Follow the Energy, Find the Path: Riemannian Metrics from Energy-Based Models
Authors: Louis Béthune
David Vigouroux
Yilun Du
Rufin VanRullen
Thomas Serre
Victor Boutin
Venue: NeurIPS 2025 poster ()
Keywords: Riemannian metric
geodesics
energy-based model
data-driven metric
energy landscape
shortest-path
Primary area: deep_learning
TL;DR: We derive Riemannian metrics from pretrained EBMs to compute data-aware geodesics. Our approach outperforms standard methods across datasets, offering a scalable solution for learning data geometry in high-dimensional spaces.

Abstract: What is the shortest path between two data points lying in a high-dimensional space? While the answer is trivial in Euclidean geometry, it becomes significantly more complex when the data lies on a curved manifold—requiring a Riemannian metric to describe the space's local curvature. Estimating such a metric, however, remains a major challenge in high dimensions.

In this work, we propose a method for deriving Riemannian metrics directly from pretrained Energy-Based Models (EBMs)—a class of generative models that assign low energy to high-density regions.
These metrics define spatially varying distances, enabling the computation of geodesics—shortest paths that follow the data manifold’s intrinsic geometry. We introduce two novel metrics derived from EBMs and show that they produce geodesics that remain closer to the data manifold and exhibit lower curvature distortion, as measured by alignment with ground-truth trajectories.
We evaluate our approach on increasingly complex datasets: synthetic datasets with known data density, rotated character images with interpretable geometry, and high-resolution natural images embedded in a pretrained VAE latent space.
Our results show that EBM-derived metrics consistently outperform established baselines, especially in high-dimensional settings. 

Our work is the first to derive Riemannian metrics from EBMs, enabling data-aware geodesics and unlocking scalable, geometry-driven learning for generative modeling and simulation.

Follow the Energy, Find the Path: Riemannian Metries from Energy-Based Models
Louis Bethune Apple
David Vigouroux
IRT Saint Exupéry, ANITI, IMT Atlantique
Yilun Du Harvard University
Rufin VanRullen CNRS
Thomas Serre Brown University
Victor Boutin" CNRS
Abstract
What is the shortest path between two data points lying in a nigh-dimensional space? While the answer is trivial Euclidean geometry, becomes significantly more complex when the data lies on curved manifold manifold-requiring Riemannian metric describe the space's local curvature. Estimating such a metric, however, remains major challenge in high dimensions.
In this work, we propose method deriving Riemannian metrics directly from pretrained Energy-Based Models (EBMs)-a class of generative models that assign low energy to high-density regions. These metrics define spatially varying distances, enabling the computation of -eodesics-shortest paths that follow the data manifold's s intrinsic geometry. We introduce two novel metrics derived from EBMs and show that they produce geodesics that remain closer to the data manifold and exhibit lower curvature distortion, as measured by alignment with ground ground-truth truth trajectories. We evaluate approach on increasingly complex datasets: synthetic datasets with known data density, rotated character images with interpretable geometry, and high-resolution natural images embedded in a pretrained VAE latent space. Our results show that EBM-derived metrics consistently outperform established baselines, especially in high-dimensional settings.
Our work is the first to derive Riemannian metrics from EBMs, enabling data- aware geodesics and unlocking scalable, geometry-driven learning for generative modeling and simulation.
Introduction
What the shortest path between two data points a high--imensional space? In Euclidean geometry, the answer is straight line. But in modern machine learning, where data often lies on unknown curved manifolds within high-dimensional space, straight lines slice through regions without data (see linear interp. in Fig. []- Capturing the true geometry of data is therefore critical in fields where distance-based -aaed analyses depend on underlying structure, such as vision 113) language (5) biology (6), and cognitive science 7 ( Riemannian geometry offers principled way to navigate these spaces by introducing a smoothly varying local metric, the Riemannian metric, which encodes how space bends and stretches [9]. Within this framework, the shortest path between two points is no longer straight line, but a geodesic a curve that follows the intrinsic curvature of the manifold. Computing geodesics requires knowing the underlying Riemannian metric, but estimating such metric for complex, high-dimenssonal data remains a major challenge in machine learning.
* Equal contribution.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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A promising strategy for deriving Riemannian metrics is to take data-driven approach gearningg the metric directly from the data itself. This approach estimates the data density and turns into a Riemannian metric that contracts high-density regions and dilates low-density ones, aligning the geometry with the data manifold [T0] (see $ $2 for more details). Existing methods, such as kernel- based estimators [ normalizing flows [2,, and density-bacn -basd constructions [3]. have succeeded in low-dimensional settings. However, their performance often degrades high dimensions, where sparse local sampling makes it hard to capture reliable geometric structure [14 [1]. Meanwhile, recent advances in generative AI (16H18] have produced models capable of capturing complex data distributions in nigh-dimensional spaces with remarkable accuracy. If these models can learn the data distribution, can they also reveal its underlying geometry?
In this article, we answer affirmatively by proposing to derive Riemannian metrics from pretrained Energy-Based Models (EBMs) [16] EBMs are a flexible class of gen- erative models that define energy function Eg. parameterized by a neural network, as- signing low energy to likely data points (i.e.. Px(x) x exp( Eg (x))). ).. We show that the en- engy landscape of an EBM encodes rich geo- metric structure and can be leveraged to derive effective Riemannian metries. Specifically, we introduce two novel conformal Riemannian met- rics-mess mes-mmsriiss that scale the identity by pos- itive scalar function: GE, proportional to the energy itself, and G1/pa. proportional the in- verse unnormalized density. We evaluate both against established alternatives (GRBF [33 and GLAND III) across datasets of increasing complexity from toy distributions with known geodesics (see $ to rotated character images where the manifold structure is partially known (see $ (.3).. and finally to high-dimensional nat- ural images where no ground truth geometry is available (see $ 4.44.. Throughout this work, we adopt the common choice equipping the data space with a density-based Riemannian metric, thereby defining the geometry of the manifold in terms of data concentration. We show that EBM- based metrics yield geodesics that (i) remain closer to the data manifold and (ii) better reflect its intrinsic eurvature (see Fig. for visualization of the geodesics).
Sample:
T
Endpoints
Geodesics
19 20
Datasel
Trajectory:
Ours
G
Ge
GyMF
GLAND
Lin. Inter
F
F
Figure I: Geodesics visualization for the URC dataset. Trajectories and samples are projected in the PCA space for visualization.
Overall, our contributions are summarized as follows:
We propose a novel framework based on pretrained Energy- Models (EBMs) to derive Riemannian metries. In partieular, introduce novel conformal metrics Ge, and G1/pe= based on the data log-like likeliiel and data density, respectively.
We demonstrate that these M-d-rvrived metries yield geodesics that remain closer to the data manifold and better reflect its curvature.
We show that the proposed EBB-based metrics scale more robustly than prior approaches.
By grounding Riemannian metrics in generative AI, we hope to initiate a new paradigm for under- standing and navigating the hidden geometry of nigh-ddmensiooal data spaces.
2
Related Work
The many facets of data geometry: A variety of approaches have been proposed to study the geometry of data:
Information Geometry: This historical approach is rooted in the work of Rao 21 and Amari
22 It connects statisties and differential geometry by interpreting the Fisher information 23
a Riemannian metric the manifold of parameters of statistical model. In contrast,

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work derives Riemannian metrics directly from the data space using the energy or the likelihood of an EBM.
Data-Space induced metrics: Closer to our work, this approach estimates Riemannian metrics directly from samples. The LAND) metric [1]] derives a local metric tensor from the empirical covariance of nearby points. The RBF metric [13] defines a conformal metric using an RBF network trained as a parametric KDE, learning centres, widths, and weights so its output forms unnormalised data density. Both serve baselines in study (see $ 4..1) and have recently been used for geodesic fitting via flow matching 24. The (unpublished) work of Perone [25]
was also key inspiration, proposing build metries from the score function of generative model- an idea also explored by Diepeveen et al. [26).
Latent-Space induced metrics: Another line of work uses pullback geometry 12 32 mapping
the Fuclidean metric from a network's latent space to the data space- tapccaly through the Jacobian of a VAE encoder (3 While our method operates in the latent space of VAE in high-dimensional settings, the metric is derived from the energy of the EBM and remains independent of the VAE encoder.
Generative modeling on a pre-defined manifold: Recent approaches such as flow-based mod- els [34 355 and Schrödinger bridges [36] (37 learn transport paths between distributions, some- times defined over Riemannian manifolds [3 These methods assume a known, fixed manifold geometry (e.g.. a hypersphere) and design generative models operate within that structure. In contrast, our approach starts from a generative model an EBM- and derives the Riemannian metric itself from the model, allowing the geometry to emerge from the data.
For a more detailed review of the related work (including topological data analysis, symmetries, computer graphics, metric learning), see Supp. A and
42 14]
Energy-Based Models (EBMs): EBMs, trained via maximum likelihood 16 (see 3.2 are particularly well-suited for deriving Riemannian metrics. Their contrastive training, combined with Langevin dynamics sampling, encourages learning a global energy landscape that assigns meaningful values across the entire ambient space, including regions far from the data manifold. In contrast, normalizing flows [3] are limited by their invertible architecture 44 45) and tend to perform poorly on outto--dist data [46], sometimes leaking probability mass outside the support 4 EBMs trained with diffusion losses [48 or distilled from diffusion models [49] generate high-quality samples, but their energy function depends on time-indexed noise scale, limiting them local rather than global energy landscapes. This makes them unsuitable for defining a consistent Riemannian metric. Prior work has used the global energy landscape EBMs trained via maximum likelihood for trajectory planning in robotics [50), though not in the context of geodesics.
3 Method
Scalars are denoted by plain lowercase (e.g., x), vectors by bold lowerçase (c.g. E IR P) and matrices by bold uppercase (c.g.. X). Let I be the identity matrix of IR Dx Sp, the set of symmetric D x D positive definite matrices. Let M be Riemannian manifold, with tangent space at E M denoted T.M. Herein, we assume that M is embedded in a D-dimensional Euclidian space (M C D)
Notation:
3.1 A primer on Riemannian geometry
Riemannian manifold (M, G) is smooth manifold M (i.e., a set locally homeomorphic to RDD) equipped with Riemannian metric G M + SL.. G defines a smoothly changing inner product on the tangent space T.M at each point M: (u, v)x u G(x)v, with u, v E T&M [9. The length of a curve 7 : [0, 1] > M linking two points Xo = (0) and X1 = 7(1) (Xo. X1 E M), is measured
I(>) -(>)= 1.
(10))
In Eq. I (t) denotes the velocity vector of the curve (t), which lies in the tangent space at that point (I.e., (t) E ). The minimizer of Eq. [ is called geodesic; represents the (locally) shortest path between Xo and X1. this work, we minimize the kinetic energy functional instead of the length (see Eq. 2). Although both functionals yield the same geodesics up to a parametrization.
3

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minimizing the kinetic energy functional results in a constant Riemannian speed parametrization This property simplifies optimization and improves numerical stability 9 13
(1) = arg min 8l>] s.t. 8l>] = 2 ! ()))))00))c) dl.
(2)
7
In the Euclidean case (M = TR. G(x) = D) equivalent to the kinetic energy of a unit-mass particle moving along 7(t,, hence the name kinetic energy functional.
To avoid the computational cost of solving Eq. 2 for each new pair (Xo, x1) at inference time, we follow (24] and approximate the geodesic with a neural interpolant ip, (with parameters 11).
X1, (1-t)xo + tx 2t(1-t)4, (x0, x1.
This parameterization satisfies the boundary conditions (xu.,)XX, X1.1) -1, In Eq. 3 S, serves as nonlinear correction to the linear path, allowing the learned path to bend toward the data manifold. We train a single interpolant network ,,, over batches of random endpoint pairs so it can approximate geodesics between arbitrary points (see Algo. [T. Intuitively, our geodesic interpolant begins with straight line between the endpoints and uses a neural network to compute a smooth curvature relative this seelinee-bedinng the path toward regions of higher data density, much like pulling string taut over a curved surface that reflects the geometry of the data. Unlike Kapusniak et al. [24], who use full autodifferentiation compute Xt, we opt for finite difference instead. We found this approach more stable and accurate when using fine-time discretization.
Although Algo. I approximates geodesics for a given metric G. the trajectories may initially deviate from early
Algorithm 1: Training geodesic interpolant
Input: Endpoints pairs: ((xo), (x1) }), Interp. net.: (P,) Metric: G. Time steps: T
the data manifold-especiale in training, when they are initialized as straight lines in the ambient space. However, if (i) the eigenvalues of G are large when off-manifold and (ii) small when on-manifold, then the interpolated points X₁ are progres- sively drawn toward the manifold during optimization [24 13]. In other words, an effective G should penalize off-manifold directions and encourage paths through high-density paths, steering the geodesics along true data geometry. This insight suggests that defining the metric as a decreasing function of the data probability (e.g., G(x) x p(x)-1 I) can effectively steer trajectories toward high-density regions. In practice, however, the true data distribution is unknown and only observed through samples. this work, we use an EBM to approximate the data distribution.
dt= T-1- t=[0:1:dt]
while training do
Xo ~ {Xo} and x1 {x1} ## sample batch of pairs
X1,1 = (1- -)=x txi + 26(1 t)sp,,(xo, x1,
Xt.n = Xc+1,9 Xt.n
## finite difference
dt
1
((T) Exxxx1
Σ
XT., G(X..))
2
Update using gradient V,C
3.2 Energy-Based Models
Let PM be the true data distribution supported on the manifold M, such that fxe PM (x) In practice, we do not have access to P.M directly. but only to a finite set of samples D = (x/ 1 drawn from it. These samples define the empirical distribution pd. which we use to train our models.
Energy-Based Models (EBMs) provide a flexible framework for modeling complex, unnormal- ized probability distributions making them particularly well-suited for data concentrated low- dimensional manifolds. Here we define the energy function Es E IR, parameterized with a neural network with weights 0. This energy induces a probability distribution of the form:
(x) = exp (x))
where Z(0) = exp (- (x)o
fcxp
Z(0)
"iit length fixed, the strictly convex energy F= 1 So x(t)² attains its minimum by Jensen's 's incqual- ity only when the speed v(t) is constant.

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Our goal is to train the EBM so that pe approximates the data distribution P.M- To do so, we minimize the negative log- likelihood w.r.t the empirical distribution: L.r. (0) Ex -ppl log pa (x)]
Although the partition function ( is intractable, previous works have shown that the gradient of
this objective can be estimated without computing (()) explicitly [51 52
(see Supp. B. for the
demonstration), loss known as contrastive divergence:
VOLML = Exttppp (Fx(x))
~pe Es (x")
(5)
where x+ are data samples and x samples drawn from the model distribution po using Langevin dynamics. We adopt the training procedure of [16], which known to scale well (see Supp. B for the
full pseudo-code). From this point on, we refer to Eg as a pre-trained energy function.
EBM can be hard to train in bigh-dimensional pixel space, especially because of the sampling procedure (33555 For complex tasks, follow standard practice and operate in the latent space of a pretrained VAE [56). where all baselines are evaluated for fairness. To improve the EBM training training stability, regularize the contrastive divergence loss with denoising term, which preserves the global structure of the energy landscape while enhancing convergence- technique we find both effective and broadly applicable.
3.3 EBM-derived Riemannian Metries
Here, we describe the EBM-derived metries GE G1/De For details on the baseline Riemannian
metrics GLAND GRBE, see $ 4.1 To ensure a fair comparison and following standard practice in
the field 42 57 58
all metrics are cast using a shared parametric form:
( h(x) 3 B3) for G1/pe GLAND, GRBE, G(z) ( (())) (ax))))++
for GE
where h(x) is metric-specific, positive-definite function (either scalar, diagonal, or matrix), and a, 3 are calibration constants. These constants are chosen so that the metric scale to I on the data manifold and to 103 I in low-density regions This allows fair comparison across metric choices without introducing significant sensitivity to hyperparameter tuning. Further details about the metric calibration procedure are provided in Supp. C.T Importantly, all EBM-derived metrics are conformal, they take the form I(x)I, where À is scalar function. In other words, they scale the identity matrix uniformly in all directions, resulting in isotropic metrics:
Ge defines a Riemannian metric by directly scaling the raw energy of a pretrained EBM. This is the simplest -yet surprisingly cnsecivee-fommuattiin we consider:
GE (x) (a Ex(x) + -3)
(6)
Intuitively, high-energy (low-density) regions receive larger metric, penalizing movement away from the data. Note that Eg is an affine rescaling of the negative log-likelihood - log PD.
G1/ps leverages the inverse of unnormalized probability estimate:
G (x) = (a exp(- + -I.
Compared to GE,, this metric applies an inverse to a decreasing exponential, forming strong barrier against low-density regions. stays small near the data manifold but rises sharply elsewhere, acting as repulsive force. Its key advantages are: (i) a clear probabilistic interpretation via the unnormalized density, and (ii) direct comparability to GLAND and GRBF as they share the same inverse form.
In the next section, introduce the baseline Riemannian metrics used for comparison. We also empirically evaluate their behavior across datasets of increasing complexity, focusing on how they capture the underlying manifold and shape geodesic paths.
4 Experiments
4.1
Baseline Riemannian Metrics
GRAE
13
24 and GLAND are established metrics from the Riemannian geometry literature:
Note that this multiplicative factor amounts to a change of unit, to ensure reasonable scaling of the lengths, but the induced geodesics only determined by the ratio a/B.
5

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Gland, also known as the LAND metric [1, is nonparametric Riemannian metric that adapts the local geometry of the dataset. Around each point x. it estimates Gaussian distribution by weighting all data points (x+) according to their distance to
N
||x-xxill 2x
GLAND = (ax (eiiia((xx) + 3I) s.t (x)
Σ (1 10))=001 2 exp
(8)
Here, 1000 measures the local variance along dimension j. weighted by a Gaussian kernel with bandwidth a. GLAND is the only diagonal (i.e., non-conformal) metric we consider, allowing it to model local anisotropy. While flexible and model-free, LAND has practical drawbacks: requires the full dataset at inference, is sensitive to the choice of a, and can behave non-smoothly near sharp neighborhood transitions (see Supp. C.2 for examples).
GrBE is a conformal Riemannian metric in which h is weighted sum of Radial Basis Functions (RBFs) centered on K cluster centroids {
) computed via K-means [3]
K
WE exp (-1 >xel/x - &c
GRBF(x) = (ax -ex)h(())))) + 11 II, (xx)=
Σ
k=
The weights wy are trained so that f(x) on the data manifold, and A is from inter-cluster distances (see Supp.
C.3 This yields a smooth, efficient approximation of the data geometry and
scales better than LAND 24 However, it may miss fine-grate aineggriined structure, especially in regions of complex or uneven density. Like other methods based on Euclidean distance (and K-means), it suffers from the curse of dimensionality. Its accuracy depends on K, AE, and centroid placement (illustrated in Supp. C,3
The scaling constants (a, (3 are introduced to ensure consistent dynamic range across metrics and have minimal impact on convergence or geodesic quality; the number of discretization steps (T' = =100) is chosen as a trade-off between efficiency and accuracy, consistent with prior work. We evaluate /t/pa GB., GRBF, and GLAND on three datasets of increasing complexity. Circular Mixture of Gaussians offers full control and ground-trun -ounnd-ruuhh geodesics. The rotated characters dataset nigher-dimensionee but still allows quantitative evaluation. Animal Faces is made of higher- digher-dimennional images but with no ground truth. This progression tests metric performance as data complexity grows. The code to reproduce all our experiments is available at https:://// / thub. wwt.///ttooooooiiii..oo RinieimmmmBBBBM
4.2 Circular Mixture of Gaussians
We consider two toy datasets built using a mixture of Gaussians arranged along semicircle. In the first, called Uniform Circular Gaussians (UCG), the Gaussian components have equal weights (see Fig. 2b) In the second, Weighted Circular Gaussians (WCG), the weights non-uniform, with higher density near the center of the arc, as reflected by the contour intensity shown in Fig, 2 For both datasets, we have access the closed-form probability distribution of the data, denoted P.M (see Supp. for details of P.M). We first train an Energy-Based Model (EBM) on each dataset to derive the metrics Gg, and G1/po (see Supp. D.2 training details). Then, we apply Algo. too both datasets using all Riemannian metrics described above. Additionally, we include two baseline Riemannian metrics derived directly from the true distribution PM:
GE (x) aa log (M) + 3 and G1/paa (x) = (a * P.M(X) + 33
Eq. [ uses calibration constants and B. computed as in other metrics. Some geodesics obtained for the 6 different metrics are shown in Fig. 2p and Fig. Z for the UCG and WCG datasets, respectively. We refer the reader to Supp. D for details on network architectures and hyperparameters.
To evaluate geodesic quality, we use two evaluation metrics. The first is the accumulated probability along the geodesic path, (M" = Σ (x It measures how closely the trajectory aligns with the data manifold the higher the better. The second is the RMSE geodesic computed using the true distribution PM, matched by metric type (e.g., GE, vs. G All quantitative results are averaged over 1, 000 geodesies with distinet endpoints (See Fig. 2 and d). GE, achieves the highest accumulated probability. indicating closest alignment with the data manifold, while G1/p. yields the lowest RMSE its baseline-best approximating the ground-truth geodesic. Both EBM-based metrics consistently outperform other methods across evaluation criteria.
baseline
To test how different metries behave when the density varies along the data manifold, we switch

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a) Geodesics UCG
b) Geodesics evaluation UCG
PM(7*)
RMSE
Metric
(t)
(4)
GEM
0.79
GI/PM
0.77
GE
0.78
0.12
Gy/pe
0.73
0.10
rt
GLAND
0.60
0.38
GRBF
0.61
0.39
c) Geodesics on WCG
-log
d) Geodesics evaluation on WCG
PM( (f)
RMSE
Metric
(4)
GEM 0.67
G3/RM
0.73
8
G
0.67
0.18
G1/po
0.67
0.14
#
GLAND 0.65
0.34
GRBF 0.47
2.2
Gipb
GLPLr
Gaar
Figure 2: Geodesics on UCG and WCG datasets. (a, e): Some geodesics obtained UCG (a) and WCG (c), for 6 different Riemannian metrics. The contour plots represent the energy landscape given by lo-oogg PM- (b, d) Quantitative evaluation of geodesies on UCG (b) and WCG (d). We report (i) the accumulated probability along the geodesic (the higher the better) and ii) RMSE between each geodesic and its corresponding baseline (i.e., G for GE and G G1/pm for G1/pe> GLAND and GRBF). See Supp. D.3 for the 2-0 error.
from the uniformly populated UCG semicircle to the Weighted Circular Gaussian (WCG), whose samples clus- ter near the are's centre. As shown in Fig. 3 log- based met- rics (GE, GE accentuate the manifold curvature more than //p-based ones (G1/pu> GRBF, GLAND, G1/pm), producing larger steps high-density regions. This is because log p diverges as p - 0, amplifying distortions and speed variations.
G₁
Guar
step size -
1.10a
GrANO
Ge
G
6.10-1
4.10-1
3.10
4.3 Rotated Characters
We use an image dataset of seven rotated, non-symmetric characters in variants: Uniform Rotated Characters (URC), with evenly distributed angles, and Biased Ro- tated Characters (BRC), concentrated near 0". In this sub- section, all computations are done in the 6-1-dimensional latent space of regularized autoencoder trained with triplet loss, ensuring that small angular differences yield short latent distances. This setup provides unique middle ground: although the underlying Riemannian metric is unknown, we can the smooth in-plane rotation between two instances of the same character as proxy for the ground-tue -truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations meaningful approximation of the true geodesic in the task-relevant transformation space. Separate FBMs and interpolant networks are trained for each dataset variant. Full experimental details (datasets, architectures, and hyperparameters) provided in Supp. E
20
40
80
100
step
Figure 3: Step size along geodesics in the WCG dataset. Log-based metrics (Ge, and GE produce sharper vari- ations, reflecting stronger sensitivity to density curvature.

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a)
b)
Q
G
D-RMSE (4)
9 -RMSE (4)
Metric
P P
F
linear
2.96
3.52
interp.
LAND
G
0.11
0.40
5
5 5
G1ip1
0.14
0.44
GLAND 0.66
2.39
5
GRBF 0.36
0.86
CL LL
Q 0 L
Figure 4: Geodesics on the URC dataset. (a) Geodesics computed with different Riemannian metrics, projected into pixel space for visualization. GRBF and GLAND often deviate from the intended path, sometimes drifting toward other characters (e.g., the letter F). (b) Quantitative evaluation using two metries: (i) D-RMSE, which measures proximity to the dataset manifold, and (ii) 7-RMSE. which measures the deviation from an ideal smooth rotation. See Supp. E.6 for the 2-a error.
In Fig. we visualize geodesics computed on the URC dataset, projected back into pixel space (see Supp. E.3 for additional results on both URC and BRC). EBM-based metries (GE, and /1/p.) yield smooth rotations that preserve character identity, while GREF and especially GLAND often deviate from the intended trajectory. To illustrate these failures, Fig. [
Gur
75
Gr,
decoded rotation (deg)
50
GLANO
25
shows all geodesics projected into PCA space for case in-
volving the letter F. While GE, and G1/ps remain on the manifold of rotated instances, linear interpolation cuts through low-density regions, and GRBP and GLAND drift toward other character classes. To quantify this, we use two metrics: D-RMSF, which measures the average distance from each geodesic point to its nearest neighbor in the dataset-lowee values indicate better adherence to the data manifold; and 7-RMSE, which evaluates how closely the geodesic follows ideal smooth rotation be- tween endpoints. All results are averaged over 1, 000 geodesics with random endpoint orientations. As shown in Fig- 4b. EBM-based metrics consistently outperform others; GRBF performs reasonably well, while GLAND shows large deviations on both metrics. Overall, these results suggest that EBBM-based metrics scale more effectively high-dimensional data than alternative approaches.
-5
-50-
-75
20
100
Figure 5: Step size along geodesics in the WCG dataset. Log-based metric (Ge,) ) produces sharper variations, re- flecting stronger sensitivity to density curvature.
As in the previous section, we examine how different metrics influence geodesic's ability to follow the manifold's curvature. We focus the BRC dataset, where orientations are biased toward 0%, creating sharper curvature near that region. To assess this, we decode the orientation at each time step along geodesics connecting fixed endpoints. As shown in Fig, 5 geodesies under Gg, rotate significantly faster near 0% than those under G1/pp and GRBF, reflecting stronger sensitivity to density variations.
At first glance, it may seem counterintuitive that trajectories following the geodesics move faster in high-density regions. However, this is consistent with minimizing the kinetic energy E in Eq which enforces constant Riemannian speed (i.c., the quantity [/((t)]l)t)) is preserved along the trajectory) but not a constant Euclidean speed (i.e.. [t)( is not constant). Since EBM- metrics assign lower Riemannian cost in high-density regions, maintaining constant Riemannian speed requires moving faster in Euclidean terms through these regions. The faster rotation near 0%, observed in Fig 5g Fig. thus reflects the lower Riemannian cost of traveling through high-density regions. These results confirm and extend our previous findings: metries based on energy (i.e., proportional to log more effecttivly capture the curvature of the data manifold.
8

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4.4 Animal Faces
We now evaluate our method on the Animal Faces High Quality (AFHQ) dataset [59], using the latent space of the pretrained Stable Diffusion v1 VAE [T8] (latent dimension: 4 x 16 16). An EBM is trained to model the distribution latent codes, and Algo. i used to compute geodesies between cat and a dog representation. We compare the resulting paths to two baselines: (i) linear interpolation and (ii) spherical interpolation (slerp) 60), which is known better preserve the structure of VAE
latent spaces under Gaussian priors (see Supp. F.6. Full experimental details are in Supp.
Fig. 6 illlustraees geodesics computed in the latent space of pretrained VAE and projected back into image space (see [.3 for additional samples as well as samples for GLAND and linear interpolation). Qualitatively, we observe that geodesics computed with the G1/pe metric best adhere to the data manifold. The Gg, metric also shows noticeable improvements over the other metrics. Despite extensive tuning, GRBF and GLAND produce trajectories only slightly better than linear interpola- tion Suggestgg these parametric metrics struggle to scale in high dimensions, consistent with prior findings
Gije
G5
Gzur
Slerp Interp.
Guv
Gg
Ggar
Slerp
Interp.
Figure 6: Geodesics on the AFHQ dataset. Each block shows an interpolated trajectory between two animal images (cats and dogs), projected back into image space for visualization. We compare geodesics computed with two EBM-based metrics (G1/pe> GE,), a parametric RBF-based metric (GRBF), and spherical interpolation (slerp). Results using GLAND, linear interpolation, and additional examples are provided in Supp.
To quantitatively assess geodesic quality, we report FID scores [61] Table [] computed over 50, 000 trajectories that interpolate from randomly chosen cat images to ran- domly chosen dog images. The results are consistent with qualitative observations: G1/ps and GE, yield the low- est FIDs, followed by the model-free slerp baseline, then GRBE, GLAND, and linear interpolation. Note that the FID measures how aligned individual samples are with the training distribution- on-manifold alignment- but does not assess whether the full trajectory respects the true man- ifold curvature. Unfortunately, AFHQ lacks ground-trun truth geometry for such evaluation.
Metric
FID (L)
Linear interp.
42.47
Slerp interp.
32.67
GE.
20.79
G1/ps- 16.47
GLAND 39.17
GRBF 37.98
Table 1: FID along geodesics for differ- ent Riemannian metries. FID is com- puted at each trajectory point to assess on manifold alignment. See Supp. F4 for the 2-a error.
Conclusion
In this work, we use pretrained Energy-Based Models (EBMs) derive conformal Riemannian metrics, GE, and G1/ps and compare them to established alternatives (GLAND | and

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GRBF [T3].. On both synthetic and high-dimensional data, EBM-derived metrics yield geodesics that stay closer to the data manifold and better capture its curvature-especialle with GE,
We focus on conformal metrics, which scale the identity by scalar field to encode density. While more complex, non-conformal and anisotropic metrics (e.g., the Stein metric (25)) are accessible from the EBM score, we found that conformal metrics offer comparable performance with simpler interpretation and reduced computational cost, justifying our focus in this work. Future work may explore these extensions with regularization or structural priors to ensure smoothness and scalability (See Supp G for discussion of limitations and Supp H for broader impact). To keep computational cost manageable, we train the EBM in the latent space of pretrained autoencoder and compute geodesics using inite-difference optimization, two design choices that substantially reduce complexity and memory use without compromising performance.
Although this article is primarily methodological, it points to promising applications. One example the mental rotation task, in which humans mentally rotate objects to match a target [62.. In such experiments, reaction times tend decrease with training [63], suggesting that repeated exposure sharpens internal representations around training examples, These refined representations may concentrate in high-density regions, where mental transformations occur more quickly, As shown in Fig. S and 5 our geodesics naturally accelerate in such high-density regions, echoing these psychophysical findings. Modeling mental simulation as geodesics on Riemannian manifolds shaped by generative model offers a principled computational framework to understand human cogmition. provides a way to formalize and test the hypothesis that the human cogmition relies on generative models to support flexible inference [64 Our approach is also particularly relevant for neuroscience, where datasets are high-dimensional. often sparsely sampled, and where understanding the geometry of neural population activity is central to scientific insight. In such settings, high-fidelity geodesics are essential for capturing the true structure of neural distort the manifold and lead to misinterpretation of brain dynamics. While training EBMs is costly, the benefits terms of interpretability and geometric accuracy make this approach compelling for applications where precision is critical.
may
As machine learning models are increasingly used capture complex data distributions, understand- ing the geometry of their latent spaces becomes essential. Our work contributes to this effort by showing that geometry can serve as a useful tool for building models that better reflect data structure, align with human perception, and shed light on cognitive processes.
Acknowledgments
This work was supported by ANR-3IA Artificial and Natural Intelligence Toulouse Institute (ANR- 19-P13A-0004). Part of this work was carried out within the DEEL project, which part of IRT Saint Exupéry and the ANITI AI cluster. The authors acknowledge the financial support from DEEL'S industrial and academic members and the "France 2030" program (NR- 1O-AT-0- and ANR-23 IACL-0002). Additional support for TS provided by ONR -200014-24-1-2026 and REPRISM MURI N00014-24-1-2603) and NSF (TS-2402875).
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A Extended related work
Several tools have been developed to study the geometrical properties of distributions. We survey some prominent approaches below.
Information geometry, initiated by the seminal works of 21 was the first to apply ideas from differential geometry to the field of statistics. Unlike our present work, the goal was not to understand the geometry of the data æ, but rather to understand the geometry of a smooth manifold I E è of parameters of an estimator Po- In particular, starting from the Taylor expansion of reverse
Kullback-Leii [69 divergence to po, in the neighborhood of po itself, with 0' = 0 + 6, we get
DKL(Pellpe) R2 (pI(pellpe) + DiDKE(Pollpe) c+ e "V (pe ppo)e.
(10)
One can show that, since 0' is the global minimum of this function, the first-order term vanishes and the second-order term Di (po (Ipo) must be a positive definite form i.e an inner product. This quantity, called Fisher information (23). gives the structure of a Riemannian manifold. The Riemannian gradient associated with this manifold yields a second-order optimization method coined natural gradient descent [70], that has been proved helpful in deep learning [71]. Our method inherits some spirit of this approach, since we define a local inner product function of the density, to give a Riemannian structure to the data manifold. However, we focus on the geometry of the data æ, not the geometry of the model's ' parameters 0.
Riemannian structure of data manifolds has already been proposed in the past. For example, the seminal LAND metric [I] is non-conformal metric built from the samples, with the intent of generalizing multivariate normal distributions to manifolds. The RBF metric [3] a conformal
metric, derived from kernel density estimator, with some learnable coefficients. More recently, Kapusniak et al. 22 proposed to use those metrics and learn a flow matching algorithm to fit geodesics in the data manifold. The Jacobian of a generative model also defines metric (33). The (unpublished) work of Perone (25) has been inspirational for our contribution. They use the Stein score function to build the metric, an approach also chosen by (26 althoouhh restricted to unimodal
densities.
Pullback geometry of latent manifolds is an active research area. [72 studies the manifold of representations of a given network, while [0] builds generative autoencoder to represent the manifold. Shortest paths are computed with fixed-point methods [31. or using a discrete graph [32] While we may rely on the latent space of VAE for some challenging tasks, studying latent representations of a neural network is beyond the scope of our work.
On-manifold generative models can be found in the literature. For example, we can mention flow and bridge matching approaches [3], which learn a flow between a source and a target distribution, including on Riemannian manifolds [8 39). In particular, the Schrödinger bridge 36 377 focuses on an optimization problem involving paths in the space of probability distributions, and was also generalized non- -on-uuciid geometries 440 4T These works differ significantly from ours: they assume the Riemannian manifold to be given, not chosen, and they build generative model on top it. To the contrary, given a special class of generative models to represent the data, we choose the metric to build the manifold.
Topological data analysis 73 studies the topological properties of the data manifold. This field aims to estimate some topological invariants such as the Euler characteristic 75) and persistent Betti numbers (76) (which are the number of connected components, number of closed loops, etc.) from a finite sample. It relies on tools such as persistent homology [77+79) to design algorithms. This approach typically focuses on the global properties: it assumes that the data accumulate well-defined manifold, from which these high-level features must be computed. To the contrary, our approach focuses on the local structure defined by the metric, while the global structure is inherited from the induced geodesics. Furthermore, we consider the whole ambient space for our manifold, tweaking only the metric to account for low-density regions.
Symmetries and geometry in representations have gathered considerable attention from the deep learning community, warranting no fewer than 3 workshops Neurips alone Symmetries are
https: :/tw::://wwwwww. www.:/.eerrrs.oo
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operations under which a structure is left invariant, or equivariant. In particular, some neural architectures are leveraged to reflect priors about the underlying symmetries of the data [80 In other cases, symmetries are discovered and learned from observations 855-7.. Unlike these approaches, we do not seek symmetries in data, and we make minimal assumptions about the model; we are mainly interested in the density to build the structure.
Non-Euclidean 2D and 3D manifolds are first-class citizens in computer graphics. The works of [88 89] define way to find shortest paths over such manifolds. However, this requires solving the Eikonal equation. which is prohibitively expensive in high dimensions or restricted Euclidean geometries [90]. Geodesics can be learned, but this is restricted to low dimensions [89]. These setups beyond the scope our work, as we focus on higher- siaher-dimensional and sparsely populated spaces, and no discrete meshes can be built from samples.
Metric learning (or distance learning) is another field whose purpose is to learn distance function between samples, typically in a -eppery-sper manner with contrastive losses [91-93]. Often, these distances cannot be realized as geodesic distance and intended for a specific task, like classification or retrieval.
B Energy-Based Model
B.1 Derivation of the Gradient of the EBM Log-Likelihood
The demonstration below is adapted from 2 to fit our notation. Even though this mathematical derivation is not crucial for good understanding our work, we include make sure our article is self-contained and complete.
We consider an Energy-Barn Based Model (EBM) defining a probability distribution via the Boltzmann form:
px(x) = xx)()E with Z(0) = exp((EE(x)) dx.
Z(0)
Our goal to minimize the negative log- lig-likel with respect to the empirical data distribution pp: CMu (0) = Ex-po ! (x()).
We first expand the log-probability:
log px Eo (x) log 2(0).
Taking the gradient with respect to 0:
VoLMI. = Exppp Ex + Vo log 2(0))
The derivative of the log- partition function could be simplified:
V log Z(0)
7()) VoZ(0)
= Z(0)
Vo
(x)()=F(x)))o dx
1
/ exp(-
dx
Z(0)
/ v dx
Exxopo [x)E
Substituting this back into the gradient the loss:
V VgLMI. Ex-pp [x)Ex(x)] - Ex-ps [x))]
In practice, denote the + the "positive" samples from the empirical data distribution pD, and T the "negative" samples from the model:
V VoLML 2 Extrop (veEs(x Ex-~pp [V )]]
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B.2 EBM training algorithm
To train our Energy-Basee Models (EBMs), we follow the approach of [16] Algo. 2 general training procedure:
details the
Algorithm 2: Training Energy-Based Model using Langevin Dynamics
Input: Training dataset :D, learning rate 1/, Replay Buffer B, Langevin step size a, noise scale number of Langevin steps
while Training do
x ~ "D sample from the dataset
x ~ sample from a replay buffer with probability 95%
## Refine negative samples using Langevin dynamics
for <- to do
x*+1 x a Ex(x
+ with (N(,,))
X = =x" .detach()
VoLML EBatc VoEo(x)) [xE }] ## Compute the ML loss
LREG Enach
Vo x V. ) ## Compute Regularization loss ooVoLnECtt update parameters with gradient descent
0 + 0 V
BUx+
In all experiments,
L 100 Langevin steps with step size 1 and noise scale 10-2
The energy function is optimized using the Adam optimizer (94) with a learning rate of 1j = 10 10-4, In addition to the maximum likelihood (ML) loss, include a regularization term that encourages the energy values to remain close to zero, technique shown to be effective in prior work [16).
We observed that training can be unstable, particularly for high- ligh-dimensional datasets. We attribute this instability the lack of gradient supervision: the loss not backpropagated through the Langevin dynamics reduce memory usage. To mitigate this, we introduce small Denoising Score Matching (DSM) loss-oni for the AFHQ This additional regularization loss is similar to the DSM loss in (95] We found this trick to strongly improve stability without degrading the performance.
-which provides weak supervision of the energy gradient.
The energy network architecture is adapted to the complexity of each dataset. Full details are provided in Appendix D.2 E.3 and F.2 Following Li et al. [95,, we design the output layer of the energy
function to take a quadratic form.
B.3 Other training procedure in literature
EBM can also be trained by minimizing the so-called Stein discrepancy 96
Denoising Score
Matching [97]. Sliced Score Matching 198), Noise Contrastive Estimation 99 A related objective
contrastive divergence energy discrepancy [100). We refer the reader 20 for complete review
of the different methods to train EBMs.
19

---

C
Riemannian Metrics
C.1
Calibration
We normalize each metric using calibration coefficients a and B, with two goals: (i) ensuring that the Riemannian metric averages to the identity matrix I the manifold, and (ii) aligning the overall scale of all metrics to allow fair comparisons. Here are more details on the calibration procedure:
First, randomly sample data pairs (Xu, x1) from the dataset D (it corresponds to the geodesics endpoints) and generate linear interpolations between them using:
(1 t)xo + tx1
(11)
Second, we define two of samples: SM. which contains the endpoints xo and xl lying on the data manifold, and S Sy. which contains the midpoints at f = 2. These sets are then used to estimate the calibration coefficients and 8:
Jmax
Omnn
G h(x))())))))))( +
a Exes h(x) 3m
SMl
Exesm h(x)
3 Imin
|SMl Σxesm h(xx)
1/gmax 1/9min
Exesge
h(x)
Ball Exesm h(x)
G(x) ((x) B
B
ExEsm 1(x)
[SM
Imiu
This calibration strategy adjusts the metric based on both on-manifold and off-manifold regions. ensures that all metrics operate within comparable dynamic range and promotes useful geometric prior: lower metric values near the data manifold and higher values farther away. As a result, geodesics are encouraged stay close high-density areas, aligning the geometry with the data distribution.
C.2 LAND metric
0.1
0.25
0.5
g=1
Figure 7: Effect of the bandwidth a on the geodesics obtained with the LAND metric. Here we have explored five values (a (0.1, 0.25, 0.5, 1, 1,5)). We observed that has major impact the shape of the geodesics.
We remind the land metric formula (see Eq.
N
exp ( xx xxx=)
GGAND(x) = (a (hag(((()))))))))))))))))))))( + +18 s.t 16)
(x)
Σ
(x)
2
(12)
20
20

---

This metric is highly sensitive to the choice of the parameter, which controls the "locality" of the metric. A small o results in a very local metric that strongly influenced by nearby points. while large smooths the metric by averaging over wider region. This directly affects the trade-off between how closely geodesics follow the data manifold and how smooth or stable they are. In practice, we observe that has major impact on the shape of the geodesics, as shown in Fig 7 confirming earlier findings by[[]] To illustrate this, we plot geodesics for five different values of (a {0.1, 0.25, 0.5, 1, 5)) and find that they closely follow the data manifold only within narrow range, particularly around 0.5.
C.3
RBF metric
We first remind the RBF formula :
K
GRBF (a h(x) + II, h(x) = Σ exp (-0.5 >
- 2x,11)) 111xx)++11))))
In the equation, the { are centroids evaluated using a K-Means algorithm. Following [3] the bandwidth (Aş) using the inter-distance to prototype (see Eq. 13):
K
A
X 8x1l
(13)
2
2K
k=1
The bandwidth, λs. controls spatial extent of each radial basis function. In Eq. 13 hyperparameter controlling how concentrated spread out the RBFs are. Intuitively, larger results narrower kernels (stronger locality) while smaller one yields wider coverage. This trade-off is explored via hyperparameter search. The weights wy modulate the relative contribution of each RBF to the resulting scalar field. These weights are optimized to ensure that h(x) remains close to on the training data, using the following loss:
k: is tunable
NV
((w)=
Σ ==
11 h(x))l1
(14)
This encourages the RBF combination to approximate constant value (here, 1) across the data distribution, ensuring consistency and stability of the field on the manifold.
In Fig- we evaluate how the number of centroids K affects the shape of the geodesics. The results show that geodesics are highly sensitive to this parameter. When is too small, the geodesics fail to follow the data manifold accurately. Conversely, when K is too large. the trajectories become overly sinuous passing through many centroids that are not necessarily aligned with the true manifold.
K- 100
K-50
K=10
Figure 8: Effeet of the number of centroids K the geodesics obtained with the RBF metric
(K {10,50, 50, 100)). We observed that K has a major impact on the shape of the geodesics.
21

---

D Experimental details on the Circular Mixture of Gaussian datasets
D.I Datasets
To design toy datasets, we have used mixture of K (2D) Gaussians. Specifically, K 200 in all our datasets. The resulting probability distribution is therefore:
K
p(x) = Σ Nox (g. I):
(15)
1=1
where N(x Mg: denotes 2D isotropic Gaussian centered at Pa- Here, I is the identity matrix of size 2 x 2. In both datasets, the centers of the Gaussians are uniformly positioned along a semi-cirele Radius R (here R 8). Specifically, the centers are given by:
u1 R cos (0x) sin (0x)
A K
with
T.
k ==
K
(16)
The only difference between the Uniform Circular Gaussian (UCG) dataset and the Weighted Circular Gaussian dataset (WCG) is the weighting coefficient {
Uniform Circular Gaussian dataset. Here all the weights are similar and equal to //K As result, the energy landscape forms a semi- circular basin with constant depth (see contour plot of Fig. 2a for an illustration of the energy landscape).
0.006
Causzian weights
Weighted Circular Gaussian dataset. this setting, the mixture weights vary, concentrat- ing the distribution toward the center of the are.
The weights symmetric with respect to the horizontal axis, producing an energy landscape with semi-circular shape and slopes symmetric around the are's midpoint (see the contour plot Fig .) Fig[I shows the weights T&& as func- tion of orientation, with all weights summing to
0.002
This setup generates curved, non-uniform density with higher mass near the center of cur- vature (i.e., at degrees), allowing us intro- duce a controlled curvature in the data manifold and assess how well different metrics capture
25
50
75
100
125
150
175
orientation
Figure 9: Profile of the Gaussians weights Ti
D.2 Neural networks architectures and Hyperparameters on the Circular Mixture of Gaussian Dataset
Here, we describe the architecture of the energy function (see Table 2}. the interpolant network (see Table 3). and the hyperparameters used for the GLAND and GREF metrics. Note that the architectures and settings are the same for both the UCG and WCG datasets.
summarizes the architecture used for the energy function of the EBM. The output is designed to follow quadratic form, similar to the approach in (95), which we found improves performance across all datasets. To assess whether the EBM successfully learns the target distribution, we visualize the learned energy landscapes for both the UCG and WCG datasets (see Fig. [ and Fig. 10p, respectively). For reference, we also include the ground-trute -nuuh energy landscapes of the target distributions (see Fig. [O]: and Fig, [Od for UCG and WCG, respectively). We observe that the EBM accurately captures the overall shape of the energy landscape for both distributions. However, in the WCG dataset, the true energy spans broader range than the EBM's learned energy. This discrepancy is partially corrected by the normalization procedure described in Appendix C.I
Energy-Based Model Table 2
22

---

a) Eg the UCG dataset
b) Eg the WCG dataset
2
Es
c) log PM on the UCG dataset
d)
log PM on the WCG dataset
20
18
14
8
PM
Figure 10: Energy Landscape on the UCG and WCG datasets. (a, b) shows the energy landscape learned by the FBMMs the UCG and WCG datasets, respectively. (c, d) shows the true energy landscape (i.e., - log PM) on the UCG and WCG datasets, respectively.
Interpolant Network Table 3 summarizes the architecture used for the interpolant network (i.e., St,,, in Algo. and Eq. 3. For all datasets, we autoencoder-like architecture for the interpolant, following similar approach to (24.
Nb. Layers
Layer type Linear (2, 32)
NB. Layers
Layer type Linear (3, 32)
SiLU
1
SiLU
Linear (32, 32)
4
Linear (32, 64)
SiLU
SiL.U
Linear (32, 32)
Linear (64, 64)
Three output heads:
1
SiLU
Linear (32, 1) for
Linear (64, 32)
Linear (32, 1) for f2
1
SiLU
Linear (32, 1) for fa
Linear (32, 3)
output fz) fa(x) + fx(")
Table 3: MIP architecture of the interpolant net- work for, for WCG dataset.
Table 2: MLP architecture the energy function on both UCG and WCG datasets.
LAND metric We performed hyperparameter search the parameter. We found that yielded the best performance. Parameters are similar for both UCG and WCG.
RBF metric We conducted a hyperparameter search to tune both the number of centroids K and the scaling factor k. The best results were obtained with K 30 and x Parameters are similar for both UCG and WCG.
23

---

D.3 Quantitative evaluation with error bars
In Fig. I1 we report the same quantitative results as Fig. 2 now including 2-o error bars. The standard deviation a is computed over evaluation metrics, each averaged on a different set of randomly
sampled trajectories (five sets total).
a) Geodesics evaluation on UCG
b) Geodesics evaluation on WCG
PM((**) (1)
RMSE
PM(7*) (1)
RMSE
Metric
Metric
(4)
(4)
GEM
0.79 + 0.02
GEM
0.67 + 0.05
G1/pM
0.77 + 0.04
G1/pM
0.73 + 0.07
GE
0.78 0.03
0
GE
0.67 + 0.06
0.18 -
G1/pe-
0.00
0.10 0.03
G1/pe_
0.67 0.09
0.14
GLAND
0.60
70.38 0.05
GLAND
0.65
0.34 0.05
GRBF
0.61 + 0.06
0.39 = 0.1
GRBF
+
2.2
Figure 11: Quantitative evaluation of the geodesics on the UCG and WCG datasets. We report (i) the accumulated probability along the geodesic (the higher the better) and ii) RMSE between each geodesic and its corresponding baseline (the lower the better). Values after the + sign indicate the 2-a error.
24

---

E Experimental details on the Rotated Character Dataset
E.1
Datasets
The Rotated Character Datasets consist of printed characters (5, G, P.J. 2), represented as whaht-and-white images of size 32 x 32. These characters were selected for two main reasons: (i) they commonly used in psychophysics experiments [63], and (ii) they are asymmetric and visually distinct, which helps avoid ambiguities in the resulting geodesic trajectories. Fig. 12 shows all characters their unrotated form.
5 J
Figure 12: Original (non-rotated) samples from the Rotated Character Dataset
The only difference between the Uniform Ro- tated Character (URC) and Biased Rotated Char- acter (BRC) datasets lies in the distribution of character orientations.
Uniform Rotated Character (URC) In this setting, character orientations are sampled uni- formly across the full range of [-179", 180%]. using a one-degree step. This ensures that each possible orientation within this interval is equally likely. Importantly, the distribution is consistent across all characters, meaning that each character appears with the same uniform spread of rotations.
0.006
density
0.004
0.002
Biased Rotated Character (BRC) Here, ori- entations follow a truncated Gaussian distribu- tion centered 0% designed mimic natural rotation statistics (see Fig. 13 Unlike the Mix-
0.000
orientation (deg)
Figure 13: Distribution of orientation for the BRC dataset
ture of Gaussian datasets, we do not have access to closed-form expression for the underlying distribution Р.M but we do control its empirical form. This setup introduces controlled curvature in the data manifold, allowing us to assess how well different metrics adhere to it.
E.2 Architecture and algorithm of the Triplet Loss autoencoder
We computed geodesics in the latent space of an au- toencoder trained with a Triplet Loss [101]. This approach is motivated by the fact that image space inherently non-Huclidean, making poorly suited for defining meaningful dis- tances. In contrast, the la- tent space of our autoen- coder is explicitly regular- ized so that Euclidean dis- tances correspond to differ- ences in orientation. By treating the latent space as the ambient space for geodesic computation, we align with the assumption that the data manifold is embedded in an Euclidian Manifold. The training procedure is described in
Algorithm 3: Autoencoder with Triplet regularization
Input: Dataset D {(x;, 0:)). Encoder Ez, Decoder D while training do Sample bath of triplet B=(xa, Xp; xn) from D
# Same character; 0, close 0ar 02 farther
za Ex(x,),
Zp Es (xp), 2n (x(x,)
f = | - x,ll
Ad, 10a Bpl. Ad,- 10g 0.1
CT EB (l|za zpll full-aai AA,)) +(llzz znl aLO)))
Ciotal Live + X Lr Update (Ø, v) using gradient V Ciotal
25

---

Algo. 3 and the encoder and decoder architectures based on the Regularized Autoencoder (RAE) framework [T02] are detailed Table 4 and Table respectively.
We trained the model using the Adam optimizer 94 with learning rate of x 10 and batch size of 128. In Algorithm 3 we
1 and À 0.1. For the architecture, the number of input features (i.c., the number of channels in the first convolutional layer) was set to / = 128. In Table 4 and convolutional layer with ne input channels,
T the notation "Conv2D(no. Rt,. 3, 1)" refers R output channels, a kernel size of 3, and padding of 1. Similarly, "ConvTr2D" denotes transposed convolution. The RAE blocks are modules introduced in [102] referred here RacBlockDown and RaeBlockUp, and are used for efficient downsampling and upsampling, respectively.
Nb. Layers
I Layer Type
Nb. Layers
Layer Type
Conv (z. 4F, 8, 0)
Conv2d (1, F, 1) RaeBlockDown (F,, 2F)
ReLU
Conv2d (4F, 4F, 3, 1)
ReLU
RaeBlockUp (4F, 2F)
1
Conv2d (2/', , 2F, 3, 1)
ReLU
RacBlockDown (2F, 4F)
Conv2d (2F, 2F, 3,
ReLU
RaeBlockUp (2F, F)
Conv2d (4F, 4F, I)
ReLU
ReLU
Conv2d (F, F, 1)
Linear (4l 8 8,Z)
Conv2d (F, 1, 4.
Table 4: Encoder architecture of the autoencoder. F is the number of features (F 128), and is the size of the latent space (= = 64).
Tanh
Table 5: Decoder architecture of the autoencoder. F is the number of features (F = 128), and z is the size of the latent space (z 64).
E.3
Architecture of the energy function and the interpolant network on the Rotated Character Dataset
The architecture of the energy function used in the EBM shown in Table 6 and the architecture of the interpolant network is provided in Table These architectures are used for both the URC and BRC datasets. The EBM was trained using the procedure described Algorithm 2 and Fig [4] samples generated by the EBM at the end of training. All EBM training hyperparameters match those described in Section B.22 For both the EBM and interpolant training, we use batch size of 128. The interpolant network optimized with Adam, using learning rate of 4
Nb. Layers
Layer Type
Nb. Layers
Layer Type
Linear (64, 128)
Linear (64"3, 3, 128)
SiLU
SiLU
Linear (128, 512)
Linear (128, 128)
SiLU
SiLU
Linear (512, 512)
Linear 128, 128)
6
SiL.U
SiLU
Linear (512, 64)
Linear 128, 128)
Three output heads:
SiLU
Linear (64, 1) for fi
Linear 128, 128)
Linear (64, 1) for f2
SiLU
Linear (64, 1) for fg
Linear 128, 64)
Output
fl()) f2(2) + fa (x))
SiLU
Table 6: Archiecture of the EBM energy function on both URC and BRC datasets
Table 7: Architecture of the interpolant network
used on the URC and BRC dataset.
E.4 Hyperparameters of the LAND and RBF metric
LAND metric We performed a hyperparameter search tne the a parameter. We found that
0.4 yielded the best performance. Parameters are similar for both the URC and BRC datasets.
26

---

2
7
7
J
J
3
7
5
P
7
5
R
2
2
J
5
5
P
F
J
4
A
R
P
2
F
P
G
P
R
5
V
J
F
4
5
F
5
S
1
G
F
R
Figure 14:
Samples from the EBM train on URC. These samples are generated by applying
Langevin dynamics to the energy function Iearned by the EBM.
RBF metric We conducted a hyperparameter search to tune both the number of centroids K and the scaling factor K. The best results were obtained with K = 300 and Λ = 0.75. Parameters are similar for both the URC and BRC datasets.
E.5
Additional geodesies
URC dataset:
In Fig.
15
we show additional geodesics on the URC dataset.
E
b) GRBF
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
6
6
6
6
5
6
6
5
5
5
6
6
G
G
G
G
4
A
L
L
p
LANN
2
2
2
2
2
2
2
2
5
5
6
6
6
5
5
5
5
6
6
5
Q
?
7
7
4
Linear interpolation
2
2
2
5
S
5
5
S
5
5
S
O 5
5
5
5
5
5
5
O
O
O
G
G
P
P
P
Figure 15: Geodesics on the URC dataset. Geodesics are computed using four different metrics: a)
G b) GREP, c) G1/pa. d) GLAND. For comparison, a simple linear interpolation is shown in e). The trajectory are computed in the latent space of the autoencoder and projected into pixel space for visualization, Each trajectory is subsampled at 20 time steps for clarity.
27

---

BRC dataset: In Fig.
16 we show additional geodesics on the BRC dataset.
a) GE
b) GRBF
1
2
2
2
2
2
2
5
5
5
5
5
5
G
G
G
G
G
G
J
F
F
F
G1 ps
LAND
1
1
2
2
2
2
2
2
2
2
5
5
5
G
G
G
G
G
5
5
5
5
4
4
4
Linear interpolation
7
7
7
2
2
2
2
2
2
2
2
2
2
2
2
5 2
5
5
5
5
5
5
5
Figure 16: Geodesics on the BRC dataset. Geodesics are computed using four different metrics: a) GE. b) GREP. <) GL/p,, d) GLAND. For comparison, a simple linear interpolation is shown in e). The trajectory are computed the latent space of the autoencoder and projected into pixel space for visualization. Each trajectory is subsampled at 20 time steps for clarity.
E.6
Quantitative evaluation with error bars
we report the same quantitative results as in Fig. now including 2-a error bars. The standard deviation a is computed over evaluation metrics, each averaged on a different set of randomly sampled trajectories (five sets total).
In Table. 8
D-RMSE (1)
7" RMMRM (1)
Metric
linear
2.96 +: 0.42
3.52 + 0.21
interp.
GB
0.40 0.01
Gifes
0.02
0.44
GLAND
0.66
0..51
GREF
0.06
0.86
Table 8: Quantitative evaluation the URC dataset with the 2a error. Quantitative evaluation using two metrics: (i) D-RMSE, which measures proximity to the dataset manifold, and (ii) RRMSE which measures the deviation from an ideal smooth rotation. Values after the + sign indicate the 2-o error.
28

---

Experimental details on the Rotated Character Dataset
Dataset
In this section, we conduct experiments on the Animal Faces High-Qualis (AFHQ) dataset introduced by [59]. The full dataset contains 13,000 images across three categories: cats, dogs, and wild animals. For experiments, approximately 5,000 images. This choice avoids introducing curvature in the data manifold that could arise from the relatively small number of samples in the wild animal category. All images cropped, aligned, and have a resolution of 5 512x512 12 pixels. AFHQ is widely used for image-to-image translation and style transfer, and its diversity in pose, breed, and appearance makes it well-suited for smooth interpolation tasks. See Fig. 17 for example images from the AFHQ dataset.
restrict the dataset to the cat and dog classes only, each comprising
Figure 17: Samples from the AFHQ dataset
59
For the experiments in this section, we compute geodesics in the latent space of a pretrained Variational Autoencoder (VAE). Specifically, use the VAE from Stable Diffusion VI [18]. The latent representations have a spatial size of 4 x 16 x 16.
F.2 Architecture of the energy function and the interpolant network on the AFHQ dataset
Energy Function: The architecture used for the energy function detailed in Table We set the number of input channels to n. = 4, matching the dimensionality of the latent representation, and use 256 feature channels in the first convolutional layer. The network follows a simple sequence of downsampling convolutional layers, which found to yield more stable training than ResNet-style architectures. The EBM is trained using Algorithm 2 with the same hyperparameters as in Section B.2 To further improve training stability, we add denoising score matching regularization term and use cosine learning rate scheduler.
Nb. Layers
Layer Type
Conv2d (n.. F,3, 1)
SiLU
Conv2d (F, F, 3, 1, 1)
SiLU
Conv2d (F,, 2F, 4, 2, 1)
SiLU
Conv2d (2F, 2F, 1)
SiLU
Conv2d (2F, 4F, 4, 2, 1)
SiLU
Conv2d (4F, 4F, 3, 1, I)
SiLU
Conv2d (4F, 8F, 4, 1)
SiLU
Conv2d (8F, 1, 0): for fi
Conv2d (8F, 1, 2, 1, 0): for fz
Conv2d (8F, 1, 2, 1, 0): for fa
Output
f1(x) (2(x) + fa(=²)
Table Architecture the energy function. F' denotes the base number of feature channels, and
ne is the number of input channels. The final energy is computed using three parallel output heads. The notation Conv2d(ni.. ny. k. s, p) refers to 2D convolutional layer with n. input channels, ny output channels, a kernel size of k. stride s, and padding p.
29

---

In Fig. 18 we show randomly selected samples generated by the EBM after training. The Fréchet
Inception Distance (FID) of the model is measured to be 9.89.
Figure 18: Samples from the EBM trained the AFHQ dataset. These samples generated by applying Langevin dynamics to the energy function learned by the EBM.
Interpolant Network: We use the U-Net architecture from 103 following the same hyperparam- eter settings.
F.3 Hyperparameters of the LAND and RBF metric
LAND metric We performed hyperparameter search tune the parameter. We found that 10 yielded the best performance.
RBF metric We conducted a hyperparameter search tune both the number of centroids K and the scaling factor k. The best results were obtained with K 1000 and 3.
F.4
FIDs with error bars
In Table. 10 include 2-o error bars. The standard deviation is computed over different sets of randomly sampled trajectories (five sets in total).
Metric
FID (L)
Linear interp.
42.47 3.17
Slerp interp.
32.67 + 2.33
GE
72.77
G1/pe
16.47 1.04
GLAND
39.173..663
GREF
37.98 2.46
Table 10: FID along geodesics for different Riemannian metries. FID is computed at each
trajectory point assess on- manifold alignment. Values after the sign indicate the 2-a error.
30

---

F,5
Additional geodesics on AFHQ
Riemanian metric: /p/
Figure 19: Geodesics on the AFHQ dataset using G Each row shows geodesic in latent space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each geodesic, from left (start) to right (end). Images are obtained by decoding the latent representations back into pixel space.
31

---

Riemanian metric: GB,
Figure 20: Geodesies on the AFHQ dataset using GE,, Each row shows a geodesic in latent space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each geodesic, from left (start) right (end). Images are obtained by decoding the latent representations back into pixel space.
32

---

Riemanian metric: GRBF
Figure 21: Geodesics on the AFHQ dataset using GREE. Each row shows geodesic in latent space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each geodesic, from left (start) to right (end). Images are obtained by decoding the latent representations back into pixel space.
33

---

Riemanian metric: GLAND
Figure 22: Geodesics on the AFHQ dataset using GLAND. Each row shows a geodesic in latent space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each geodesic. from left (start) to right (end). Images are obtained by decoding the latent representations back into pixel space.
34

---

Linear interpolation
Figure 23: Linear interpolation on the AFHQ dataset. Each row shows an interpolation in latent
space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each interpolation, from left (start) to right (end). Images are obtained by decoding the latent representations back into pixel space.
35

---

Slerp interpolation
Figure 24: Spherical interpolation (Slerp) on the AFHQ dataset. Each row shows an interpolation in latent space between a randomly sampled cat image (start point) and a dog image (end point). Columns correspond to time steps along each interpolation, from left (start) to right (end). Images are obtained by decoding the latent representations back into pixel space.
36

---

F.6 About the Spherical interpolation
Given two points Xo. X1 RP lying on the unit hypersphere (i.e., ||x0|| ||x]]]
1), the spherical
interpolation between them is defined as:
sin((1 t)0)
sin(to)
sterp (t; Xo, xi)
X0 +
t [0, 1], X1.
sin 0
sin 0
where 0 is the angle between Xn and XI, given by:
(xo,
arccos
||x0||| ||]]]]
In practice, when interpolating latent codes from Variational Autoencoder (VAE), the latent vectors Xo and X1 are typically drawn from a standard normal prior and do not lie on the unit sphere. To apply slerp, we first normalize the vectors:
X0
X1
X1
|xxll] |l
||||||
and compute as:
arceos ((xo, x1))
This interpolation method, introduced by [60]. has proven particularly effective for interpolat- ing in the latent space of VAEs. The intuition be- hind its success is that it implicitly assumes the data manifold lies a hypersphere. While this may seem restrictive, the assumption is reason- able practice. In VAE, each latent coordinate z; is drawn from a standard Normal distribution: Iz 10, (1
0.0025
0.0020
D). As a result, the D
Density
0.0015
squared norm, ||x||2 Σ
de follows chi-
0.0010
squared distribution with D) degree of freedom. This distribution is known to concentrate tightly around D, effectively placing most latent codes near the surface of hypersphere. To validate this empirically,
0.0005
visualize the distribution of |xx| for all latent codes of the AFHQ dataset (see Fig. 25)). We observe that this distribution concentrated on D-D = 1024 for VAE with latent space of size x 16 16.
0,0000
500
750
1000
1250
1500
1750
2000
2250
[[]]|
Figure 25: Distribution of |x| on the AFHQ dataset
To conclude, slerp interpolation well-suited for VAE latent spaces because aligns with their underlying geometric structure.
37

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F.7 Physical interpretation
We refer the reader to [04 or 105 for detailed background on differential geometry.
Assume that the manifold M is the ambient D-dimensional Euclidean space (M = P). We equipped the manifold M with a conformal Riemannian metric G(x) p(x) I,
Geodesic equation.
with the probability density of the data, and I the identity matrix of Dx Let 7(t) be geodesic (i.e. 7 : [0, 1] - DD.). We denote the instantaneous speed of the geodesic at time I, Y(),, and its acceleration (4). Said otherwise, +() and (t) denote 07 Øt (t) and 8F (1) respectively.
-R²).
('7
The geodesic equation is the 2nd- ODE written as:
7 (t) Eits(r(t) (²()) (t) =0 1
(17)
In Eq 17 (t) (t) and (t) denotes the k-th coordinate of (t)) and 7(t), respectively (here 1 < < D). T.j are the Christoffel symbols, they are derived from the Riemannian metric and encode how it bends and curves the space. r tells how much the change in direction in the i-th and j-th coordinate causes acceleration the k-th coordinate (1 < i, k < D). Said differently, i, j refer to the coordinate direction along which the particule is moving, and k refers to the coordinate direction where the motion causes effect (i.e. curvature induces acceleration).
Christoffel symbols for conformal metric. A(x) I (with À scalar function):
The Christoffel symbols for a conform metric G(x)
(x)
21(x) 0,Ax) + 6r, 05A(x) 6ig axx)
(18)
aA(x) dz (i.e. the partial derivative of X(x) with respect to the i-th coordinate),
In Eq. 18 0;A(x)
and 65. is the Kronecker symbol d55.= = if = and 65, = otherwise). If one plugs Eq. 18 in the right hand side of Eq. 17
Σ 21)6000)) 2A(7(1))
Σ a (a)
+ Σ 0,A(+(()) +1))
Σ (aA(7(0)) 7'00))
2x(r(t)) 27(0) -ax((7(0) MA(O)P]
(19)
where ( and ||-|| are the usual Euclidean inner product and norms, respectively.
So Eq. 17 becomes:
*1(1 (V)()()))5())) + 2x(x(1) (c((x(t) 1111
(²())
(20)
X(>(1)
Pulling everything together.
If one plugs our definition the Riemannian metric (i.e. X(>(t))
=
p(>(t)) and therefore vA(r(t)) Vlog Eq. 20 becomes:
X(x(t)
((t) = (V, (gg)((()))))))))) (t)) 11001) V, logr(>(t))
(21)
Eq. 21
similar in form to Newton's second law. The acceleration of a particle (of unit mass)
governed by veloeity-dependet force built from the Stein Score (i.e. V, log)()(())) More speficically:
38

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V log (t)),,)), xt)) (t) describes a "force" aligned with the particle velocity direction. This term acts like an anisotropic drag or propulsion term: it speeds up the particle when it goes toward high-density region and ii) it slows down the partiele going the other way.
(1) V log (orp((((()) is force in the direction of the stein score (pointing toward low density regions). It behaves like repulsive force, pushing the particle toward areas with low probability. The faster the particle moves, the stronger the force.
The "force" seems depend on the velocity (t)., which typical of inertial forces (i.e, forces that depend on given frame). This is an artifact from the affine parametrization of the geodesic, which ensures constant speed along the trajectory.
Newtonian formalism. Note that the variable t in previous equations is the geometrical "time". This variable stems from the affine parametrization (e.g. see Eq. ( and is not related to the physical time. To make Eq. 21 compatible with the "physical" time, denoted one can consider the following change of variable:
Ös
Ōt (s) p(>(((s)))
Ot (t) - p()(t)) equivalently
(22)
as
This change of variable implies that when moving through space according to arc-length s, the geometric time runs more slowly low-density regions and faster in high-density ones. This change of variable is particularly handy to interpret Eq. 21 Newtonian motion. Let's therefore consider the following reparametrization: (((s)) = x(s), where x is the new trajectory parametrized by the physical time So:
a
(t) = ô
dx Ôs
(tt)
xs()))
x(()) p(((ss) (x(s) pix(s)))) ds
=
(23)
Ot
Üs Øt
0
(t) = dt
(x() p(x(s))))
Oi x(s) p(x(()) Vv(((s)) x(s)) x(s) p(x(s))
0s
x 2x
x) x
(24)
Now plugging Eq. 24 and Eq. 23 in Eq
pxx) + p(x) (Vp(x), &
1 11(01)1 v llgg(x)
(ogg)((), +()) +(1) (Vlogp(x), p())x) (p(x)x) [11(x)x11 V Vlogp(x)
p(x)² (V (x), x) x) [(x))11x11 V Vogg(())
||||
log (())
Stein score
This equation can interpreted through Newton's 's second law: it describes the motion of a particle following& a geodesic in the Riemanannian manifold (M, p(z)) where p(x) denotes the data density. The particle experiences a force - |||| 2 logg((x), pushing away from regions of high probability. The term ||x| modulates the forces magnitude and plays a role analogous to momentum, strengthening the pull when the partiele moves quickly. While this is not literal physical system- here the particle is data point, and has no mass, it provides useful analogy for understanding the dynamics of trajectories shaped by data geometry.
39

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G
Limitations
While our approach provides promising framework for deriving Riemannian metrics from EBMs. several limitations should be acknowledged:
First, we restrict our study to conformal metrics, which uniformly scale the identity matrix and thus cannot capture directional (anisotropic) structure in the data manifold. While this simplifies optimization, it limits expressivity in settings where geometry varies across Prrections-sometine
more expressive, score-based metrics may help resolve.
Second, our method relies pretrained EBMs that assign meaningful energy values across the entire space. Training such models is challenging in high- sigh-dimensional settings due the computational cost of sampling (e.g.. Langevin dynamics), and performance can degrade if the energy landscape is poorly shaped or overfitted.
Third, although demonstrate improvements geodesic quality remains largely indirect-relying on alignment with proxy measures (e.g.. density, rotation smoothness, FID). In complex datasets like natural images, the absence of ground-truth geometry makes rigorous evaluation difficult.
strong baselines, our evaluation of
Fourth, our approach assumes that the data distribution is adequately captured by the EBM, yet in practice, misestimation of distort the metric and lead to suboptimal paths.
in underrepresented regions may
Finally, while we demonstrate promising results on several datasets, our experiments are constrained to pretrained generative models and fixed feature spaces (e.g., VAE latents), and generalizing to end-to-end learnable architectures remains unexplored.
Future work may address these limitations by developing scalable score-based metrics, improving EBM training stability, integrating richer evaluation protocols, and extending the framework to broader model classes and learning settings.
H Broader Impact
This work advances our understanding of data geometry by connecting generative modeling and Riemannian geometry, with potential implications across machine learning, neuroscience, and cognitive science. By enabling principled geodesic computation in high- diee-niieniinall spaces, our approach could support safer interpolation generative models, improve motion planning in robotics, or inform models of human cognition. However, care should be taken when applying such methods sensitive domains, learned energy landscapes may inherit biases present training data.
I Computational ressources
All experiments conducted NVIDIA RTX 3090 GPUs (32 GB memory). Training the toy dataset was fast, with both the EBM and interpolant completing in a few minutes. For the Rotated Characters dataset, EBM training took 8 GPU hours and the interpolant 30 minutes. On the AFHQ dataset, training required 6 GPU days for the EBM and 24 GPU hours for the interpolant. Including extensive hyperparameter searches and aril--and-error development, the total compute usage amounted to approximately 123,000 GPU hours.
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Reviews:
[{"review_id": "DMQb7DbByC", "reviewer": "Reviewer_yiCX", "summary": "The authors propose to use (unnormalized) energy-based models to create a conformal metric for Riemannian geometry, e.g calculating geodesics (conformal refers to the metric being isotropic). The important observation is that normalization is not needed for computing the metric. Two alternative forms are investigated and compared with two state-of-the-art approaches. The energy based models are generally better at capturing the Riemann structure, quantitatively they show better ability to track high-density regions.  The approach is tested on toy data, digits, and more realistic images", "questions": "Question 1: Please discuss limitations e.g. computational effort\n\nQuestion 2: Please reflect on convergence / learning curves for hyperparameters\n \nQuestion 3: Could simulated data (computer graphics etc.) be generated? - possibly parametrically, to provide ground truth for more realistic problems?", "limitations": "Could not locate a discussion of limitations in the manuscript", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strength 1:  Clear research question; i.e., to use energy based models' unnormalized density estimate for defining a conformal metric. \n\nStrength 2:   Clear and simple development of the innovation (i.e., the idea to use EBMs for definition of metric). \n\nStrength 3:  Convincing experimental evaluation on \"toy\" data (i.e., the results shown in figures 1-5) \n\nWeakness 1: No ground truth for more realistic data (i.e., figure 6 and table 1). \n\nWeakness 2.  Learning curves for hyperparameter optimization missing (i.e., showing the convergence of hyperparameters as functoin of sample size) .", "was_revised": true, "final_justification": "I enjoyed the reading the paper as reflected in my initial score, the paper provides for significant originality and convincing evaluation. Rebuttal confirmed this impression in my view, I maintain my score.", "rebuttal": "We thank reviewer yiCX for the positive feedback, and we are particularly grateful for the recognition of the clarity of our research question, the novelty of our contribution, and the strength of our experimental evaluation. \n\n* __Q1:__ Our method involves two main stages: (i) training an Energy-Based Model (EBM), and (ii) training a geodesic interpolant network. The primary computational cost lies in stage (i).  __To mitigate the expense of training EBMs in pixel space, we follow a common strategy from generative modeling (e.g., Stable Diffusion) and operate in the latent space of a pretrained autoencoder. This significantly reduces dimensionality (e.g., 1024 for AFHQ) and lowers overhead__. Stage (ii) (i.e. training the geodesic interpolant) is efficient and converges quickly. In particular, the use of finite difference to estimate velocities in Algorithm 1 (instead of full auto differentiation) significantly reduces computational cost while maintaining accuracy. Overall, we found the computational effort of the method to be reasonable and not limiting in practice. To clarify this in the manuscript, we propose adding the following sentence at line 343: _“To keep computational cost manageable, we train the EBM in the latent space of a pretrained autoencoder and use finite differences for geodesic optimization, both of which significantly reduce complexity without compromising performance.”_\n\n\n* __Q2 (related to Weakness 2):__ The main hyperparameters in our method are the energy scaling ratio alpha/beta and the number of discretization steps along the geodesic. We found the model to be highly stable across a wide range of scaling values. __These constants are introduced primarily to ensure fair comparison across metrics by matching their dynamic range; they have negligible impact on convergence speed or geodesic quality.__ Regarding the number of steps, smaller values lead to faster convergence but coarser geodesics, while larger values improve geodesic fidelity at the cost of training time. Our choice of T=100 offers a good trade-off and is consistent with prior work (e.g., [13, 24]). To clarify this in the paper, we will include convergence plots for different values of T in the appendix and add the following sentence at line 230:  _“The scaling constants (\\alpha, \\beta) are introduced to ensure consistent dynamic range across metrics and have minimal impact on convergence or geodesic quality; the number of discretization steps is chosen as a trade-off between efficiency and accuracy, consistent with prior work.”_ \n\n\n* __Q3 (related to Weakness 1):__ We agree that our high-dimensional experiments lack access to ground-truth geodesics. __This is a deliberate choice: unlike prior work that evaluates on synthetic manifolds with known metrics, our aim is to learn the Riemannian metric—via an Energy-Based Model—in complex domains where no closed-form metric exists, such as natural images.__ While this makes evaluation more challenging, it also highlights the originality and applicability of our approach. To partially mitigate this limitation, we include the rotated letter experiment, where the transformation space is well understood. The latent space, shaped by a triplet loss, ensures that nearby points correspond to small in-plane rotations of the same character. As a result, the smoothest rotation between two character poses serves as a meaningful proxy for the geodesic. To clarify this in the manuscript, we propose adding the following sentence at line 273: _“This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”_\n\nPlease note that limitations are stated in appendix G. We sincerely thank reviewer yiCX for the constructive feedback, which helped us improve the clarity and completeness of the manuscript."}, {"review_id": "pqyFkar9s3", "reviewer": "Reviewer_jD3y", "summary": "The paper proposes using Energy-Based Models (EBMs) to derive the Riemannian structure from data. The authors introduce two metrics derived from EBMs based on the energy and likelihood, respectively. Experiments on synthetic datasets with known data density, rotated character images, and high-resolution natural images encoded in latent space show the empirical performance of the method.", "questions": "1. How do the authors select the values of $\\alpha$ and $\\beta$ for the proposed Riemannian metric? The explanation provided around line 191 appears somewhat unclear. Additionally, is the performance sensitive to the choice of $\\alpha$ and $\\beta$?", "limitations": "The authors have introduced the limitation of the work in  Appendix G.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strength\n1. The paper is well structured and easy to follow.\n2. The proposed method, which uses an Energy-Based Model (EBM) to induce a Riemannian metric, is reasonable and well motivated.\n\n## Weakness\n1. The proposed method appears to rely on a pretrained, high-quality EBM. However, to the best of my knowledge, EBMs are notoriously difficult to train, often suffering from slow convergence and instability. Moreover, designing an effective energy function that balances expressiveness and trainability is a challenging task. Therefore, I am concerned that the proposed method may lack scalability due to its dependence on a high-quality EBM.\n2. The experiments seem to be conducted mainly on relatively low-dimensional datasets. The authors have not demonstrated the performance of the proposed method on high-resolution image datasets. Given the issue mentioned above, I am uncertain whether the proposed method would generalize effectively to high-dimensional data.", "was_revised": true, "final_justification": "Since directly training EBMs on high-dimensional pixel data is a challenging task, this limitation prevents the proposed method from deriving the Riemannian structure directly from pixel data. After the discussions with the authors, I believe that addressing the challenge of training EBMs is not the primary goal of this work. The authors also acknowledge this issue. After thorough consideration, I lean toward accepting the paper.", "rebuttal": "We thank the reviewer for the thoughtful and constructive feedback. Below, we address the main concerns regarding the clarity of our hyperparameter choices, the scalability of our approach, and the stability of EBM training.\n\n* __Q1:__ On the selection of the $\\alpha$ and $\\beta$ parameters: Thank you for pointing this out. The parameters $\\alpha$ and $\\beta$ are used to scale the energy (and likelihood) derived Riemannian metrics for comparability across methods. __Specifically, we set $\\alpha$ and $\\beta$ so that all tested metrics share the same minimum value on the data manifold (normalized to 1) and a maximum value of $10^3$ off the manifold.__ This normalization ensures that observed differences in performance (e.g., FID, smoothness) are not due to scale differences between metrics, but rather to the structure they induce. Empirically, we found that performance was robust to a wide range of $\\alpha$ and $\\beta$ values, as long as this normalization was preserved. To clarify this in the manuscript, we propose adding the following sentence at line 194: _“We normalize all metrics by choosing $\\alpha$ and $\\beta$ such that the minimum Riemannian energy on the data manifold is 1, and the maximum energy in off-manifold regions is $10^3$. This allows fair comparison across metric choices without introducing significant sensitivity to hyperparameter tuning. Note that only the ratio alpha/beta influences the geodesics.”_\n\n* __About the training of the EBM:__ __We first emphasize that the core contribution of our paper is primarily methodological, and serves as an additional justification for the development of EBMs.__ Our contribution is not only to propose interpolants while using EBM and metrics, it is also to demonstrate that EBM can be used to define a geometry over the data manifold. Geodesics interpolants are merely a consequence of this modeling choice. We are optimistic that further developments in EBM training will enhance our results (backed by the elements below). We agree that training EBMs can be challenging, particularly in high-dimensional settings. However, recent advances—some of which we leverage in this paper—have made EBM training significantly more stable and scalable:\n   * __Latent space modeling:__ Rather than training EBMs on raw pixels, __we operate in a pretrained latent space that is smoother, lower-dimensional, and semantically structured—greatly improving training stability and efficiency.__ For our high-dimensional experiment (AFHQ), we used the same VAE encoder as Stable Diffusion, which maps natural images into a 1024-dimensional latent space. This latent space has been shown to capture rich, fine-grained structure in large-scale datasets like LAION-5B. As such, the use of a 1024-dimensional latent does not reflect a limitation of our method, but rather a design choice aligned with modern generative modeling practices.\n   * __Denoising Score Matching regularization:__ While not widely documented in the literature, __we found that adding a denoising score matching (DSM) term to the contrastive divergence loss significantly improves training stability and convergence of the EBM.__ This regularization smooths the energy landscape and reduces sensitivity to small-scale noise in the latent space—without compromising the model’s expressiveness. Importantly, it does not impair the key advantage of contrastive divergence: the learned energy landscape remains global (i.e., not tied to a specific noise level), which is essential when using it to define a Riemannian metric. Empirically, this regularization appears to improve both performance and robustness. We are currently preparing a dedicated article that systematically benchmarks its impact across tasks and architectures. To make this point explicit in the manuscript, we propose adding the following sentence at line 185: _“To improve stability, we regularize the contrastive divergence loss with a denoising term, which preserves the global structure of the energy landscape while enhancing convergence—a technique we find both effective and broadly applicable.”_\n   * __Established EBM training techniques:__ In addition to the above, we apply well-known stabilization techniques from the EBM literature [16], including __support clipping__ (to constrain latent codes during Langevin dynamics) and __replay buffers__ (to initialize Langevin sampling from previously generated examples, with periodic resets). These practical tricks further contribute to stable training dynamics.\n\n   Together, __these strategies allow us to train stable and expressive EBMs at scale__—without compromising the quality of the learned energy landscape. We believe that these practical insights will facilitate the adoption of our method in more demanding applications. We thank the reviewer again for these valuable comments, which led us to clarify key aspects of our approach and improve the overall quality of the paper."}, {"review_id": "4M9TAIClBG", "reviewer": "Reviewer_UjCm", "summary": "The authors present a method for finding geodesic interpolations in image latent spaces through the use of pretrained energy based models as a proxy for the Riemannian Metric. They present quantitive results on toy datasets that their method matches the true ground truth geodesics better. On higher dimensional data, they provide qualitative results that their method scales better than existing alternatives and produces more realistic samples along the interpolated trajectory", "questions": "- Does asserting that $\\mathbf{G}(\\mathbf{x}) \\propto p(\\mathbf{x})^{-1} \\cdot \\mathbf{I}$ assume an a priori independence of the interpolation directions in latent space? Does this assumption match the true data generating process, or can you provide motivation for why it might hold? In otherwords, I believe this is the same as asking, what are the advantages or disadvantages of using a conformal metric parameterization?\n- If authors could release code that would greatly improve the strength of their submission.\n- For the RMSE metric of Figure 2, it is not stated to which 'ground truth' metric the LAND and RBF Models are compared. How is this decided? Wouldn't it make sense to compare all models with both 'ground truths'? \n- Is the use of the triplet loss for the autoencoder of section 4.3 truly necessary? Shouldn't the infintessimal rotations already be the ground truth geodesics of the dataset if the model is trained that way? Second, why is an EBM not directly trained on these simple images? If you are already training an autoencoder, why not train a VAE and use importance sampling to estimate the data density and use this to compute a metric?", "limitations": "The limitations section should be included in the main text, not the appendix, in our opinion, especially given the relevance of the limitations which are not discussed elsewhere in the text.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n- The related work is very thorough and it is appreciated that the authors cite unpublished work (such as that of Perone) which inspired their work. \n- The finding that metrics based on energy more effectively capture the curvature of the data manifold is interesting and welcome. \n- The idea and mathematics of the paper are well introduced and motivated, and it is an interesting topic for the community to explore. \n- The paper is well organized and presented cleanly.\n\n\n**Weaknesses**\n- The motivation for some of the experimental choices remains unclear, for example, the reason for the use of the triplet loss trained encoder in Section 4.3 is not clear. Furthermore, it is not justified why an autoencoder is needed at all, and why the EBM cannot be trained on raw images.\n- The intuition behind the use of (3) over alternative interpolants is not clear.\n- 'Code released upon acceptance' is not convincing. \n- Despite the authors claiming that their approach is scalable, the highest dimensional space they model is 1024, slightly larger than a raw MNIST image. \n- The choice 'large scale' dataset that the authors choose to evaluate on in somewhat questionable.\n\t- For example, why should we assume that the ground truth animal face geometry even has connected paths between these presented images? Wouldn't it make much more sense to try to interpolate something such as natural videos or motion where the 'correct' interpolation is much more apparant?\n- The Limitations section is relegated to the *seventh section* of the appendix, when it should be included in the main text.", "was_revised": true, "final_justification": "The paper is a solid theoretical and empirical exposition of an interesting idea that is relevant to the generative modeling community. It has the potential for significant follow up work. It is novel, precise, and well clearly presented, and therefore demonstrates a valuable contribution to the community. The authors have clarified all my concerns related to scalability, evaluation, and limitations.", "rebuttal": "We thank the reviewer for the thoughtful and constructive feedback. We greatly appreciate the recognition of the paper’s originality, clarity, and mathematical formulation. Below, we address each question and concern in detail, while also proposing concrete edits to clarify and strengthen the manuscript.\n\n* __Q1 : About the form of the Riemannian metric:__ Thank you for raising this important question. The conformal metric we use, defined as $G(x) \\propto p(x)^{-1}⋅I$, indeed assumes isotropy at each point in space—it scales all directions equally, without modeling directional preferences or anisotropies. While this does not imply full independence of latent factors, it does neglect directional correlations that could be captured by a full, non-conformal metric tensor. This modeling choice is grounded in a long-standing tradition in Riemannian geometry and manifold learning, where density-based conformal metrics are used to contract high-density regions and expand low-density regions (e.g., [13, 24, 33, 38]). __This heuristic offers a compelling balance between expressiveness and interpretability.__ In our early experiments, we also explored non-conformal alternatives,  particularly the metric explored in the blog of Perone [25] ( $s_{\\theta}(x).s_{\\theta}(x)^{T}$ with $s_{\\theta}(x) = \\nabla_x E_{\\theta}(x)$). However, we found that (i) it offers no performance gain, (ii) it is harder to interpret, and (iii) it is significantly more expensive to compute due to nested gradients. For these reasons, we focused on the conformal case. We propose to clarify this in the conclusion by adding: _“While more complex non-conformal metrics (e.g., the Stein metric) are accessible from the EBM score, we found that conformal metrics offer comparable performance with simpler interpretation and reduced computational cost, justifying our focus in this work.”_\n\n* __Q2: Code release:__  We fully agree that open-sourcing the code improves transparency and reproducibility. Although the codebase was not ready for public release at the time of submission, we have since prepared a clean version. Due to NeurIPS rebuttal guidelines, we are unfortunately not allowed to share any links at this stage, including an anonymous GitHub repository. Nevertheless, __we want to reassure the reviewer of our strong commitment to open science, and we will publicly release the full code upon acceptance.__ \n\n* __Q3 : About the RMSE in Fig 2:__ Thank you for this comment. In Figure 2, the RMSE is computed with respect to the ground-truth geodesics defined by a known Riemannian metric, which in turn is derived from the known data density used to generate each synthetic dataset (e.g., $G(x)\\propto p(x)^{-1}.I$). All models—including LAND and RBF—are evaluated against this same reference metric to ensure fair and consistent comparison. We will clarify this point in the caption and main text.\n\n* __Q4 : About the latent space (also related to weaknesses 1):__ This is an insightful question. While it is theoretically possible to train an EBM directly in pixel space, doing so is both computationally intensive and geometrically problematic: __Euclidean distances in image space are not perceptually meaningful. For this reason, we project images into a learned latent space where distances are more semantically aligned with task-relevant transformations.__ In the rotated character experiment (Section 4.3), we specifically use a triplet loss to regularize this latent space. __The goal is to ensure that the shortest path between two instances of the same character with different orientations corresponds to a smooth in-plane rotation.__ The triplet loss enforces local structure such that small angular changes are reflected by small latent distances, while large rotations and identity changes are discouraged. This gives us a meaningful approximation of ground-truth geodesics that we have leveraged to benchmarked the different metrics. In general, the use of a latent space makes EBM training significantly easier, as the latent space is lower-dimensional and smoother than pixel space. To clarify this in the manuscript, we propose adding the following sentence at line 273: _“This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”_\n\n* __A word about scalability of our method:__ We respectfully disagree with the concern that our method lacks scalability. While the latent dimensionality used in our AFHQ experiment is indeed $1024$, this corresponds to a spatial code of size 16×16×4—identical to the pretrained autoencoder used in Stable Diffusion, which is trained on the LAION datasets. __This dimensionality has been shown to capture rich visual features across highly diverse image distributions, indicating that it is not a limiting factor.__ Importantly, our method operates directly on this latent space without architectural changes, demonstrating that it is readily compatible with state-of-the-art, large-scale generative pipelines. Our goal here was not to push to the largest possible models, but to validate the method in a regime already relevant to modern generative pipelines. The method itself imposes no architectural limitations and can easily be scaled to more complex datasets or higher-dimensional latents given appropriate compute resources..\n\n* __Concerning the geodesic interpolant:__ Our geodesic interpolant follows the formulation introduced in prior work [28], which decomposes the trajectory into two components. The first is a fixed linear interpolation, $(1-t)x_0 + t x_1$, which ensures that the curve starts at $x_0$​ and ends at $x_1$​. However, this linear path is only a geodesic in Euclidean space, which does not apply in our setting. To account for the underlying geometry, we add a learnable deformation term, $ 2 t (1-t) \\phi(x_0,x_1, t) $, which vanishes at the endpoints and peaks at $t=0.5$, allowing the network $\\phi$ to flexibly bend the curve while preserving boundary conditions. This design balances geometric expressiveness with stability and ensures that the interpolant satisfies geodesic constraints in the learned Riemannian space. To give more intuition about the geodesic interpolant, we propose adding the following sentence (l. 142): _\"Intuitively, our geodesic interpolant begins with a straight line between the endpoints and uses a neural network to compute a smooth curvature relative to this baseline—bending the path toward regions of higher data density, much like pulling a string taut over a curved surface that reflects the geometry of the data.\"_\n\n* __About the limitations section:__ We agree with the reviewer. The limitations of our approach (e.g., assumptions about smoothness, dependence on latent space quality) are important and deserve more visibility. We will move the content of the Appendix Section 7 into the main paper and streamline the discussion to make it fit within the page limit."}, {"review_id": "RoVsbxPG5I", "reviewer": "Reviewer_RvU6", "summary": "This paper proposes a Riemannian metric on the data manifold and demonstrates the geodesics obtained using this metric. They propose to define a conformal metric using energy-based models (EBM). Specifically, the EBM estimates the probability density function of the data, and it is calibrated and used to rescale the identity matrix to get a metric matrix. This metric is then used to compute geodesics by minimizing the length of a neural-network-parametrized curve. The geodesics are evaluated on toy and real data on the closeness to known ground truth and closeness to the data manifold, outperforming model-free baselines.", "questions": "1. The metrics are only evaluated indirectly through the geodesics derived from them. Why not directly evaluate the metrics against the ground truth on a known toy manifold with known analytic-form metrics?\n2. For the toy dataset, the ground truth geodesic is obtained from a metric of the same form as the proposed metric (isometric, rescaled from the ground truth density). As a result, the evaluation comes down to how accurately the EBM approximates this probability distribution function. How to justify the assumption that the ground truth follows the same restricted form?\n3. The geodesics are only evaluated for “closeness to the manifold”, but not for the length minimization, which is a key characteristic and motivation for computing geodesics. Specifically, the evaluation metrics are based on the cumulative probability density, FID, RMSE vs ground truth derived from the density or points in the dataset. Each each of them quantify if the geodesics goes through high density areas, hence close to manifold. While this is an important characteristic and are useful for getting realistic samples, the actual length with respect to the ground truth metric is not evaluated. In the proposed method, the length is minimized with respect to the proposed metric, but how much does it minimize the length of the ground truth geodesic is not evaluated.\n4. Could you compare with the simple baseline of the path from Dijkstra’s algorithm? It connects points on the manifold, so it would probably be a strong baseline for the evaluation metrics used in this paper, which quantify how close the points are to the manifold.", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengthes:\n\n1. the method draws a novel connection between EBMs and Riemannian metrics\n2. The paper is well-written and clear.\n3. The proposed method is easy to compute and can be a plug-in replacement for generative methods requiring metrics (such as [24, 38]), although there is a separate challenge to fit a high-quality EBM. \n4. The method has shown good performance for deriving geodesics that are close to the manifold, which can have potential for improving real application.\n\n# Weaknesses:\n\n1. There lacks theoretical results that supports the proposed method, only a heuristic for encouraging geodesics along the data manifold (line 157). This means the contribution of the paper might not be sufficient for a neurips paper.\n2. The evaluation only covers closeness to the data manifold, but did not cover length minimization, a key aspect of geodesics, or directly compare the metrics. (see questions)\n3. The experiment on the toy dataset is not sufficient, Because for this manifold with only one intrinsic dimension, the curve is the geodesic as long as it is on the manifold. However, in a more general case (for example, a 2-sphere) the curve being on the manifold does not imply it has the shortest length. It would be more convincing to also evaluate the curve length on a toy dataset with intrinsic dimension larger than 1. More baselines such as Dijkstra’s algorithm could also be added.", "was_revised": true, "final_justification": "I still have the concerns on the theoretical properties of the proposed method on the ideal case, but I agree it is a more minor one after the clarifications on the problem setup. I appreciate the clarifications and the promise to include additional baseline experiments.", "rebuttal": "We thank the reviewer for the feedback. \n\n* __About the “lack of theoretical support of the proposed method” (weakness 1):__ We would like to clarify that __the notion of a “data manifold” is not an observable ground truth but a modeling assumption introduced by the experimenter. Consequently, there can be no general theoretical guarantee on the optimality of any chosen Riemannian metric, since manifold recovery from finite, noisy data is an inherently underdetermined problem.__ Defining a metric requires imposing structure—just as in Topological Data Analysis, where researchers rely on persistent homology.  Our approach follows a well-established principle [13, 24, 33, 38]: that data density can guide the design of a meaningful Riemannian geometry. We show that this modeling choice leads to empirically useful geodesics that capture structure in the data. The strength of our empirical validation supports the relevance of this choice. As noted by other reviewers: Reviewer yiCX described our evaluation as “convincing,” Reviewer z3pm highlighted the “impressive performance” of our method, and Reviewer RvU6 acknowledged both its novelty and relevance.\n\n* __About the experiment on the toy dataset (weakness 3):__  In the toy experiment (Section 4.2), the data are sampled from a 2D mixtures of Gaussian arranged along a semi-circle. __While we agree that the shortest path lies along a 1D structure (lying on the $\\mathcal{S}^1$), we emphasize that our geodesic optimization takes place in the full ambient space ($\\mathbb{R}^2$)__, and the initial trajectory is a straight line between the endpoints—thus not aligned with the manifold $\\mathcal{S}^1$. The optimization must therefore rely entirely on the learned Riemannian metric to steer the path toward the data manifold, without any explicit constraints enforcing manifold adherence. This setup makes the problem more challenging than optimizing directly on $\\mathcal{S}^1$, and demonstrates the method’s ability to recover geodesics that align with the data structure from off-manifold initializations.\n\n* __Q1-2 (benchmarking geodesics):__ As stated previously, the notion of a \"data manifold\" is not an observable ground truth but a modeling assumption introduced by the experimenter. In this work, we adopt a common and well-supported choice: equipping the ambient space with a density-based Riemannian metric, following prior work [13, 24, 33, 38]. This assumption defines both the manifold’s geometry and the corresponding geodesics. Under this modeling framework, our evaluations are internally consistent. In Section 4.1, __we construct toy datasets with known density functions, which allows us to define a ground-truth metric and geodesics analytically. This setup enables a principled evaluation of geodesic accuracy using RMSE—not merely of density estimation, but of how well the learned metric captures the induced geometry.__ We agree that this modeling assumption could have been more clearly stated in the paper. To address this, we will add the following sentence at line 165: _“Throughout this work, we adopt the common choice of equipping the data space with a density-based Riemannian metric, thereby defining the geometry of the manifold in terms of data concentration.”_\n\n* __Q3 (related to weakness 2):__ Section 4.3 was specifically designed to address this concern. In high-dimensional settings—as in AFHQ (Section 4.4)—it is indeed difficult to obtain ground-truth geodesics. To mitigate this, we introduced __the rotated character setup, where we can define a meaningful proxy for geodesic paths.__ The latent space is regularized using a triplet loss to ensure that small rotations of the same letter are mapped to nearby points. As a result, the shortest path between two rotated versions of the same character corresponds to the smoothest in-plane rotation, which serves as a valid approximation of the true geodesic in this space. The $\\gamma^{\\star}$-RMSE metric then quantifies how closely our predicted geodesics align with this ground-truth proxy. To clarify this in the manuscript, we propose adding the following sentence at line 273: _“This setup provides a unique middle ground: although the underlying Riemannian metric is unknown, we can treat the smooth in-plane rotation between two instances of the same character as a proxy for the ground-truth geodesic. Thanks to the triplet loss, the latent space is structured so that nearby points correspond to slight rotations of the same character, making the shortest path between two orientations a meaningful approximation of the true geodesic in the task-relevant transformation space.”_\n\n* __Q4 (about the Dijkstra baseline):__ We agree that Dijkstra’s algorithm can serve as a strong baseline in low-dimensional settings, and __we will include it in Section 4.2__. However, it’s important to note that Dijkstra operates on a discretized graph and must rely on the Riemannian metric derived from our EBM to reflect manifold structure. As such, it does not directly assess the quality of the learned metric, but rather the quality of geodesics we compute. __Extending Dijkstra to high-dimensional spaces is computationally infeasible due to the curse of dimensionality.__ To approximate a geodesic with ( $T = 100$) discrete steps, a naive discretization requires evaluating a graph with ( $\\mathcal{O}(d^T) $) nodes, where ( $T$) is the number of discretization points per dimension. Even if the intrinsic manifold has dimension ( $d$ ), the search frontier at each step spans a $(d - 1)$-dimensional surface, resulting in memory costs of $ \\mathcal{O}((d - 1)^T)$. This makes Dijkstra intractable beyond very low-dimensional settings (see, e.g., [87, 88] for examples in 3D).\n\nWe hope this clarifies our methodological choices and reinforces the coherence of our evaluation framework, demonstrating that our work offers a meaningful contribution to the community."}, {"review_id": "r8CgAZmnLu", "reviewer": "Reviewer_z3Pm", "summary": "This paper proposes two conformal Riemannian metrics defined from a pre-trained Energy-Based Model (EBM) for a given dataset. The metrics leverage the EBM's learned energy landscape to define a geometry where paths through high-density regions are shorter. Through experiments on datasets of increasing complexity, the authors provide evidence that their metrics produce geodesics that stay closer to the data manifold compared to established methods.", "questions": "My recommendation for this paper could be raised based on the authors' responses to the first two questions, which address the paper's scope and practical significance. The subsequent questions are for my interest and could hopefully help strengthen the paper further.\n\n1.  **On the Restriction to Conformal Metrics:** The study is limited to conformal metrics. Could the authors please elaborate on the reasons for this choice? Was it primarily for computational benefits, or because estimating a full, non-conformal metric tensor from sparse high-dimensional data is fundamentally more difficult? Furthermore, could you provide some intuition on how one might expect geodesics to differ if they were computed with a non-conformal metric (e.g., say one derived from the Hessian of the energy function) versus the proposed conformal ones?\n\n2.  **Envisioned Applications and Cost-Benefit Analysis:** Training an EBM is the most computationally expensive part of the proposed pipeline. Could the authors provide concrete examples of applications where the high fidelity of the resulting geodesics is a critical requirement? Specifically, what are some real-world problems where the benefits of these more accurate paths would warrant the significant upfront cost of training a large-scale energy-based model?\n\nThank you for considering these first two points, which are my main questions. The following are a few smaller suggestions that I hope will be helpful in further polishing the paper.\n\n3.  **Clarification on Geodesic Behavior with Manifold Gaps:** The evaluation on rotated characters assumes an \"ideal smooth rotation\" as the ground truth. However, what is the principled or \"correct\" behavior if the data manifold has significant gaps (e.g., training data contains images at 0-30 degrees and 60-90 degrees, but nothing in between)? Should the geodesic cut across the empty latent space while preserving identity, or should it detour to a \"cheaper\" path, perhaps via another character's manifold? How do the proposed metrics behave in such a scenario?\n\n4.  **Details on Figures 1 and 3:** The visualizations in Figure 1 are quite dense. A more detailed walkthrough of one example trajectory in the caption would improve readability. For instance, the caption notes the trajectories are projected into PCA space; a brief elaboration would be helpful. Similarly, for Figure 3, could you please clarify what \"step\" (x-axis) and \"step size\" (y-axis) represent? The y-axis appears to be the Euclidean norm of the velocity vector, but this could be explicitly stated. The paragraph explaining Figure 3 is also a little vague, and a more detailed explanation would strengthen the point being made.\n5. **Question on Finite Difference Stability:** You make an interesting practical claim that using finite differences was \"more stable and accurate\" than full autodifferentiation for computing the geodesic path's velocity. Could you provide a little more detail or intuition as to why this was the case in your experiments?\n6.  **Minor Suggestion:** Many of the plots are very effective in color, but their key details may be lost when printed in grayscale.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Quality and Clarity:**\nThe paper is exceptionally well-written, clear, and easy to follow. The authors do a good job of building intuition for complex topics in Riemannian geometry. For instance, the paragraph explaining why geodesics accelerate in high-density regions is insightful and helpful for readers less familiar with the topic. The experimental setup is convincing and showcases the method's capabilities.\n\n**Originality and Significance:**\nThe primary strength of this work lies in its originality. The proposal to derive Riemannian metrics directly from pretrained Energy-Based Models (EBMs) is novel and an interesting connection between generative modeling and geometric data analysis. It may be that this specific approach has not been explored previously because the significant computational expense of training an EBM could outweigh the benefits for many common applications. Nonetheless, the two proposed metrics, $G_{E_{\\theta}}$ and $G_{1/p_{\\theta}}$, are intuitive, well-motivated, and demonstrate impressive performance in the experiments.\n\n**Weaknesses:**\nThe main weaknesses of the paper relate to the practical limitations and the scope of the contribution.\n\n* **Computational Cost and Practical Applicability:** The proposed method hinges on having a well-trained EBM. Training EBMs, especially on high-dimensional data, is computationally intensive and challenging. This significant upfront cost may limit the method's practical adoption, especially when compared to the much cheaper baseline methods. The paper does not provide strong motivation for a real-world application where the improved geodesic quality would justify the cost of training an EBM from scratch, raising the question of whether the method is primarily intended for scenarios where a suitable EBM already exists.\n\n* **Focus on Conformal Metrics:** The paper's choice to focus on conformal metrics is a key aspect of its scope. This design choice assumes isotropic geometry and cannot capture more complex, anisotropic structures that may be present in the data.\n\nOverall, this is a strong paper with a novel idea that is well-executed. My \"Weak Accept\" recommendation is based on the fact that while the contribution is elegant and effective, its practical significance is somewhat diminished by the high computational cost and the lack of a clear, compelling application.", "was_revised": true, "final_justification": "My initial score was borderline, based on concerns about the method's practical applicability. The authors' rebuttal successfully addressed these points by providing compelling use cases and discussing the restriction to conformal metrics. This has alleviated my main reservations.\n\nWhile the desirability of the geodesic's behaviour across manifold gaps remains a nuanced, context-dependent issue, this is not a fundamental flaw of the method. The paper is original, clear, and makes a solid contribution to data-driven geometry. I have raised my score as a result and recommend acceptance.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We particularly appreciate the recognition of our paper’s originality, clarity, and experimental design. Below, we address the two major questions, which both touch on the foundational motivations behind this work. We also answer the additional concerns of the reviewer.\n\n\n* __Q1: On the restriction to conformal metrics:__ This is an excellent and timely question. Our initial experiments were not limited to conformal metrics. We tested the Stein metric $s_\\theta(x).s_\\theta(x)^T$, s.t. $s_{\\theta}(x) = \\nabla_x E_{\\theta}(x)$, which has been proposed as a valid Riemannian metric for EBMs [25], and also explored the Hessian-based metric in early trials. While both define valid geometries, we found that conformal metrics matched the performance of the Stein metric while offering clearer interpretability and significantly lower computational cost. Furthermore, for the Gaussian case, the Hessian of the energy  yields geodesics that are straight lines—ignoring mass concentration around the mean—which we found undesirable. Finally, computing and storing the full metric tensor incurs a quadratic cost in dimension. __Conformal metrics align with density level sets and produce isotropic, interpretable geodesics. In contrast, non-conformal metrics can introduce anisotropy and directional bias, offering flexibility but at the cost of control and interpretability.__ For these reasons, we chose to focus on the conformal case. To clarify this design choice, we propose adding the following sentence to the conclusion (line 346): _“While more complex non-conformal metrics (e.g., the Stein metric) are accessible from the EBM score, we found that conformal metrics offer comparable performance with simpler interpretation and reduced computational cost, justifying our focus in this work.”_\n\n* __Q2: On envisioned applications and cost-benefit tradeoff:__ We agree that assessing the cost-benefit tradeoff is essential when evaluating practical impact. Training EBMs is indeed computationally expensive, but we believe the cost is justified in domains where (i) data are high-dimensional and sparse, and (ii) precise control over the geometry of the latent space is critical. In these domains, high-fidelity geodesics ensure that interpolated paths (e.g., geodesics) remain meaningful and do not cut through implausible regions. __A prime example (and the original motivation for this work) is the modeling of neural manifolds__: the geometry traced by high-dimensional population neural activity recorded over time. In this context, a low-fidelity approximation of the data geometry is insufficient, as downstream interpretations rely on fine-grained geometric structure. We are exploring this in an ongoing work in which we show that geodesic length in learned neural manifolds correlates with human reaction times in object recognition tasks, suggesting that energy-derived geometry can capture behaviorally relevant structure. __This paper lays the theoretical and empirical foundation for such applications by validating the method on controlled benchmarks.__ To better highlight the cost-benefits tradeoff of our method, we propose adding the following sentence to the discussion (l. 356): _“This approach is particularly relevant for neuroscience, where datasets are high-dimensional, often sparsely sampled, and where understanding the geometry of neural population activity is central to scientific insight. In such settings, high-fidelity geodesics are essential for capturing the true structure of neural trajectories—approximations may distort the manifold and lead to misinterpretation of brain dynamics. While training EBMs is costly, the benefits in terms of interpretability and geometric accuracy make this approach compelling for applications where precision is critical.”_\n\n* __Q3: On geodesic behavior with manifold gaps.__ This is an important and insightful question. When the data manifold contains significant gaps—such as missing intermediate orientations—our method does not impose a fixed strategy but instead lets the learned energy landscape shape the geodesic. Since the Riemannian metric is derived from the EBM and inversely related to data density, geodesics are encouraged to follow high-density (i.e., plausible) regions. __However, if no high-density path exists between two endpoints, the geodesic may cut across a low-density gap if doing so yields lower accumulated energy than detouring through unrelated regions (e.g., another character’s manifold).__ This behavior reflects a key strength of our framework: geodesics adapt to the structure learned by the EBM and naturally trade off between identity preservation and plausibility. In ambiguous cases, the path reflects the most efficient solution under the current energy landscape. We agree that this behavior would be best illustrated with a concrete simulation. While this was beyond the scope of our initial evaluation, __we will include a follow-up analysis in the supplementary material to examine geodesic behavior on discontinuous or sparse manifolds__. This will help clarify when and how shortcuts or detours emerge under different density configurations.\n\n* __Q4: Clarification on Figures 1 and 3:__ Thank you for pointing this out. We will revise the captions to improve clarity. For Figure 1, we will explicitly mention that the trajectories are projected into the first two PCA components of the latent space. For Figure 3, the x-axis represents optimization steps, and the y-axis is the Euclidean norm of the geodesic velocity vector at each step. We will also expand the paragraph describing Figure 3 to clarify the stability behavior and interpret the drop in step size over training.\n\n* __Q5: On finite difference stability:__ We observed that computing velocities via finite differences was more stable and accurate (and also much faster) than using full auto-differentiation. __This is likely due to the fact that finite differences avoid computing second-order gradients through the geodesic interpolant network, which can be noisy or unstable in practice.__ Additionally, the network typically takes the time in input with a time embedding layer (with high frequency sine functions, a la flow matching), whose first order derivative can be ill-conditioned. Finally, finite differences act as a form of local smoothing, making the velocity estimate less sensitive to local curvature irregularities. This contributes to both numerical stability and faster convergence.\n\n* __Q6: On grayscale robustness of plots:__ Thank you for the suggestion. We will update key visualizations to improve grayscale robustness by using colorblind-friendly palettes, distinct line styles, and/or markers to ensure readability when printed in black and white.\n\nWe hope these clarifications address your concerns and will contribute to a positive reassessment of the paper. We remain fully open to further discussion and sincerely thank you again for your thoughtful and constructive review."}]

Author rebuttal:

Meta-review: This paper proposes a method for deriving Riemannian metrics from pretrained Energy-Based Models (EBMs) to compute geodesics that follow a data manifold's intrinsic geometry. All reviewers recommended acceptance, highlighting the originality of the approach and the clarity of the manuscript. Initial reviews raised concerns about the scalability of EBM training (jD3y, z3Pm) and the evaluation on complex data without ground truth (yiCX, RvU6). The authors' rebuttal addressed these points by clarifying their use of latent spaces for scalability and by discussing why the lack of ground truth data is not a problem in areas where no closed-form metric exists. I agree with the reviewer consensus and believe this work would be a valuable contribution to the conference. I recommend acceptance.
Decision: Accept (poster)