Jacob Bamberger* Institute of Artificial Intelligence
Iolo Jones'
Dennis Duncan
Institute of
Institute of
Artificial Intelligence
Artificial Intelligence
Medical University of Vienna, University of Oxford
Medical University of Vienna, Durham University
Medical University of Vienna
Michael Bronstein
Pierre Vandergheynst EPFL.
Adam Gosztolai
University of Oxford, AITHYRA
Institute of Artificial Intelligence
Medical University of Vienna
ABSTRACT
Deep generative models often face a fundamental tradeoff: high sample quality can come the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carré du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisttion tradeoff by regularising the probability path with geometry-
noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covari- ance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers better quality- generalisation tradeoff even when used as a latent space generation model. We observe significant improvements over standard FM in data-scarce regimes and highly non non-uniformly sampled datasets, which are often encountered in AI for sci- ence applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can readily integrated into existing flow matching pipelines.
INTRODUCTION
Deep generative models aim to sample from an unknown probability density v(z), given finitely many training points D = { N
M=1 C Rd. Prominent paradigms include variational autoencoders
(Kingma & Welling, 2014), generative adversarial networks (Goodfellow et al., 2014), diffusion pro- cesses (Sohl-Dickstein et al., 2015; Ho et al., 2020), and methods based on continuous normalising flows (Chen et al., 2019; Song & Ermon, 2019; Lipman et al., 2023; Albergo et al., 2023) (CNFs). Among these, CNFs have had striking recent success across domains from image generation to molecule design and weather prediction, owing to their ability to generate high-quality samples. However, trivial and undesirable way of achieving high quality through reproducing training points or close variants, known as memorisation. Thus, in addition to high quality, another desirable property of generative models is their ability generate novel examples, known as generalisation.
Recently, there has been an increasing concern that the observed generative quality has been partly achieved through memorisation (Somepalli et al., 2023; Škrinjar et al., 2025), but possibly on large
"Equal contribution. ordered alphabetically.
Coorespondencc: adam. goantol a @meduni wi een actat

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scales (Graber al., 2025), undermining novelty, diversity, and data privacy. Recent works have shown that, geometrically, memorisation coincides with the sudden drop or complete vanishing of intrinsic dimensionality of the data manifold (Achilli al., 2024; Ross et al., 2025; Ventura et al., 2025). In other words, during memorisation, the learned distribution degenerates towards an empirical measure supported on isolated training points, rather than smooth, finite-dimensional manifold. This observation suggests that way to address memorisation is by stabilising the intrinsic dimensionality and preserving non-degenerate tangent spaces.
This paper reports an advance on mitigating the quality-generalisttion tradeoff in flow matching (FM, Lipman et al. (2023)), a unifying framework of CNFs, that models a deterministic probability path (z) between a source density at t = 0 (often Gaussian), and a target density of arbitrary complexity at = 1, and subsumes the probability paths modelled by other generative models, such as diffusion processes and score matching (Lipman et al., 2023). The standard, widely used FM construction induces, near 1, a homogeneous and isotropic Gaussian kernel approxima- tion that concentrates around each training point. In practice, most implementations consider small-bandwidth limit to maximise accuracy, thereby relying on architecture and training loss for regularisation. Empirically, we find that the quulity-generaliin while models stopped early during training typically generalise well but yield subpar sample qual- ity, training longer improves sample quality at the cost of memorisation and, consequently, reduced generalisation. This result remained consistent across datasets and for different neural network ar- chitectures (MLP, UNet, Transformer). We further demonstrate that the quality- generalisation trade- off does not simply depend on the dataset size, but on a balance between local geometry and data sparsity, indicating that memorisation can also occur in large-scale datasets.
tradeoff defines frontier for FM:
To improve the geniralaggneraiiattiinn tradeoff, we introduce Carré du champ flow matching (CDC- FM), which explicitly regularises the FM probability paths through an anisotropic and inhomoge- neous diffusion term that, for Po = .N(0, I), yields the conditional probability path (see Appendix for the general case for arbitrary initial density po)
Pr (x][x)) = N( x5 txi, [
The matrix field I controls the local Dirichlet (carré du champ) energy, and can be efficiently and robustly estimated from data using diffusion geometry (Jones, 2024a;b; Jones & Lanners, 2026), providing explicit geometrie noise regularisation aligned with the data manifold. We demonstrate that across diverse synthetic and real-world datasets and neural network architectures, CDC-FM achieves comparable or better quality than FM, preserving the fine-grained details of the data, while substantially reducing memorisation and increasing generalisation. Our work provides theoretical framework for the geometry-awar regularisation of flow-based generative models and a practical method that can be readily used in existing FM pipelines.
2 BACKGROUND
In this paper, we are interested in modelling a set of data points in Rd whose density p(x) con- centrates around an unknown lower-dimensional manifold. We begin by revisiting the standard FM formulation (Lipman et al., 2023). We highlight that the FM probability path (x)) induces at / - a homogeneous, isotropic Gaussian kernel approximation of v. This will motivate our generalised framework in Section 3, where we incorporate an anisotropic and inhomogeneous kernel approxima- tion to strike a balance between faithfully modelling the data, substantially reducing memorisation, and improving generalisation. To illustrate the distinction between the two frameworks, we use a toy problem of learning to generate a circle from equidistant training points (Fig. 1).
2.1 FLOW MATCHING
FM learns a velocity field vx(x) [0, x Fd + Rd (vanishing at the domain boundaries) that generates the probability path p((x), satisfying the continuity equation
at pc (1) = V (x))
po(x) u(x), pi(x) v(x)).
(2)
We consider u(xx) = N (2; 0, I) for simplicity, and generalise it to arbitrary μ in Appendix A. Con- ceptually, FM takes bottom-up approach by designing the flow path of particles X, == the (Xo)
2

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Conditional flow path design
b
Conditional velocity fields, (r(2((2))
Generated distributions
Flow matching
Flow matching (FM)
(FM)
samples
Flow time,
manifokd
x
image of samples
770
d
CDC-FM (ours)
CDC-FM
(ours)
30
21
Po(- [g
Ptf- 2g)
Figure I: Carré du champ flow matching. FM conditional path design is oblivious to the man- ifold structure, which can result in off-manifold samples, shown by the black arrows (amin > 0). Conditional velocity fields (blue arrows) in FM transport mass training points. Generated density by FM trained on eight samples of a unit circle (amin 0). FM memorises, concentrating likelihood around training points. d CDC-FM conditional probability paths are the displacement (optimal transport) interpolants between local covariances and are thus aligned with the geometry. e CDC-FM conditional velocity fields flow perpendicular to the manifold. CDC-FM regularises along the manifold, mitigating memorisation and facilitating generalisation.
Pt. The velocity field u is related to 4/ (Xo) by the characteristic ODE
d vx(x) = ve(V)(x)), di
vo(x) = æ.
(3)
The velocity uy in turn, induces (x), the probability of finding particle at x at time f, via (2). Specifically, FM takes the final particle position X, ~ V as an auxiliary variable (a training point) and specifies the conditional flow (/x(XXx Although the choice of (XIx) has implications for the regularity of the density v(x) (Albergo et al., 2023), the standard choice is the affine flow
(X = 1X1 + O,
(1 1) + la
This choice of flow induces mapping between Gaussian distributions via the probability path
(x(+)))) N (x tzz, of1).
(5)
which linearly interpolates the position and convolves with an isotropic Gaussian, and therefore is oblivious the manifold geometry (Fig. Ia). Given the conditional probability path, the marginal velocity field u, (x)) is learnt via a neural network üf (I) with weights @ by minimising the loss
= = (10)()) -u (x|x))) E.X.,.X / (X) dt 2e (Xoox1111 (6)
where L ~ 210 1], Xo ~ (O, I), X, ~ U, and X ~ (x After training, oe will approxi- mate u, (2) (Theorem Lipman et al. (2023)), generating the marginal probability path pt via (2). Note that, in the second equality, we used (3-4) to specify closed-form expressions for the condi- tional target velocity to lead directly to training points. Our numerical experiments corroborate this, showing that as t + 1 the learnt fields concentrate mass around training points (Fig. 1b).
Flow matching risks memorisation of training data. FM's flexibility lives in the transport, i.e., the learnt velocity field (z). Yet, in the limit - 1, it produces a fixed-bandwen bandwidth, isotropic approximation of v. Indeed, marginalising (5) over the target distribution v((1) yields the mixture:
N
V ~~ <>pi(x) = { No (x (I|21, (x( i ((11)d1
N
x
N
Thus, in the limit + 1, Jmin the probability path converges to the empirical density. While setting Gmin > 0 can achieve a fixed-bandwidth regularisation, in practice, is common to take Tmin 0, risking memorisation (Fig. Ie) as shown by simulations of toy model.
3

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3 REGULARISED FLOW MATCHING ON GENERALISED DATA MANIFOLDS
3.1 DIFFUSIVE REGULARISATION. CARRÉ DU CHAMP FLOW MATCHING (CDC-FM)
Motivated by the fact that the data density is concentrated around the manifold, introduce principled regularisation into FM by replacing the conditional flow path (4) with
of ( = 1X (X))
ΣT(x) - [(1 () +f=
(8)
where [(x) is local anisotropic covariance around We have chosen this flow path because it replaces the homogeneous, isotropic covariance in the conditional probability path (5) by
(x(((11) = Wi tæı, Σ/ (x1));
the displacement (optimal transport) interpolant between μ = NO I) and an anisotropic Gaussian centred at 2 ~ y that is geometrically aligned with the data manifold. In Appendix A (Proposition 1), we also derive the conditional probability paths valid for general μı, v, illustrated in Fig. 1d.
Note that as an alternative to (8) one could consider the naïve data augmentation that replaces the training points by perturbed samples, x(() + _N((1)) ((x())), and then uses FM flow paths (5). However, as we prove, this approach generates strictly suboptimal paths (Appendix B, Theorem 1).
Armed with the new conditional path (9), we may directly use the FM loss (6) obtain the regu- larised velocity field. Compare Algorithms 1 and in Appendix C for a condensed summary. A simple approximation shows that (8) provides an inductive bias for the learnt velocity field to ap- proximate the data manifold. To see this, we substitute our flow path (8) into the FM loss (6), noting that the target velocity is de (Xoo = xι) œ1 + [[(z, 1/2 - Xo. The random part (second term) can be approximated to leading order as [f(x))1/2-]] [ Xa
(20 +f = (O, I {(=1)). This means that if T(x1) approximates the projection map onto the local tangent space, the dominant contributions to the velocity are approximately perpendicular (Fig. le), min-
~
imising tangential flows (Fig. 1c), which are associated with memorisation (Achilli al., 2024).
To understand the mechanism behind the geometric regularisation induced by the flow path (8), we may first focus on the target distribution v v(01). As before, marginalise (9) obtain
1 N
Σ N (a/a(); f(x()))
(10)
We see that our flow path replaced the FM approximation of the data manifold v in (7) with an anisotropic Gaussian mixture. As F(x1) approximates the projection map onto the local tangent space, we numerically observe that CDC-FM faithfully learns the data manifold (Fig. If) in contrast FM (Fig. 1c).
To obtain a deeper understanding of the regularisation mechanism, one may study marginal probabil- ity paths Pt. We prove (see Appendix D. Proposition 2) that our flow path (8) induces geometry- aware anisotropic diffusion term to the continuity equation (2) to obtain drift-diffusion process (22). For diffusion processes, one may use the Dirichlet energy (32) to quantify the smoothing in- troduced by the diffusion term arising from our flow paths. However, the Dirichlet energy is just the CDC field ()) measuring the local smoothness around training point (9) integrated over the data manifold. This justifies the construction of our flow path.
3.2 ESTIMATING THE CARRÉ DU CHAMP
To compute π that optimally captures the local geometry, we follow Jones (2024a;b); Jones & Lan- ners (2026) and provide a local kernel density estimate using the diffusion maps Laplacian (Coifman & Lafon, 2006; Berry & Harlim, 2016). We compute a variable- bandwidd Gaussian kernel up the
kth neighbour of node i, w. (xx(()) .z((i) exp(- (=)()))=)))) x00) 1|2 /(GG))) where G, is the distance to the hi nearest neighbour of a0()) ,00,, respectively. We use this to obtain a local estimate of the transition probabilities of a Markov process generating the data:
W. (z0(0),)(0)
(P)((()))= == = Pus f(ab0)) BY~R[f(Y)),
PJ = Ee w. (x((), x(1))
(11)
4

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where f is well-behaver test function and P.y represents the one-step transition probabilities from sample (2()) neighbour x(6), Using the local Markov kernel estimates (11); compute
[(x!)) = Ex=P, (x-m"(a())) (x m mm(x""" (z (z(1))
(12)
which is the local covariance of the random variable X P. (Bakry et al., 2014). We prove in Appendix E (Theorem 2) that (12) is the optimal Gaussian covariance at x:(*) given the Markov kernel (11). In practice, we downscale T(x(() to ensure that the added Gaussians (10) add only small first-order correction to the FM path and not distort the training distribution (Appendix E). We then take the rank-dede approximation of I, optimising dede using grid search. To globally scale the effect of regularisation, we also introduce a hyperparameter multiplying r.
Computational complexity. Our algorithm scalable to large datasets. The additional compo- nent compared to FM is the computation of T, which has a complexity of (N log(N)) and a memory requirement of O(N) with respect the training set size (Appendix F). Further, in periments below, we also report the number of function evaluations (NFE) required for the adaptive ODE solver reach prespecified tolerance during inference. We find that CDC-FM has compara- ble or lower inference-time complexity than FM.
4 EXPERIMENTS
We now present a series of experiments to quantify the advantages of geometric regularisation over FM, particularly concerning: (i) low memorisation of training data; (ii) good generalisation test data; and (iii) high sample quality. We quantify (i) by marking a generated sample from the model as memorised its nearest-neighbour ratio (Yoon et al., 2023), Myy = lly 1''l/// x |. with z(1) and (2) the first and second nearest training neighbours of y, falls below a cutoff. To allow analysis at the level of training points,
compute the percentage of memorised samples per nearest training point and average for a global memorisation measure. To measure (ii), we use the negative log-likelihood (NLL) of test dataset, which is equivalent the cross-entropy loss between the data and the model predictions. While both (i) and (ii) indirectly measure quality (iii), we, in (DtM) when possible. For details on network architectures
addition, use the and hyperparameters, see Appendix G.
4.1 IMPROVED REPRESENTATION OF GEOMETRIC DATASETS
Given that I approximates tangent spaces, we expected our method to be well-suited for data with strong geometric structure.
Three-dimensional geometry inference. Light Detection and Ranging (LiDAR) scans provide point clouds of complex two- dimensional (2D) surfaces embedded in 3D space from limited samples. We consider topographic LiDAR data from Mt. Rainier, WA, previously used in geometric applications of FM (Liu et al., 2024; Kapusniak et al., 2024). We trained FM and CDC-FM on 40-200 uniformly sampled points, with velocity fields parameterised by multilayer perceptrons.
Training data
FM
CDC-FM (ours)
Test data
Figure 2: Visual comparison of FM vs CDC-FM for LiDAR data.
Early in training, both FM and CDC-FM samples covered the target manifold, achieving low mem- orisation and good generalisation but poor geometric fidelity (DrM, Table A6). As training pro- gressed, their behaviour diverged. FM had consistently higher quality than CDC-FM, but achieved this by memorising training points, at the expense of generalisation. In contrast, CDC-FM improved quality with substantially better generalisation (Table A7). Qualitatively, FM terrain reconstructions appeared patchy and disconnected, while those of CDC-FM were smoother and coherent (Fig. 2).
5

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Flow matching (FM)
CDC-FM (ours)
1.75
100
Quality (DtM)
1.5
Generalisatinn
% memorlsed
75
1.25
CDC-FM
80
0.75
-1
25
2
5
Epochs 10*
Epochs 10
Epochs x10*
Figure 3: Early stopping for spatially heterogeneous data. Samples from FM and CDC-FM trained on the two-circles dataset at an epoch late in the training (40k), when FM captures the small circle and memorises samples the larger one. Quality, generalisation, and d memorisation against training epoch for the two methods, presented separately for the two circles. Lines represent means over samples.
Comparison of FM and CDC-FM single-cell data. Earth mover distance between predicted and held-out snapshots, mean over 5 runs.
Table 1:
Inference of single-cell gene expression trajectories. We evaluated the impact of geometric regularisation on two single-cell gene expression benchmarks (CITE-seq "Cite' and Multiomics 'Multi') from Lance et al. (2022). These datasets comprise temporal snapshots of low- oww-iiensional, spatially complex trajectories in a high- dimensional space, where points define the gene expres- sion state of cells. Interpolation between snapshots is challenging because cell sampling methods are destruc- tive, meaning cells between time points are unpaired.
Method
Cite 4
Multi 4
LFM
48.276 1 3,281
$7.262 1. 3,855
LCDD-FM
46.657 + 3.412
54.419 0.629
OTEM
1. 0.416
5,858
OT-CDC-FM
44.410 +: 0.993
$2,043 + 1,948
Following Kapusniak et al. (2024), we used PCA to reduce dimensionality to 100-dimensions, and performed leave-one-out interpolation by assessing the models' ability to reconstruct from total of four snapshots one of the two intermediate snapshots. We modelled the velocity field using an MLP and trained FM and CDC-FM until the validation loss plateaued. We found that CDC-FM resulted in consistently better reconstruction than FM, both when samples from snapshots were unpaired (I-FM) paired using optimal transport (Tong et al. (2024), OT-FM, Table 1).
These experiments demonstrate that CDC-FM can improve the quality-generalisation tradeoff in domains with underlying geometric structure.
4.2 QUALITY-GENERALISSS
TRADEOFF FOR SPATIALLY HETEROGENEOUS DATA
Choosing the number of training epochs trades off sample quality against generalisation and memo- risation. We hypothesised that for spatially heterogeneous data, different regions converge at differ- ent rates: any fixed epoch, FM may memorise sparse regions and generalise in dense ones.
Early stopping for spatially heterogeneous data. To illustrate this, we trained FM and CDC-FM on two circles with different diameters, each with eight training points. We observed two training phases. In the initial phase (S10* epochs), both methods broadly covered both circles (Fig. Ala) with low quality (Euclidean DIM; Fig. 3b), corroborating our earlier findings. In the second phase 2214 epochs), sample quality improved rapidly (Fig. 3a.b). For FM, this improvement sparser, larger circle came via collapse onto training points (Fig. 3a), reflecting the loss of gener- alisation (Fig. 3c) and memorisation (Fig. 3d). By contrast, CDC-FM quality increased without overeeepresintttoonn training points, yielding markedly less memorisation (Fig. 3d), stable gener- alisation (Fig. 3c) and better inference-time numerical efficiency (Fig. Alb) across both circles.
the
Overall, this example illustrates that there no single optimal early-stopping point for FM. At any epoch, FM tends to produce mix of high-quality yet memorised samples and novel yet low-quality samples. CDC-FM is substantially less sensitive to this heterogeneity, allowing training to proceed until the desired quality is reached without incurring memorisation.
Limiting spatially localised memorisation in animal motion capture data. To reinforce our sults on the two-circles data, we considered animal motion capture data the fruit fly, Drosophila melanogaster (DeAngelis et al., 2019). Points in this data are 31-frame pose sequences where each snapshot represents the 2D position of the six legs relative to the body centroid (Fig. 4a, inset), yielding 372-dimensional (2 6 x 31) state space. A 3D UMAP embedding of this data permits

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b
Walking speed (Hz)
x10:
CDC-FM. d.a.
7.5
Generalisation (NL
10
memorised
Pai
2.5
Epochs
UNAPP
2.8
3.2
3.3
3.3
stereotyped
complex
Quality (manifold coverage)
Quality (manifold coverage)
Sparsity
x10
7.5
GGneralisation (NLL)
matching (FM)
2+
memorised
CDC-FM (ours)
15
CDC-FM
2.5
+ memo
10
Epochs
UNAP
Epochs 10
Sparsity (distance n.n.)
Figure 4: Quality-generalisttinn tradeoff for animal motion capture data. Inset: an example fruit fly pose sequence. Point cloud, each point representing 31 31-frame pose sequence, visualised in 3D UMAP coordinates. Shading indicates walking speed. b Generalisation against quality for CDC- FM for different Î ranks, dede, and for FM for different minu Points are averaged over eight seeds. Black circles indicate epochs analysed in Same as but for memorisation. d Generalisation and memorisation against epochs. Shaded error bars show one std from the mean. e Percentage of memorised samples nearest to a training point (FM: miin =0, CDC-FM: dede 16). f Variation of train data sparsity. Average memorisation against sparsity for different epochs.
the visualisation data manifold, which is parametrised by the walking speed on the longitudinal axis and phase the cyclic coordinate (Fig. 4a). We also performed sensitivity analysis for the parameters kow (bandwidth) and k (number of nearest neighbours), finding that only & had measur- able influence on the estimation of the CDC field (Fig. A5). We obtained better generalisation and lower memorisation than FM for orders of magnitude changes of k. Nevertheless, optimal regulari- sation can be achieved for values large enough to provide an accurate kernel density estimate, but small enough to avoid short-cut connections in the manifold (Fig. A5).
Using the pose sequences as training points, we trained transformers, leading architecture for character motion synthesis (Hu et al., 2023), to approximate the generative velocity field of CDC- FM and FM (with different min)). We measured quality by the faithfulness with which samples covered the manifold, quantified by the mean distance of test points to the nearest samples. We found that FM with amin = traced out frontier at increasing training epochs, trading off sample quality with generalisation (Fig. 4b) and memorisation (Fig. 4c). We used memorisation cutoff of M = 0.6, confirmed by the movement traces (Fig. A2). Naïvely regularising FM by increasing generalisation (memorisation) quality.
Tmin did not exceed this frontier, achieving either By contrast, adding CDC regularisation (7 0) surpassed the FM frontier, simultaneously improving sample quality, generalisation and memorisation. While the advantage was strongest around = (Fig. 4b.c), other values also lead improvements (Fig. A3). Increasing dede rank(T) led to better generalisation with slight drop in quality, possibly due to noise leakage in off-manifold directions.
Amongst the models in Fig. 4b,c across different epochs, which one should one choose? When plotted against epochs, we see that generalisation in FM monotonically deteriorates (Fig. 4d) and memorisation monotonically increases. This means that FM models require early-stopping strategies to optimise. By contrast, CDC-FM test-set performance plateaus and memorisation remains lower, meaning that early-stopping strategies are less relevant. Our results show that for given sample quality, especially when high, there is CDC-FM model with comparable or better generalisation and memorisation than FM, and vice versa.

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FM
CDC-FM
FM
CDC-FM
Generalisationn (NILL)
FM
CDC-FM
Quality (DIM)
%% memorised
100
100
0.2
200
75
20
400
0.1
1600
800
8
10
0
2
3
4
5
2
3
5
51
Dimension
Dimension
Dimension
Figure 5: Synthetic experiment on toroidal manifold. Effect of data dimension on a sample quality, b memorisation and generalisation.
The data manifold is not covered uniformly with data points. Rather, fast behaviours are generally more stereotypical and most densely represented (left side of manifold) than slow, complex move- ments (right side of manifold, Fig. 4a). We expected that this sparsity pattern would lead to varying trade-offs in quality, generalisation and memorisation. In visualising these patterns, we took a late epoch (100k) for both FM and CDC-FM (Fig. 4b,c, black dots), finding that memorisation predomi- nantly occurred in sparse regions (Fig. 4e). Memorisation correlated with the sparsity of the training points (Fig. 4f.g), estimated based on the distance to the closest neighbour, and occurred first at the sparsest points (Fig. 4g), corroborating our earlier findings (Fig. 3). By contrast, CDC-FM showed memorisation (Fig. 4g). Overall, these
substantially lower (Fig. 4f.g) and less pparsity-deenddee results indicate that geometry-awan regularisation is especially valuable when data varies heteroge- neously over the data manifold. In the next section, we examine its limits as the intrinsic dimension and dataset size increase.
4.3 LIMITS OF EXPLICIT GEOMETRIC REGULARISATION
Until now, we have demonstrated that CDC-FM achieves better quality-generalisationn tradeoff for different architectures (MLP, UNet, Transformers) and diverse domains (LiDAR, single-cell, motion capture). We now test the limits of CDC-FM on two axes, which are known challenge manifold methods: (i) increasing dimensionality. (ii) scaling to large data.
Influence of manifold dimension. To study the influence of dimension, we generate tori Tdd (S")d C 2d2 varying the dimensions d and the number of training points. As the dimension in- creases, the effective data density decreases exponentially (curse of dimensionality), and we ex- pected the performance of generative models decrease. We present the results for FM and CDC- FM trained to 4k epochs, but found that qualitatively similar results hold for different epochs.
As hefore, found that FM samples attain a comparable quality, it required an increasingly larger number of training points as the dimen- sion increased (Fig. 5a). Yet, strikingly, this increased quality by FM was result of memorising almost all of the dataset when the dimension was high enough (Fig. 5b). This shows that mem- orisation is more likely to occur for higher dimensions for fixed data size. By contrast, while memorisation in FM and CDC-FM was comparable in dimensions one or two, CDC-FM memorisa- tion decreased with dimension and remained low (Fig. 5b). CDC-FM also attained generally higher generalisation (Fig. 5c). Taken together, these experiments show that, unlike FM, CDC-FM pre- vents memorisation and facilitates generalisation irrespective dimension, an effect that remained robust when adding small Gaussian noise to the data (Fig. A4). Yet, as the dimension increases, the desirable sample quality may only be achievable with a sufficient amount of data, likely due to the knn graph construction of the kernel.
of consistently high quality, and while CDC-FM could
Addressing the curse of dimension through latent diffusion: Celeba-HQ. We then trained FM and CDC-FM in latent space, and decoded the output back to pixel space. To test whether our findings extend to such a setting, and following the setup of Dao et al. (2023), we evaluate FM and CDC-FM the latent space of the Stable Diffusion VAE (Rom- bach et al., 2022). We train both models on a sub- set of 1000 Celeba-HQ images, and report FID and
Table 2: FID and NLL per epoch for FM vs. CDC-FM in latent space. CelebA-HQ subset of size 1000.
Epoch
L
NLL. CDC-FM
FM
CDC-FM
FM
1000
15.60
12.72
6.80
7.18
2000
13.51
13.42
6.78
6.83
4000
13.82
11.70
6.69
6.53
5000
13.10
10.85
6.68
6.48
8

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NLL across epochs in Table 2. Despite the dimensionality reduction to size 32 x 32 x 4, observe that the CDCFM regularisation improves both quality and generalisation after 3k epochs once both model performances stabilise.
Influence dataset size. Finally, we studied the scaling of CDC-FM to large data in image syn- thesis, where generative models have enjoyed considerable success, although memorisation is still reported, particularly for a low number of training points (Kadkhodaic et al., 2024; Gu et al., 2025).
We trained FM on increasing subsets of CIFAR-I0 for miin = 0, as we found that this choice yielded the best performance (Fig. A6), and similarly, CDC-FM, using a UNet network architecture. We measured sample quality by the Fréchet inception distance (FID) found that, for small training set sizes (<10k), all training points became abruptly memorised (M 0.2) training progressed, consistent with a phase transition as reported in diffusion models (Pham et al., 2024), while the fraction of memorised points steadily increased with larger training sizes (Fig. 6b). By comparison, Lower memorisation was accompanied by substantially better generalisation and sample quality at late epoch after the FM phase transition (4k, Fig. 6c,d, bottom), but only comparable generalisation and quality for epochs before the phase transition (2k, Fig. 6c,d, top). Yet, as the number of training points increased, quality and generalisation in FM and CDC-FM were comparable across epochs, indicating that implicit regularisation, from the architecture and loss function, becomes dominant.
test set of 10k samples. We
found only few per cent of memorised points with CDC-FM (Fig. 6b).
Our results demonstrate that the benefit of geometric noise highest for low, heterogeneous or geometrically structured data settings.
5 RELATED WORK
Manifold hypothesis and generative modelling. Our approach is motivated by geometric ap- proaches of generative modelling under the so-called manifold hypothesis, surveyed in Loaiza- Ganem et al. (2024), which presumes that high-dimensional data often have low-dimensional struc- ture. Generative models have also been used to estimate intrinsic geometric quantities, like dimen- sion (Stanezuk al., 2024). Conversely, related line of work constrains generative models to prede- fined manifolds (Chen & Lipman, 2024; Huang et al., 2022; Mathieu & Nickel, 2020; De Bortoli, 2022), or to manifolds that are learnt from data (Kapusniak al., 2024; Peach al., 2024; Gosztolai et al., 2025). Our use of the carré du champ is based on diffusion geometry (Jones, 2024a), which generalises classical Riemannian geometry complex geometries that do not meet the strict defi- nition of manifold. Rather than using the geometry as a constraint, we use it as a form of model regularisation by modifying the probability path with an anisotropic diffusive term aligned to the data geometry.
The quality generaistnnn tradeoff in diffusion models. Memorisation and generalisation have been studied in the context of diffusion models, where empirical evidence showed a strong depen- dence on training set size and architecture (Yoon al., 2023; Gu et al., 2025). From a geometric standpoint, Ross et al. (2025) argues that memorisation occurs when the learnt manifold's dimen- sion is too low due to either overfitting-drivv memorisation, when tangent directions are not fully captured, or data-driven memorisation, when the underlying data manifold itself is degenerate. In this regard, Achilli et al. (2024) observed failures to capture tangent space dimensions and Ventura et al. (2025) studied the dynamical regimes of diffusion models from geometric perspective. On the theoretical side, De Bortoli (2022) studied the convergence of denoising diffusion models under manifold assumptions. While FM and diffusion share strong formal analogies indeed, FM uni- fies score- score-based and diffusion training - there has been much less work on memorisation in FM. Bertrand et al. (2025) shows that the optimal FM vector field memorises and argues that generali- sation does not come from the stochasticity of the FM objective. Our results thus complement and extend the existing understanding of memorisation phenomena in generative modelling.
Geometric regularisation. Early work applied geometric regularisation to supervised learning by introducing tangent information either via hand-crafted invariances (Simard et al., 1991) tangents estimated from neural network Jacobians (Rifai et al., 2011). CDC-FM differs from these by ap- plying geometric regularisation in generative modelling setting: rather than modifying the loss,

---

Flow matching (FM)
CDC-FM (ours)
400
Epoch 2k
Epoch 2k
100
300
Generalisation (NLL)
memortsed
80*
+ Quality (FID)
100
CDC-FM
60-
Epoch 40k
Epoch 40k
40-
Training
40
20-
20-
0
Epochs
Training set sizee 10
Figure 6: CIFAR-10 in the low data regime. Percentage of points memorised for a FM and CDC-FM for increasing training epochs. e Quality and d generalisation against training set at representative early (2k, orange triangle) and late (40k, orange star) epoch.
we regularise the generative path. We use geometry-guided, anisotropic diffusion that enforces smoothness aligned with the local data geometry, and prevents collapse onto training points.
6 DISCUSSION
We introduced Carré du Champ Flow Matching (CDC-FM), a principled modification of flow match- ing (FM) that injects geometric regularisation into the conditional probability path. Our generalisa- tion be understood as adding a spatially-modulated, geometry-aware noise the deterministic FM generative path to minimise its Dirichlet energy. This noise term encourages transport normal the data manifold and suppresses tangential collapse onto training points. We show that this noise term can be rigorously justified as the optimal transport path in the space of probabilities and can be optimally estimated from data.
Empirically, we find across synthetic and real data that sample quality and generalisation are in tradeoff. Namely, FM models require specified number of training epochs to reach a desired sample quality, which comes cost of memorisation and decreased generalisation to the test set. CDC-FM could mitigate this tradeoff by achieving higher generalisation for the desired sample quality for diverse datasets, including geometric point clouds (LiDAR) and continuous trajectories (single-cell time courses; motion capture). Notably, CDC-FM reduced localised memorisation that plagues FM under heterogeneous sampling densities, even for moderately large data.
Our model uses a form of the manifold hypothesis based on local heat kernel approximations, This is weaker than the typical Riemannian approximation, it allows for the intrinsic dimension to vary and is robust to noise. Yet, we found that as the manifold dimension rises, our method needed exponentially more samples to maintain accuracy, due the need estimate tangent spaces from local neighbourhoods. Further, for non-geometric data, the benefit of geometric regularisation di- minished, on average, the training set size increased, because the neural network architectures and the loss function already confer inductive biases. Yet, we showed that citiitecturrinduuce regularisation can depend heavily on the local complexity and sparsity of the data, while the use of CDC-FM remained robust and resulted in better generalisation and, in some cases, also higher quality diverse data settings than FM.
Overall, we expect that even in overall high data settings, local memorisation is likely common occurrence, driven by local sparsity patterns (Skrinjar et al., 2025). In these scenarios, we expect that CDC-FM can provide a robust tool to reduce or eliminate memorisation and improve generalisation. Thus, our approach is not competitor to FM but plug-in regulariser that can be scheduled, adapted, or even learned and can be readily used in conjunction with existing latent space pipelines. While we used our geometric framework to generalise FM, being one of the most broadly adopted generative models, future work could explore the effect of our geometric regularisation on other generative models and stochastic regularisation strategies. Our work thus opens path to geometry-ave aware flow- based generative models with stronger guarantees, better sample efficiency, and improved robustness to privacy risks.
10

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7 ACKNOWLEDGEMENTS
This work is supported by an ERC grant (NEURO-FUSE, Project DOI: 10. 00.3030//01163040) MB and JB partially supported by the EPSRC Turing AI World-Leading Research Fellowship No. EP/X040062/1 and EPSRC AI Hub No. EP/Y028872/1.
8 AUTHOR CONTRIBUTIONS
The work was conceived and performed at the Institute of Artificial Intelligence of the Medical Uni- versity Vienna, except refining results and manuscript writing. JB, IJ, and AG conceived the study. JB, IJ and AG developed the mathematical formalism. JB, IJ and DD performed the experiments. JB, IJ, DD and AG wrote the manuscript with comments from PV and MB.
REPRODUCIBILITY STATEMENT
All details necessary to reproduce the experiments are present in the paper. Code is available at Sura---euual-Ssteen Lab/ cdc-fm.
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Appendices
USE OF LLMS AND OTHER TOOLS
We used Grammarly to facilitate writing and GPT-5, Gemini, and Claude to check for mathematical and notational inconsistencies, as well as facilitate writing code.
A ARBITRARY SOURCES AND TARGETS
In the main text, we described our method in the special case of generating flow paths from Gaussian μ _N(0,1) I) complex density v. Here we show that our framework can be readily extended to model flow paths between two arbitrary densities μ., In addition to the single-cell dataset Section 4.1, this setup relevant for many other applications (Liu, 2022; Albergo al., 2023). FM naturally lends to this setting as the conditional paths can be conditioned on both source and target samples (Tong et al., 2024). First, to approximate the manifold structure, we compute CDC matrices T(xo), ((x1), for training points zo ~ 14,51 ~ v, respectively, as described in Section 3.2. We incorporate these local tangent spaces in the conditional probability path by using the displacement interpolant between M(ro, (xx)) and (x,,, following proposition.
We formalise this in the
Proposition Given that po .n) Nan ((x0) and pı (2/ v1) Nar,, (x))) are Gaussian, the displacement interpolant between them, Pe ( x1) = [Po, P1]. is also Gaussian, with mean and covariance given by
14 = (1 - 11)20 + ta
(13)
E. = (
where A: (1- tB and := -(x0)=3 ((=))))) f(2o)= b with the cor- responding conditional flow being
(2(22221,00) (1 t)æo + ta Ai (x zo).
(14)
Proof. Follows from the optimal transport interpolation of two covariance matrices (xxo), ((m) (Bhatia et al., 2019).
B CARRÉ DU CHAMP FLOW MATCHING GENERATES OPTIMAL TRANSPORT FLOW PATHS
Since we aim to improve the approximation the underlying density as v(z) 1
N E ((x(0) (()()) using the carré du champ tangent space approximations (z) naive approach might be to randomly augment each training point x(()) by some perturbation from (0 (()()) )) take the standard conditional OT paths (5), and train against the FM loss (6). How- ever, we show that this approach is equivalent to FM between NO,, I) and N ((x())()())) with non-optimal transport probability paths.
Theorem 1. Training FM model with source points Xo 1) and target points Z NX ((X1) for X1 ou v equivalent, expectation, to training with the conditional probability path
pe(x(111) -l (1 (1
(15)
which differs from the optimal probability path obtained via displacement interpolation
Pa(z)z1) == [N(O, N(,, (Naaai))ett((nntt))aa Nto,,, (01-th t)I [
(16)
whenever ((x1) # 0.
15

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Proof. We first show that training with conditional flow paths leading to naively augmented train- ing set leads to the correct marginal probability paths. We start by considering the FM loss with augmented data, which results in the loss
Cagg (0) = X1 6f((1 x)xx + +Z)
- -x)ll
(17)
fort ~ U(00 Xo p. X1 v, and Z ~ (1 [z))). Differentiating with respect to 8, get VoLaug = Ex, Volss
((1 Xo + 12) - - -x)11.
(18)
The expectation on the right is just the FM loss (6) for an affine flow (x(2) -1- t) tZ between (N I) and 5z. By Theorem 2 in Lipman et al. (2023), we may marginalise over the conditional flows c Z))
d 1 (1- (z) xllff (Xo))
dt x (Xo)ll (19)
where t,,, denotes the marginal flow between IO. and M(x1 [x11). This tells that can view data augmentation as a distinct FM problem for each training point X1. We may then take the expectation these points obtain
V o Luu Ex, Vo (X,,, - Vi (Xo)ll
Vo
(1, (Vi,x, (Xo)) V1xxx (Xo)112
we define conditional flows between IO,II and N(x ((x1)) by (x1 == vi,,x, then
VoLing = No. | ( (Xa)x -Xx,
(20)
which shows that training with Laug is the same, in expectation, training with L (6) using the conditional flows (
Although the above shows that naïve data augmentation is equivalent FM problem, we show that the resulting conditional flow paths are different from the CDC-FM paths, and so are suboptimal. We can derive closed-form expression for the conditional path px(-)x;) as
= -
(21)
because, X ~ pe) then X = ta + (1 - t)b where ~ (O, and b ~ N(11, f(:1)) are independently sampled, but the optimal probability path between /(00 I) and N(11, [(x1)) is given by the displacement interpolant
(- 1, ((x)))
=N(IX, () 1/x13
=NV(IX,, (1-(X, 1 (f(x)) + 21(1 ())))))
which differs from (i) whenever f(Xi) + Therefore, whereas the probability paths used in CDC-FM are displacement (optimal transport) interpolants, those in FM with data augmentation are not displacement interpolants, except for when f(x;) 0, where they both collapse to FM. C
This means that, unlike the displacement interpolant paths in CDC-FM, data augmentation is not conditionally optimal. In Lipman et al. (2023), the authors show that an FM model trained with deterministic displacement interpolant probability paths requires fewer solver steps to compute solution.
C
PSEUDO CODE
We present pseudo code for CDC-FM the case of noise to data and compare it to standard FM in the following algorithms.
16

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Algorithm Flow Matching
Algorithm 2 CDC Flow Matching
I: Input: Samples D {x(9) Jiml TRd from
Input: Samples D {- C Re
(()) parameterised vector field (x,
from v(x) together with their local covari- square-root (ax( parameterised vector field v (ax, t).
2: Output: Gradient VoL(O) for updating pa-
Output: Gradient VaC(O) for updating pa-
rameters
rameters A
3: Sample 2 f- x(() v(x)
Sample 21 f x(0) v(x)
4: Sample & ~ Uniform(0, 1]
4: Sample ~ Uniform [0.
5: Sample 20 ~ N (O.
5: Sample zo ~ N (0,1)
6: Define
6: Define
Et - taj + ((1 - () + min) xp
2t t trı + ( (11-t)I+ of (x1) xo
7: Define the target conditional vector field at
Define the target conditional vector field
ty (32 m1) 21 (1 Omunn) 20
us (are (1))
F(r ju.) v0
8: Compute the loss
8: Compute the loss
[(0) = 10 (ax, 1) - vx( mm)]
[()) = || (x (ate I (n))
9: Compute the gradient V0C(0) return Vg((6)
9: Compute the gradient Ve((O)
10: return Vg((
D INTERPRETATION OF CDC-FM AS AN ANISOTROPIC DIFFUSIVE REGULARISATION
Fundamentally, our framework seeks identify the Markov process, whose density at matches the data density v. The CDC-FM flow path (8) introduces data-driven regularity in this Markov process. In this section, we justify that the CDC-FM flow path is equivalent adding space- dependent, anisotropic diffusion term into the marginal probability path (2). This leads to the Fokker-Planck equation
① ae (L')))) - (wx())))))) 1 a, Pc(z))
(22)
4.r
Here, L" is the adjoint of L, the infinitesimal generator of the stochastic process, and A, is the diffusion tensor, which adapts to the local geometry of data.
Note that the mean and covariance in (8) do not depend on x, only on the end-point T1. We may therefore prove the following result.
Proposition 2. Let pr())) define Gaussian probability path
(=(x) = (x; m,, F.,,, m. E Rdd, En sd
(23)
where my E, are differentiable and independent of Then, p((x) satisfies the following Fokker- Planck equation
i ge ((())) V (x))) + Σ (,0g Pr(x))
(24)
Proof. Because (xx)) Gaussian, we have
(ogp((x) = 1/ log det E, - [x - =
Tx, (2 m.).
(25)
Differentiating with respect to L, have
log = tr, Σe) + m. Σ, (x - m.) + (x- = E, ' (x -m)).
P. 0t Pt
0t
(26)
17

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Further, for Gaussian density pe(z), may compute the drift term
(ri pi) = ri Σ - x - mm)p:-
(27)
Likewise, for the diffusion part
Σ (E Pt) = Σ L0,0,Pt 22 [1z mi) TE 'E,'E, (x m.) tr(E, Σ.)
(28)
Adding (27) and (28), we obtain the right-hand side of (26), which is the desired result.
Using Proposition 2, we that the conditional probability path satisfies
Ot (x|z1) = V (a(az))) + Σ 0,0, (z(22))))
(29)
where, using our CDC-FM flow path in (8), we have
(x1 F1
(30)
Σx(x)) = (1 +)I (x1) (f())) I).
We may then marginalise by taking expectations over v in (29) to obtain (22) with (x(x) = [x|X, = æ| A.(x) = where we used the fact that, by Bayes' rule, f F Fan )r (2t (1) 1((,)I,, for any integrable function F.
(31)
= =]]
Dirichlet form and carré du champ. The global smoothness introduced by the diffusive term induced by the flow path is measured by the (weighted) Dirichlet form E(f, g). which monotonically decreases along the evolution Pt- For
test functions f. 9 IRd - this reads
E(f.g)== = { ((f.g)))) dr J (L(fg) fLg - glf) dr
f Vf A (32) Jzd
In contrast to E, the integrand [(f,g) g). known as the carré du champ (CDC), measures the local, point-wise smoothness, and encodes the geometry of the underlying manifold (Jones, 2024a). Note, we recover FM in the deterministic limit, (x) = 0 as t + 1. Consequently. = O, meaning the associated generator has intrinsic mechanism dissipate energy, thus smoothing.
Taken together, the spatially varying covariances introduce a source of randomness into the sample paths, which can be interpreted anisotropic diffusion. The CDC quantifies this diffusion and relates to the smoothness the data itself.
E OPTIMAL ESTIMATION OF THE CARRÉ DU CHAMP MATRIX
In Section 3.2, we estimate the local geometry at sample æ using the carré du champ matrix (ar). which we define the mean-centred covariance of the sample's neighbours. We now prove that this choice optimal in the sense that is the best local Gaussian approximation to the data. Specifically, the Gaussian that maximises the expected log-likelinhe likelihooood of the kernel Pg is the one whose mean and covariance match the empirical mean and covariance.
Theorem 2. Let Pay be transition kernel on Rdd. Let Ps denote the probability measure supported on the k nearest neighbours of x obtained by restricting and renormalising Pay (11). Assume P. has finite second moments. Then the unique maximiser over m E Rd and Σ over the space of positive definite matrices of the expected log-likelihood of sample Y from a Gaussian density N N(m, E)
(m, Σ) - Ey-P, [log (Y; E)]
is given by matching the first moments of P,: m² m²(x) = Ey [Y].
(x)x) = f(x) = Ep. [(Y m²)(Y -m")T Equivalently, ((m", Σn) minimises KL((P, [mm, E)).
18

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Proof. For Σ > 0, the log- logeliieihhod of a sample Y Nmm, Σ) is
d (2 log det
1 5V - m)) - - m).
lY)
2
Taking the expectation with respect to Y ~ Pa gives
d log(
1
Ex log (YY; m, Σ) log (T)
log det E - BBypP [(Y -m) - -m)]
We can differentiate with respect to m to get
① Ey-p, (Y m. Σ) EB m)),
dm
which is zero at the maximum m" m=(x) = Ey [M]. To find the optimal E, let S₂ = B[(Y
m²) (1)]-m"]TT m²) We can rewrite the expectation term as
Ellog (Y; Σ)] log det Σ- - ttr S.) + const,
which is strictly concave in E-1 0. Differentiating with respect to Σ-1 we obtain
i By By-r, [log/N(Y Σ) E S0
+Σ
This quantity is zero the maximum E" Eynp,
- m² [
-Y m=) and unique-
ness follows from strict concavity. Thus, the optimal covariance is given by f m=)"]. the centred covariance of the neighbours of x.
Ey.r, [(Y- (00-mm)((-- -m")
This theorem justifies using the mean-centred local covariance of the diffusion kernel's neighbour- hood as the statistically optimal covariance for a local Gaussian fit.
Rescaling the carré du champ matrix If the bandwidth of the kernel is set appropriately, the carré du champ matrix Tzx()) will capture the local shape of the density near a training point (9) However, depending on the bandwidth, samples from .N (t) T(x())) may be far from x(0). To quantify this, we may diagonalise f Q diag(A1...., Aa)], where Q is the orthonormal matrix of principal components of F, then Î represents Gaussian noise with variance X in the direction of the component Q,, Then, if samples are drawn randomly from Nxx F), their expected squared distance from x: is
E(I ==1²) = tr(f).
This means that excessive noise in directions normal to the manifold can reduce the fidelity of the geometric regularisation. Thus, control the maximum amount of noise added the data by I in any direction, we need rescale I such that the largest eigenvalue Å₁ of the right scale.
As a heuristic, we would like the noise added by ((x(0) to small enough that is contained in the gaps between the training point (9) and neighbours. This way, the training signal still dominated by the location of the training data, with the carré du champ adding small, unintrusive amount of directional training top of that. If (()(()) is the nearest neighbour of (00), then we choose ensure that all the eigenvalues of (a) are no larger than ||a(() (2x (0) ]|2 This means that, in any given direction, the noise added has standard deviation most (xx - 11/3,
so at least 99.7% of it will be closer to (() than its nearest neighbour (2)(
This local heuristic is effective except in the case that (+))) is very isolated, in which case ||x(0) T (x:(0]] is very large. To avoid adding too much excess noise to these points, we will also cap the maximum noise level at Cmax. defined to be the 90th percentile of all the local constraints (|x(0) T (2) This limits the scale of the noise at the 10% most outlying points in the data, which we can guarantee by ensuring that all the eigenvalues are below 2
To meet both the local and global constraints, we rescale ((x(()) by
min [[x(1) - x(x()))
03 = X
9
max
(33)
19

---

where Xj is the largest eigenvalue of )( The largest eigenvalue of the rescaled matrix will then be the smaller of ||x()) [12 / and c "max"
The above method describes a heuristic for setting the size of the carré du champ at each point, but, in particular cases, we may attain better performance by further globally rescaling (z:)). We therefore use the carré du champ yc; (xx()), where is from (33) and 7 is tuneable hyperparameter that defaults to 1.0.
F COMPUTATIONAL COMPLEXITY AND RUNTIMES
The conditional paths in CDC-FM require (/x2) for which we have to compute and diagonalise T(æ1). Since truncate the diffusion kernel to have only k non-zero entries per point (the neigh- bours of the point), the carré du champ at each point has rank at most k, so we can avoid working with project the neighbour differences into their k-dimensional span and compute f(x1) in this reduced basis. The resulting k matrices capture the full spectrum while being far cheaper to store and diagonalise. The dominant costs are therefore ( d) for the projection step and (1.3) for the eigen-decompositiie
full d d covariance matrices. Instead,
rather than (d In practice, even when the dimension is high, so the computation effectively takes (loNgl + d)) time and ((Nd) space. The complexities of the different steps are given in Table A.. We also report the empirical preprocessing time for the CIFAR-10 experiments in Table A2.
small
Table A1: Compute and memory complexity. Here, N is the number of training points, kt is the number of neighbours, and d the ambient dimension.
Step
Compute
Memory
k-NN graph construction Diffusion kernel
O(N log(N)) O(NK)
O(Nk Nkd) O(NK) O(Nkd)
Form covariance terms in IRd
O(Nkd)
Project covariance terms to R
d(N
O(Nkd + NK?) O(NK2)
Diagonalise [(x,) in g
O(NK*)
Overall complexity
O (Nolg((N) +dd+ 12)) O NI2)
Preprocessing empirical runtimes. Runtimes are reported in seconds for CIFAR-10 subsets of varying sizes and were run on NVIDIA A10s.
Table A2:
Dataset size
KNN graph (s)
cigendecompooition (s)
Total (s)
1000
0.45
19.27
19,72
2000
0.55
38.03
38.58
3000
1.19
56.14
57.33
4000
1.10
73.56
74.66
5000
1.89
92.09
93.98
Computational Resources All experiments were feasible and primarily performed on NVIDIA A10s (24 GB). Some experiments performed NVIDIA H100s (80GB), Drosophila experi- ments were performed on NVIDIA A100s (40GB).
G EXPERIMENT DETAILS
Numerical solver. Unless stated otherwise, for inference and likelihood computations, we the adaptive step size solver dopri5 with atol=xtol=le-1 using the torchdi ffeq library (Chen et al., 2019).
20

---

Circle, two circles, LiDAR, and torus experiments. These experiments the same basic MLP architecture from Lipman et al. (2024) with 4 hidden layers of dimension 512 and Swish activations. We used the Adam optimiser with a learning rate of 10 3, We summarise the CDC-specifie hyper- parameters in Table A3. For the memorisation metric, we used nearest-neighhour ratio cutoff of 0.2.
Single cell experiment. For the single cell experiments, we followed the setup in Kapusniak et al. (2024) and used a 3-layer MLP with width 1024 and SeLU activation to learn the vector field. We used the AdamW optimiser with learning rate of 10-3 and weight decay of 10-5. For inference, we used the Fuler solver for 100 steps. We summarise the CDC-specifie hyperparameters in Table A4. We found that adding isotropic noise with variance min in addition to the anisotropic CDC noise helped stabilise training.
Drosophila experiment. For the Drosophila experiments, we learn the vector field with trans- former architecture previously used for flow-matching-hast based human motion synthesis (Hu et al,, 2023). To accommodate the simpler motion of Drosophila compared to human motion, we down- scaled the network and used 4 transformer blocks with feedforward and key/value dimensions dæ = dgy = 256 with attention heads and GELU activation. The model was trained on 2k training points for 10 epochs using the Adam optimiser with a constant learning rate of 10-4 and batch size of 512. We summarise the CDC-specific hyperparameters centage memorised, and sample quality, we used a test set of 25k points well as 25k generated samples. For the memorisation metric, we used a nearest-neige neigesttteigbbbuur ratio cutoff of 0.6 (Fig. A2). As in DeAngelis et al. (2019), we fitted UMAP with 30 nearest neighbours first subtracting the per-limb-coordineg ing each of the 372 dimensions across all data points to zero mean and unit variance. UMAP was fitted on the larger test set, and the resulting model was subsequently used to project the training points into the UMAP embedding space.
Table A3. To evaluate NLL, per-
standardised data points, mean across time steps from each trajectory and then scal-
CIFAR-10 experiment. For the CIFAR-10 10 experiments, we followed the setup in Tong et al. (2024) with the exception that we did not use the standard flip transform for data augmentation to focus on the fundamental differences between FM and CDC-FM losses. We used a UNet (Dhariwal & Nichol, 2021) with 128 channels, depth of 2, channel multiples [1, 2, 2, 4), J 4 heads, 64-channel heads, attention resolution of 16, and 0.1 dropout. Following Tong et al. (2024), we used a constant learning rate x 10-4, gradient clipping with norm 1.0, and exponential moving average weights with decay 0.9999.
We summarise the CDC-specific hyperparameters in Table A5, To tune the dede and y hyperparam- eters, we computed the empirical covariances described in Section 3.2, used the closed-form P1 to generate 10k samples and computed the FID on a validation set consisting of 10k images. For negative log likelihoods, we report bits per dimension regular uniform dequantization with K = 1. FID is computed using the inception v3 embeddings (Szegedy al., 2016) and the cleanfid package (Parmar al., 2022). For the memorisation metric, we used sairest-neighbour ratio cutoff of 0.2.
done in Lipman et al. (2024) using the
Celeba-HQ experiment. For the Celeba-HQ, we followed the setup in Dao et al. (2023) with the exception that we did not use the standard flip transform for data augmentation to focus on the fundamental differences between FM and CDC-FM losses. We use a UNet (Dhariwal Nichol, 2021). For NLL report bits per dimension of a test set in latent space. For FID computation we use the inception v3 embeddings (Szegedy et al., 2016). The CDC-FM -specifii parameters were k 256, kiw 8, dede 4, 7 0.5.
21

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Table A3: Model parameters for different experiments.
Experiments
Parameter
Description
Circle
LiDAR
Two-circles
Drosophila
d-Torus (Fig. 5)
(Fig. 1)
(Fig. 2)
(Fig. 3)
(Fig. 4)
k
Maximal nearest neighbours
3
32
3
128 8
32
Kow
Bandwidth of wr
8
8
8
8
dede
Rank of T
1
2
1
2, 4, 8, 16
d
Y
Scaling of r
0.3
1.0
0.7
0.3
1.0
Table A4: Model parameters for the single-cell experiment.
Single Cell Experiments
Parameter
Description
Cite
Cite + OT
Multi Multi + OT
k
Maximal nearest neighbours we
256
256
256
256
kbw
Bandwidth of we Rank CDC dimension
8
8
8
8
dede
8
4
2
?
Scaling of r Isotropic noise
0.5
0.4
0.4
0.
mmin
0.4
0.3
0.2
0.4
Table A5: Model parameters for the different CIFAR-10 experiments.
Dataset size
Parameter
Description
250
500
750
1000
2000
3000
4000
5000
k
Maximal nearest neighbours
256
256
256
256
256
256
256
256
kbw
Bandwidth of w.
8
8
8
8
8
dede
Rank CDC dimension
16
8
y
Scaling of r
2.0
0.9
0.8
1.0
1.0
0.7
0.5
0.7
22

---

H SUPPLEMENTARY TABLES
Table A6: Comparison between FM and CDC-FM on LiDAR landscape after 4k epochs.
Dataset Size
Generalisation (
Memorisation J
Numerical efficiency (NFE) 4
Distance to manifold
FM
CDC-FM
FM
CDC-FM
FM
CDC-FM
FM
CDC-FM
40
2.26
1.92
5.1
3.2
74
74
105
122
80
1.51
1.52
2.1
1.8
68
68
116
122
120
1.25
1.28
1.9
1.8
68
68
99
106
160
1.20
1.22
1.5
1.5
62
68
101
106
200
1.10
1.12
1.8
1.6
74
74
76
80
Table A7: Comparison between FM and CDC-FM on LiDAR landscape after 16k epochs.
Dataset Size
Generalisation (NLL) +
Memorisation. J
Numerical efficiency (NFE)
Distance to manifold
FM
CDC-FM
FM
CDC-FM
FM
CDC-FM
FM
CDC-FM
40
3.50
2.23
32.4
7.5
92
80
65
101
80
2.16
1.66
15.4
6.5
98
86
56
72
120
1.45
11.0
6,3
104
50
160
1.30
1.22
4.8
3.3
92
56
65
200
1.34
1.24
5.6
3,5
98
98
49
56
I NEGATIVE LOG-LIKELIHHOD COMPUTATION
To evaluate the generalisation capability of our models, we compute the negative log-likelihool (NLL) on held-out test data. Recall that the flow matching model defines a vector field (x that generates probability path (z) transforming source distribution Po (typically (0, I)) to the model distribution Pi Rs v. The log-density test point under the model is computed using instantaneous change of variables formula for Continuous Normalising Flows (Chen et al., 2019):
log P1 (x) log po S ((((((VT dt,
(34)
where th, is the flow map generated by u,, and the integral is taken along the characteristic path from t = 0 to to
In high dimensions. explicitly computing the divergence V ue tr(0u/0zz) is computationally expensive (O(d)). To ensure numerical stability and scalability, we employ the Hutchinson trace estimator (Hutchinson, 1989), which approximates the divergence using random noise vectors A/ I):
)x)( dr
ue (x) E,
(35)
This reduces the complexity to ((d) using vector-Jacobian products. In our experiments, we solve the combined ODE for the particle trajectory and the log-density change using the torchdi ffeq library with an adaptive step-size solver (dopr 5) and tolerances of 10-5
23

---

J SUPPLEMENTARY FIGURES
Flow matching (FM) CDC-FM (ours)
Inference- Inference-time efficiency (NFE)
time
100
75
50
FM CDC-FM
25
3
6
Epochs 10
Figure AI: Supplementary figures for the two-circles experiment. a Samples from FM and CDC-FM, trained on the two-circles dataset at an epoch early in the training (10k), when the quality (distance to manifold) is still low, but generalisation is high. Inference- time numerical efficiency of FM and CDC-FM.
Sample
Closest point
2nd closest tr. point
Limb coordinate
0
10
20
30
0
10
20
30
0
10
20
30
Frames
Figure A2: Memorised samples in the Drosophila motion capture dataset. Three sample time
series showing the longitudinal motion for the six limbs for M 0.6, b M = 0.6 and c M = 0.58.
24

---

10
7 0.1
7 0.5
% memorised
2500
CDC-FM, dede
Genoralisation (NLL)
2000
16
1000
500
-500
2.9
3.0
3.1
3.2
3.3
2.9
3.0
3.1
3.2
3.3
Quality (manifold coverage)
Figure A3: Quality-generalisation and quality-memorisation for the Drosophila motion cap- ture data. Same as Fig. 4b,c, but with CDC rescaling parameter ? 0.1, 0.1.
FM
100
CDC-FM
FM
CDC-FM
Generallsatioo (NLL)
FM
CDC-FM
200
100
30
Quality (DeM)
400
% memorised
0.2
800
75
20.
1800
60
0.1
10
25
5.1
2
3
2
Dimension
Dimension
Dimension
Figure A4: Synthetic experiment on toroidal manifold with additive Gaussian noise. Effect of
data dimension on generalisation, b memorisation and e sample quality. Noise level: = 0.02.
25

---

b
Point on dense manifold region
Point sparse manifold region
0.4
0.4
Relatve (Fronbenius norm)
(Fronberius norm)
Rolative dift
0.3
0.2
Cumulative
0.4
0.2
0.2
6
7
9
Bandwidth, (kg,(
neighbours, (kg,(k)
Nearest
Nearest neighbours, log,(k)
x10'
Last epoch (100k)
Generalisationn (NLL)
2
7.5
Generalisation (NLL)
38 memorised
3.4
3.2
Epoche
Epoche
0-
200
2.6
2.9
3.0
3.1
2.9
3.0
3.1
Quality (manifold coverage)
neighbours, log,(k)
Nearest
Figure A5: Ablation studies corresponding to Fig. 4. Effect of doubling the bandwidth kon on the CDC field at a given value (A: 128). Minimal effect observed. Effect of doubling the number of nearest neighbours A: on the CDC field at given value (kow 8). The CDC field is less sensitive to changes in for higher & values. e Neighbourhood of point
dense (left) and sparse (right) manifold region. Sparse regions can lead to shorteut connections that do not respect the manifold geodesic distance. The kernel density estimate assigns large mass to these points, biasing the CDC estimate. d Influence of the choice of le the overall generalisation and memorisation performance. CDC-FM provides gains over FM for all choices of k. e Generalisation and memorisation at the final epoch (100k, mean and std over 3 seeds). The overall performance gain (although improved compared to FM for all choices of k) is in a tradeoff between generalisation and memorisation. Small leads better tangent space estimates and thus better generalisation, but yields less strong regularisation. Large & leads to worse tangent space estimates, so generalisation drops, but overall reduces memorisation.
(a) FM
(b) CDC-FM
Figure A6: Visual comparison of generated images from FM and CDC-FM models. Models
were trained 40k epochs on 5k images from CIFAR-10. Both models have similar visual quality.
26


Reviews:
[{"review_id": "9CiaO9kh5Z", "reviewer": "Reviewer_7Zem", "summary": "The paper proposes CDC-FM, a geometry-aware variant of flow matching that replaces the standard isotropic conditional path with an anisotropic, spatially varying Gaussian whose covariance estimates local tangent structure via diffusion-geometry tools. The resulting conditional path is the displacement (OT) interpolant between standard Gaussian and the spatially varying Gaussian. The authors show is equivalent to adding a space-dependent diffusion (Fokker–Planck) term. Across geometric datasets (circles, LiDAR surfaces, single-cell trajectories, animal motion), CDC-FM aims to mitigate memorisation while preserving or improving sample quality and test NLL, especially in data-scarce and heterogeneous regimes.", "questions": "1. How is NLL computed across models?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Clear replacement of the FM conditional path with an OT-consistent Gaussian path. The equivalence to adding anisotropic diffusion (Fokker–Planck form) is neat and well-derived. \n\n2. It works with MLP/UNet/Transformer and diverse domains, including both low-dim synthetic and higher-dim biological/motion data and scaling discussion.\n\n3. The paper uses nearest-neighbour ratio to quantify memorisation; separates quality, generalisation, and the percentage memorised, showing heterogeneous behaviour over manifolds/regions.", "weaknesses": "1. The test NLL is model-dependent and sometimes hard to compute reliably for FM variants; \n\n2. Key assertions (e.g., improved convergence/quality or robustness) are not tested across standard benchmarks, strong baselines, or multiple seeds with confidence intervals.", "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": "Z3wMwzFjHC", "reviewer": "Reviewer_bx3v", "summary": "The paper addresses a key challenge in deep generative modeling via flow-matching methods: achieving very high sample quality often comes at the cost of over-fitting or memorization of the training data, rather than genuine generalization to the underlying data manifold. The authors propose a novel method termed Carré du champ flow matching (CDC-FM) that augments the standard flow‐matching (FM) framework with a geometry‐aware noise regularization.", "questions": "1. Is there a practical guidance on when to use Carré du champ FM vs regular FM?\n\n2. In practice, people use latent diffusion models (or flow-based models) for high-dimensional data.Can Carré du champ FM work in the latent space? Is it possible to show some results on data like CelebA-HQ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well written. Fig. 1 provides an overview of the proposed method and clearly shows its difference with classic flow matching.\n\nThey introduce the novel idea of aligning the noise covariance in the conditional paths of FM with the local geometry of the data manifold. It leverages diffusion geometry / local covariance estimation to regularise generative flows. The authors also show the connection with anisotropic diffusion, providing nice theoretical justification. \n \nThe authors show how one can estimate the local covariance (noise field) from data in a scalable way (via k-NN/diffusion kernel methods) and integrate it into the FM pipeline. The method can be plugged into existing flow matching implementations. \n\nThe experiments span a nice variety of domains: synthetic engineered manifolds, point clouds, genomics, motion capture, and images. o\tAccording to the results, CDC‐FM achieves similar or better sample quality while reducing memorisation and improving generalisation.", "weaknesses": "The authors point out the scalability issue due to the use of manifold hypothesis. When the underlying manifold dimension grows, the samples required to estimate the tangent space grow exponentially. \n\nThe practical benefit seems strongest in specialized regimes (maybe low-dimensional manifold data, low data) rather than the large‐scale image domain that most practitioners focus on; the implementation adds complexity and hyper-parameters.", "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": "yjFda126PF", "reviewer": "Reviewer_gnWi", "summary": "The paper introduces Carré du Champ Flow Matching (CDC-FM), a generalization of the Flow Matching method designed to enhance generalization while maintaining generation quality. The approach incorporates a geometry-aware noise term as a regularizer. Its effectiveness is demonstrated through extensive experiments across diverse datasets, validating the model’s properties and performance.", "questions": "**Experiments**\n- In Tables A6 and A7, I noticed that the DtM for FM is consistently lower than for CDC-FM, which is quite interesting. Do you think this is just due to memorization?\n- The experiment \"Early stopping for spatially heterogeneous data\" is very interesting.\n\n**Paper structure and writing**\n- In the introduction, the motivation for using geometric regularization as a means to improve the quality–generalization trade-off is not clearly stated. While it is clear that it introduces a more geometry-aware noise, it remains unclear how this would contribute to better generalization.\n- Providing an intuitive explanation of the method - similar to the excellent one you give in the Conclusion - before presenting the technical details in the introduction would make the paper easier to follow.\n- Making your loss function explicit could also improve clarity.\n- The paragraph preceding Section 3.2, which introduces the connection to the Dirichlet energy and carré du champ, is quite dense but conceptually important. I believe this explanation should be expanded and made less synthetic.\n- The explanation of the experiment on animal motion capture data is also rather dense and not very smooth to read. In this case, the solution might not be to add more text, but rather to move some details to the Appendix. Of course, this is just a suggestion to improve readability.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-structured and clearly written, with a rich set of figures that effectively illustrate the concepts and accurately present the results.\n- The work presents a novel, interesting, and mathematically sound solution to an open problem in flow matching, demonstrating strong practical performance.\n- The model is extensively evaluated to properly experimentally show the benefits provided by the method, as well as to test and show the limitations, which are the ones usually affecting geometric models.", "weaknesses": "- The main weakness of the paper, also acknowledged by the authors, lies in its reliance on the manifold hypothesis. The experiments show that as the dimensionality of the data manifold increases, CDC-FM struggles to maintain the same sample quality as FM when trained with the same number of data points. I consider this a significant limitation, though the paper remains of high quality overall.\n\n- Another area for improvement is the writing. Although the paper is detailed and mathematically precise, the motivation for adopting this specific approach to address the generalization–memorization problem in FM is not clearly articulated in the introduction. Moreover, while the method is rigorously developed in the introduction and Section 3, providing an intuitive explanation before delving into the technical discussion (e.g., before Equation (1)) would make the paper more accessible. I will elaborate on these points in the Questions section.", "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": "cgtJK7uoza", "reviewer": "Reviewer_M4z6", "summary": "This paper introduces **Carré du Champ Flow Matching (CDC-FM)**, a *geometry-aware generalization of Flow Matching (FM)* designed to improve the tradeoff between **sample quality** and **generalisation** in deep generative models.  \n\nStandard FM often suffers from *memorisation*, in which the model reproduces training examples rather than sampling broadly from the underlying manifold. CDC-FM addresses this by introducing a **spatially varying, anisotropic diffusion term** aligned with the data manifold’s local geometry.  \n\nThe method provides both a **theoretical framework** linking geometric regularisation and generative transport, and a **scalable algorithm** (O(N log N)) applicable to large datasets.", "questions": "1. How sensitive is CDC-FM to the choice of kernel bandwidth and neighbour count in the diffusion map estimation of \\(\\hat{\\Gamma}\\)?  \n2. Could the authors explore combining CDC regularisation with learned adaptive manifolds or latent diffusion embeddings?  \n3. In higher-dimensional settings (e.g., CIFAR-10), can CDC-FM’s benefits be retained through hierarchical or local manifold approximations?  \n4. How does CDC-FM interact with implicit architectural regularisation (e.g., transformers vs UNet backbones)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. **Conceptual Novelty and Theoretical Depth**  \n   - Reformulating flow matching as a geometry-regularised stochastic process via the *Carré du Champ* operator is mathematically elegant and well justified.  \n   - The theoretical analysis clearly connects the anisotropic diffusion term to Dirichlet energy minimisation and optimal transport interpolants.  \n\n2. **Empirical Breadth**  \n   - Demonstrations span multiple domains (geometric point clouds, biological data, motion capture, image synthesis), highlighting robustness.  \n   - Consistent improvements in *generalisation* metrics (NLL, Earth Mover’s Distance, memorisation ratio) show the practical benefit of geometry-aware noise.  \n\n3. **Quality–Generalisation Tradeoff Analysis**  \n   - The paper systematically studies how FM overfits under data sparsity and heterogeneous sampling and shows CDC-FM mitigates memorisation even without early stopping.  \n   - Figures such as Fig. 3–6 effectively illustrate the dynamics of quality, memorisation, and generalisation over training epochs.", "weaknesses": "1. **Limited Theoretical Generality**  \n   - The approach depends on accurate tangent-space estimation via diffusion maps, which scales poorly with high-dimensional data and may break for non-manifold or mixed-domain datasets.  \n   - The analysis assumes a strong form of the manifold hypothesis; the method’s behaviour in non-geometric or noisy data regimes is not deeply studied.\n\n2. **Empirical Limitations**  \n   - Although CIFAR-10 results are included, they remain moderate; improvements appear strongest in low-data or geometric domains rather than large-scale image generation.  \n   - Some evaluations (e.g., motion-capture experiments) are largely qualitative; more statistical metrics (e.g., variance, confidence intervals) would improve credibility.\n\n3. **Ablation and Hyperparameter Sensitivity**  \n   - The role of parameters such as the scaling factor \\(\\gamma\\) and rank \\(d_{cdc}\\) is discussed but not systematically ablated.  \n   - The influence of kernel bandwidth and nearest-neighbour size on \\(\\hat{\\Gamma}\\) estimation could be quantitatively explored.", "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)