The Spacetime of Diffusion Models: An Information Geometry Perspective
Authors: Rafal Karczewski
Markus Heinonen
Alison Pouplin
Søren Hauberg
Vikas K Garg
Venue: ICLR 2026 Oral (2026)
Keywords: diffusion models
information geometry
Primary area: generative models

Abstract: We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic probability flow ODE decoder, is fundamentally flawed. It provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits a stochastic decoder via the reverse SDE, which enables an information geometric treatment with the Fisher-Rao metric. However, a choice of $\mathbf{x}_T$ as the latent representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime $\mathbf{z}=(\mathbf{x}_t,t)$ that indexes the family of denoising distributions $p(\mathbf{x}_0 | \mathbf{x}_t)$ across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation-free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at: https://github.com/rafalkarczewski/spacetime-geometry.

Published as a conference paper at ICLR 2026
THE SPACETIME OF DIFFUSION MODELS: AN INFORMATION GEOMETRY PERSPECTIVE
Rafal Karczewski', Markus Heinonen', Alison Pouplin', Spren Hauberg', Vikas Garg! Aalto University, Technical University of Denmark, Yai Ltd {rafal karczewsk,, markus o www.innnnnnn (asoonn)@aaalle /aallto alison, poupl comupiin@gmaiii com, sohauêdtu. sgargecsail.min edu
ABSTRACT
We present a novel geometric perspective on the latent space of diffusion models. We first show that the standard pullback approach, utilizing the deterministic prob- ability flow ODE decoder, is fundamentally flawed. provably forces geodesics to decode as straight segments in data space, effectively ignoring any intrinsic data geometry beyond the ambient Euclidean space. Complementing this view, diffusion also admits stochastic decoder via the reverse SDE, which enables information geometric treatment with the Fisher-Raa metric. However, a choice of ør as the la- representation collapses this metric due to memorylessness. We address this by introducing a latent spacetime 2 (æt, f) that indexes the family of denoising dis- tributions p p( (z)) across all noise scales, yielding a nontrivial geometric structure. We prove these distributions form an exponential family and derive simulation free estimators for curve lengths, enabling efficient geodesic computation. The resulting structure induces a principled Diffusion Edit Distance, where geodesics trace minimal sequences of noise and denoise edits between data. We also demonstrate benefits for transition path sampling in molecular systems, including constrained variants such as low-variance transitions and region avoidance. Code is available at https:: hub. com Aalto QuML/ httl:/spaceiimmi tine geometry.
INTRODUCTION
Diffusion models have emerged as a powerful paradigm for generative modeling, demonstrating remarkable success in learning model and sample data (Yang et al., 2023). While the underlying mathematical frameworks of training and sampling are well-established (Sohl-Dickstere et al., 2015; Kingma al., 2021; Song et al., 2021; 1; Lu al., 2022; Holderrieth al., 2025), analysing how information evolves through the noisy intermediate states p, for E [o, T remains an open question. Our work addresses this by defining and analyzing the geometric structure of diffusion models, which provides a principled framework for understanding their inner workings.
In generative models, a common way to study the intrinsic geometry of the data is to pull back the ambient (Euclidean) metric onto the latent space (Arvanitidis et al., 2018; 2022). Equipped with this pullback metric, shortest paths (i.e., geodesics) in the latent space decode to realistic transitions along data that lie on a lower-dimensional submanifold.
X
In a diffusion model, natural choice for the de- coder the reverse ODE 2o which allows us to derive the pullback geometry of the latents ær. Interestingly, we prove that this leads to latent short- est paths always decoding linear interpolations in data space, which have little practical utility.
Xo
X1
0
Figure A geodesic in spacetime is the short- est path between denoising distributions.
We then turn our attention the denoising posterior distribution p( (x)) given by the reverse SDE, We propose an alternative Fisher- geometry. which measures how the denoising distribution p(a)olar.) changes when manipulating the latent æt.

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We introduce the Fisher-Ree Rao metric G that varies with both state and time over the latent spacetime (z), t) (Fig. 1).
Estimating geodesics in information geometry is usually tractable only for analytic families. Although denoising distributions in diffusion are complex and non-Gaussian, we show that they form an exponential family. This simplifies the geometry and yields practical method for computing geodesics between any two samples through the spacetime. In the Fisher-Rao setting, curve lengths can be evaluated without running the reverse SDE, which significantly reduces the computational cost.
We demonstrate the utility of the Fisher-Rao geometry in diffusion models in two ways. First, induces principled Diffussio Edit Distance on data that admits clear interpretation: the geodesic between x" and æ" traces the minimal sequence of edits, adding just enough noise to forget infor- mation specific to x and then denoising introduce information specific x². The resulting length quantifies the total edit cost. Second, spacetime geodesics allow generating transition paths in molecular systems, where we obtain results competitive with specialized ttate-of-the-ar methods and can incorporate constraints such as avoidance of designated regions in data space.
2 BACKGROUND ON DIFFUSION MODELS
We assume data distribution defined on R., and the forward process
p (x)) = N aato I).
(I)
which gradually transforms into pure noise pr = (O, of at time T, where ag, oi define the forward drift fr and diffusion gr. There exists a denoising SDE reverse process (Anderson, 1982)
dæ = (fix oe (x)) dt + g:dW,,
Reverse SDE:
2T PT,
(2)
where Pt is the marginal distribution of the forward process (Eq. 1) time I, and W is reverse Wiener process. Somewhat unexpectedly, there exists a deterministic Probability Flow ODE (PF-ODE) with matching marginals (Song al., 2021):
1 PF ODE: dæ = (fre - 5 loggr(2))) dt,
TT PT.
Assuming we can approximate the score V log Pa (Karras et al., 2024), we denote by ær + x xo(x)) the deterministic denoiser of solving the PF-ODE from noise ær, while we denote by p( x. the denoising distributions induced by stochastic sampling of the reverse SDE (Karras et al., 2022).
3 RIEMANNIAN GEOMETRY OF DIFFUSION MODELS
Riemannian geometry equips latent space 2 with smoothly varying metric tensor Gz2) ) for 2. This metric defines inner products and induces the notions of distance and curve length (Do Carmo Francis, 1992). Several works have developed diffusion models top of Riemannian manifolds, such as spheres, tori and hyperboloids (De Bortoli et al., 2022; Huang et al., 2022: Thornton et al., 2022). In this paper, instead study what kind of Riemannian geometries implicitly induced by the denoiser within a real vector space D (c.g., images).
In Euclidean geometry, the space is flat, with distances given by the length of straight lines connecting points. In Riemannian spaces, the shortest path between two points is no longer straight, but a curved geodesic. A smooth curve 7 it minimizes the length
2 between fixed endpoints Yoo V1 is geodesic if
((>) /' 1illldds= L''' V GTGGGm))sss
(4)
or, equivalently, the energy E(y) = 2 1o 1 11.
We introduce two interpretations of Riemannian geometry G for diffusion models, depending whether the decoder is deterministic or stochastic. both cases, we first assume the latent space is the noise space æT, and later relax this cover the entire noisy sample space #t--
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Deterministic sampler: pullback geometry. Let æy H zu(or) be deterministic map given by the PF-ODE (Eq. 3) mapping noise to data. We propose the pullback metric (Arvanitidis et al., 2022; Park et al., 2023)
axo
Ozo fxr
GrB(æT)
E RDDx
3o : xo(xT) E IR
(5)
Ber
which measures how an infinitesimal noise step dxy changes the decoded sample:
(2r + dxr) =>(>>)|l dx Grg(Er)drr + o(Iddær113).
(6)
Stochastic sampler: information geometry. A Alternatively, consider stochastic decoder that, for each latent 2y defines a denoising distribution p(æoo ( by solving the Reverse SDE (Eq. 2). We propose the Ceoommtiiongeommnnne viewpoint via the Fisher-Rao metric (Amari, 2016)
GIG E (p(aaa))))))
V log pl V ver log (2r
E IR DxD
which measures how infinitesimal noise step dæy changes the entire denoising distribution:
KL p( For a helpful tutorial on information geometry, we refer to Mishra et al. (2023).
.T)
p(
ær + dxr)
2 da d d(x
(8)
PULLBACK GEOMETRY COLLAPSES IN DIFFUSION MODELS
Both pullback and information geometries are, in principle, applicable. We will first show the pullback geometry has fundamental theoretical limitations in diffusion models, rendering it practically useless.
Assume we estimate a geodesic y in the noise space Zy such that its endpoints decode to (0(%) = x and zo (7)) = æ². The pullback energy E(y) (Eq, 4) can be shown to only de- pend on the decoded curve (o(7)) in data space (See Appendix B):
Data
Neise space
Decoder: 2o (o()))
Pullback
goodesic
Y
EPB(T) 2 /./²
00 ds. ds
(9)
Straight-line decoding
The unique minimizer the constant-speed straight line I. (1 (1-s)æe 3)x² sx in data space. Since the ODE is bijective, this line has a unique latent preimage >/- 20 z (x,) which is thus a pullback geodesic, and the en- ergy reduces Euclidean distance in data space:
Figure 2: The pullback geodesics curve in noise space, but decode to straight lines in data space.
(x,),
EpB(Y) = -ea--x"1 x 213
(10)
Hence, all pullback geodesics decode to straight segments, ignoring the curvature the data manifold and undermining downstream applications (See Fig. 2). The same pathology applies for denoised geodesics in the intermediate space as well. The core reason for this is that, in diffusion models, the latent and data spaces have the same dimension. The decoder operates directly in the ambient space and, without further dimensional constraints, it cannot capture the intrinsic structure of the data, even if the data lie on a lower-dimensional submanifold. As result, the standard pullback metric provides no meaningful geometric information. A formal proof and discussion are in Appendix B.
5 INFORMATION GEOMETRY WITH DENOISING DECODERS
Under the stochastic view, the decoder is the denoising distribution p( (zr) obtained by reversing the diffusion process (Eq. 2). This yields family of distributions the data space parametrized with noise vectors Vy. The information geometry assigns the Fisher- Rao metric to the latent domain, and geodesic energies/lengths computed in Section 3.
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The latent spacetime.
Diffusion models are "memoryless" (Domingo-Enrich et al., 2025):
p(ær
2o) 2 Pr (2r)
+ p( xr x g(2to).
(11)
Hence p( #r) is (approximately) independent of æy, implying Var log p collapse of the Fisher- Rao metric, GIG (Eq. 7). Consequently, if we identify the latent space with x = æy. all øT become metrically indistinguishable. This could be avoided by choosing x æt for some T; however, instead of choosing an arbitrary noise level noise levels simultaneously by considering points in the (D + 1) dimimennsoonal latent spacetime
Ør) R and
propose model all
= (zz, t) E R x (0,
(12)
which define the family of all denoising distributions
{v(20) [x:)] across all noise levels (Fig. 1).
Why include time? The resulting Fisher-Rao metric GIG (z) varies with state and time, restoring nontrivial geometry and enabling navigation across noise levels within unified structure. Identifying clean data with spacetime points (x, 0). for which p(æg Bg x) 8g. lets geodesics connect clean endpoints through noisy intermediates. This yields (i) principled notion of distance between data as the length of the shortest spacetime path (Diffusion Edit Distance), and (ii) mechanism for Pransition-pati sampling via spacetime geodesics; both are demonstrated empirically in Section 6.
Tractable energy estimation. Usually, the informattion-eoomeriii energy of discretized curve Y = {zn ( is approximated via the local-KL approximation (Arvanitidis et al., 2022):
N-2
E(r) = (N - 1) Σ KL pl- 2n)) || zn+1)] ==0
(13)
but such KLs are generally intractable, unless p(-|z) is a simple analytic distribution such as multino- mial or Gaussian, which not the case for denoising distributions p( ((og)z,). Nonetheless, we show that in the specific case of the diffusion spacetime, the energy can be tractably estimated.
Proposition 5.1 (Spacetime energy estimation informal). The energy of discretized spacetime curve y (2n) JN with Zn (It,, fn) admits approximation
E(7)= N N-
(((2nt))m7(zm)) (M(zntz) u(zn)):
Σ
(14)
where
at st,
at p(xrtt) = [B[20 [t]]
(xx,, t)
1|
(15)
20
The proof (Appendix C) consists of showing that denoising distributions form an exponential family, which admits a simplified energy formula. In practice, we calculate pa t) with Tweedie's formula over the approximate denoiser (x;;) (See Appendix C.2 for details),
E
(xx))
of div to(xt).
(16)
[ ||
x
ai
where both 2. and div 2. computed efficiently via Hutchinson's 's trick (Hutchinson, 1989; Grath- wohl et al., 2019), enabling the esimation of p(z; t) with a single Jacobian- vector product (JVP).
Spacetime geodesics are Limulation-freee the energy calculation requires only N JVPs of the denoiser d for a curve discretized into N points.
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PV -DD
Snacetime Goolnic
No inages 3o
The sice leage, RENUN) Goodenii
1
Figure 3: PF-ODE paths are similar to energy- geodesics Left: Geodesics move in straighter lines than PF-ODE trajectories in 1D toy density. Right: Geodesics are almost indistin- guishable to PF-ODE sampling in ImageNet-512 EDM2 model.
Figure 4: Spacetime geodesics between images. Each row shows a geodesic between clean images. The path passes through noisy states and then denoises, realizing the minimal total edit between endpoints. Its length {(>) is the Diffusion Edit Distance (DiffED), which measures how much the denoising distribution changes along the optimal traversal.
6 EXPERIMENTS
6.1 SAMPLING TRAJECTORIES
We compare the trajectories obtained by solving the PF-ODE (o(x)) (Eq. 3) with geodesics between the same endpoints 2n, æT. For toy example of ID mixture of Gaussians, we observe the geodesics curving less than the PF-ODE trajectories the early sampling (high 1), while being indistinguishable for lower values of t (See Fig. left and Appendix G.1 for details).
We find only marginal perceptual difference between the PF-ODE sampling trajectories and the geodesics in the EDM2 ImageNet-512 model (Karras et al., 2024). The geodesic appears generate information slightly earlier, but the difference is minor (See Fig. right, and Appendix G.2 for details).
We note that spacetime geodesics the endpoints beforehand. An investigation into whether our framework can used to improve sampling strategies is interesting future research direction.
alternative sampling method since they require knowing
6.2 DIFFUSION EDIT DISTANCE
The spacetime geometry yields principled distance the data space. We identify clean datum x E Rdd with the spacetime point (x. 0), corresponding to the Dirac denoising distribution 8 Given two points a vb, define the Diffusion Edit Distance (DiffED) by
DiffED( a²)) {(>))
(17)
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where Y is the spacetime geodesic between (x" 0) and (a", 0). For numerical stability, anchor endpoints at a small fmin rather than at O. See Algorithm 4 for DiffE pseudocode.
A spacetime geodesic links two clean data points through intermediate noisy states. It can be interpreted as the minimal sequence of edits: add just enough noise to discard information specific to æ". then remove noise introduce information specific to x The path length is the total edit cost, which is measured by how much the denoising distribution changes along the path. Fig. 4 visualizes the spacetime geodesics: endpoint similarity decreases, the intermediate points become noisier.
We quantitatively evaluate DilTED on image data. First, we ask whether DifED correlates with human perception as approximated by Learned Perceptual Image Patch Similarity (LPIPS) (Zhang et al., 2018). We randomly selected 10 classes in the ImageNet dataset and sampled 20 random image pairs for each. We then evaluated the DiffED and LPIPS for each image pair, and found the correlation to be very low at approximated -7%, suggesting that perceptual similarity and geometric edit cost capture different notions of closeness. We found DiffED to more closely related the structural similarity index measure (SSIM) (Wang et al., 2004), which correlates at 53% with DifIED,
To qualitatively compare different notions of image similarity, we order image pairs by their similarity evaluated with multiple metries: DiffED, LPIPS, SSIM, and Euclidean. We show the results in Fig. 8.
6.3 TRANSITION PATH SAMPLING
Boltzmann distribution
Spacetime geodesic
Transition paths
low energy states
90
80
70
60
50
X
40
30
10
Figure 5: Spacetime geodesics enable sampling transition paths between low-energy states. Left: Alanine Dipeptide energy landscape wrt two dihedral angles, with two energy minima zy a Middle: Spacetime geodesic 7 connecting zg and zg. Right: Annealed Langevin transition path samples.
Another application of the spacetime geometry is the problem of transition-path sampling (Holdijk et al., 2023; Du et al., 2024; Raja et al., 2025), whose goal is to find probable transition paths between low-energy states. We assume Boltzmann distribution
q(x) xx exp(-U(x:),
(18)
where U is known energy function, which is common assumption in molecular dynamics. In this setting, the denoising distribution follows tractable energy function (See Eq. 60)
p(xoo<x)) q(zo)) ((x,))((a,)))) (xo) exp -U(2x) (NN(()) xo z2/aae| 1
(19)
U (mtalmm))
To construct transition path between two low-energy states 2n} and ax estimate the spacetime geodesic 7 between them using a denoiser model tn (o.x) == E[zo|22 [2:] with Proposition 5.1, as shown Fig. 5, At each interpolation point [0, 1], the geodesic defines denoising Boltzmann distribu- tion p(#) where U (x 7s) is the energy at that spacetime location. See Appendix G.3 for details.
Annealed Langevin Dynamies.
To sample transition paths, we use Langevin dynamics
dæ V=U((>)))) /2dW,.
(20)
whose stationary distributions are p( 7x) x (xp(-U(xlys)) for any s. To obtain the trajectories from an to a² we gradually increase from ( to using annealed Langevin (Song & Ermon, 2019).

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MCMC variable length
MCMC fixed length
Doob's lagrangian
Spacetime geodesic + ALD
Figure 6: Transition paths generated with a spacetime geodesic avoid high-energy regions without collapsing to single path. Compared with MCMC baselines, the spacetime-geodest method yields transition paths that better avoid high-energy areas, whereas Doob's Lagrangian collapses to generating nearly identical trajectories. Ten sample paths are shown for each method.
After discretizing the geodesic into N points Yus we altemnate between taking K steps of Eq. 20 conditioned on 7nn and updating "/n - nn+1, deseribed in Algorithm 1. This approach assumes that ra) is close to p(=)) and thus a ~ pa)) is good starting point to Langevin dynamics conditioned on Ya+11
Table 1: Spacetime geodesics outperform methods tailored transition path sampling. Paren- theses denote extra energy evaluations used to generate training data for the base diffusion model, which do not scale with the number generated paths. Baseline details in Appendix H.
MaxEnergy (L)
# Evaluations (1)
Lower Bound
36.42
N/A
MCMC-fixed-lent MCMC-vaariable-lennth
42.54 + 7.42
1.29B
58. + 18.51
21.02M
Doob's Lagrangian (Du et al., 2024)
66.24 # 1.01
38.4M
Spacetime geodesic (Ours)
37.36 + 0.60
16M (+16M)
Alanine dipeptide. We compute a spacetime geodesic connecting two molecular configurations of Alanine Dipeptide, as in Holdijk et al. (2023). In Fig. 5, the energy landscape is visualized over the dihedral angle space, with a neural network used to approximate the potential energy U. Using our trained denoiser (æ:)), we estimate the expectation parameter μa, which allows us to compute and visualize a geodesic trajectory through spacetime. Transition paths were generated using Algorithm 1. See Appendix G.3 for details.
Baselines. We considered Holdijk et al. (2023); Du et al. (2024); Raja et al. (2025) and adopt Doob's Lagrangian (Du et al., 2024); the others were excluded due to reproducibility issues (sce Appendix H). We also evaluate two MCMC two-way shooting variants (Brotzakis & Bolhuis, 2016)- uniform point selection with variable or fixed trajectory length-using transition paths from the official Du et al. (2024) code release. For each method generate 1,000 paths and report mean Max MaxEnergy (lower is better) and its numerical lower bound min. max, Urr), along with the number of energy evaluations needed for 1,000 paths. To train base diffusion model for method, we generated data using Langevin dynamics (16M) energy evaluations), a one-time cost that does not scale with the number of generated transition paths.
Results. We show in Table that our method outperforms the baselines in the MaxEnergy obtained along the transition paths. It is also considerably closer to the lower bound than to the next best baseline (MCMC-fixed length) while requiring several orders of magnitude fewer energy function evaluations. In Fig. 6, we show a qualitative comparison of transition paths generated with our method and the basclines. Our proposed method shows improved efficiency in avoiding high-energy
4 16M is the number of energy function evaluations to generate the training set with Langevin dynamics for the hase di diffusion model. We did not tune this number, and fewer evaluations may yield comparable performance.
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regions compared to MCMC. In contrast, the Doob's Lagrangian method converged to suboptimal solution, producing nearly identical transition paths. We discuss this more detail in Appendix H.
Algorithm Transition Path Sampling with Annealed Langevin Dynamics
Require: Raa 2y RD. endpoints, Ny 0, T 0, fminn dt & Approximate spacetime geodesic with Algorithm Initialize chain 7 æe Iterate over the points on the geodesic "/2
1: 7 f SPACETMECEODESIC((;: 2%b)
2: T < {x = {a.
3: for n E (O....,NN
N₂ 1} do
4:
fort (l.....7 do
5:
E~ (0,
> Sample Gaussian noise
6:
x VaU(<)l)))d + v2dte Tu {œ}
D Langevin update Append state æ to chain
7:
8:
end for
9:
end for
10: return T
>
Return chain
6.4 CONSTRAINED PATH SAMPLING
Suppose we would like to impose additional constraints along the geodesic interpolants. This corresponds to penalized optimization (Rygaard al. (2025) also explore regularized geodesies)
min {(>) +A/ h(rr)ds,, s.t. Yo (x)) 0), 7/ -x. (z]:0)
(21)
where h : x RD + R is some penalty function with > O. We demonstrate the principle by (i) penalizing transition path variance, and (ii) imposing regions to avoid in the data space..
Low-variance transitions. Suppose we want the posterior p( Y.) to have a low variance. This concentrates the path around narrower set plausible states, more repeatable trajectories, albeit at the cost of reduced coverage. By Eq. 56, higher SNR(t) yields lower denoising variance, so we implement this by penalizing low SNR via (xx L) max( log SNR((), p) for some threshold p.
Avoiding restricted regions. Suppose we want to avoid certain regions in the data space in the transition paths. We encode the region avoid as denoising distribution p(-12") for some z" = (=) t") where larger the t", larger the restricted region. We encode the penalty as KL distance between the denoising distributions (See Appendix D for the derivation)
f(( (á (m(T))) mx(z))) du +
KL p(-12 (p(()))))]
(22)
1(7)) = min (p. - KI. ((T.(((1=))ll(
(23)
In Fig. 7, we compare spacetime geodesics (unconstrained) with low-variance, and region-avoiding spacetime curves. We visualize both the curves and the corresponding transition paths generated with Algorithm 1. This demonstrates that framework with the penalized optimization (Eq. 21) can incorporate various preferences on the transition paths.
RELATED WORKS
We review three directions of research related to ours: (i) studies of latent noise in diffusion models, (ii) applications of information geometry in generative modeling, and (iii) geometric formulations for sampling efficiency.
Latent-data geometry. Several works analyze the relation between latent noise x; and data Zg. Yu al. (2025) define geodesic density in diffusion latent space; Park et al. (2023) apply Riemannian geometry to lover-dimensional latent codes; Karczewski et al. (2025) study how noise scaling affects log-densities and perceptual detail. Our work also investigates the to Tu relationship but (a) uses the Fisher-Ras Rao metric rather than an inverse-densite metric, (b) retains the full-dimensional latent space without projection, and (c) analyzes the complete diffusion path across all timesteps.
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Optimized spacetime curves
Sample transition paths
SNR(()
Unconstrained
owl vaniacce
Region-avoiding
Unconstrained
Low
Reg on ovoiding
Resincied region
Figure 7: Vanilla transition paths can be constrained to have lower variance, or successfully avoid restricted region p(-(2"). Left: geodesics 7. Right: transition paths T.
Information geometry in generative models. Lobashev et al. (2025) introduce the Fisher-Raa metric on families p( () to study phase-like transitions, where is a cow-dimensional variable parametrizing a microstate In contrast, we place the geometry on diffusion 's explicit spacetime coordinates 2 = (ax,, t), induced by the denoising posterior p(ato xx)..
Geometric approaches to sampling. Two recent works also formulate diffusion models geomet- rically to improve sampling efficiency. Das et al. (2023) optimize the forward noising process by following the shortest geodesic between Po and Pt under the Fisher-Rao metric, assuming Po(2)) to be Gaussian. Ghimire et al. (2023) model both the forward and reverse processes as Wasserstein gradient flows. Our contribution differs: on the reverse process (not the forward), and only require to admit a density.
information geometry (not optimal transport), focus
8 LIMITATIONS
Although our framework defines geodesics between any noisy samples, optimizing between nearly clean ones numerically unstable because their denoising distributions collapse Dirac deltas, making Fisher-Rao (via local KL) distances effectively infinite. Therefore, consistent with diffusion practice (Song et al., 2021; :; Lu et al., 2022), we choose endpoints with non-negligible noise for tractable optimization (details in Appendix G).
The proposed distance metric DiffED (Section 6.2) considerably slower (details Appendix G.2) than established image similarity metrics such as LPIPS (Zhang et al., 2018), or SSIM (Wang et al., 2004). Exploring a distillation strategy involving training a separate model trained to predict DiffED possible future research direction.
CONCLUSION
We proposed novel perspective the latent space of diffusion models by viewing it (D + 1)- dimensional statistical manifold, with the Fisher-Rao metric inducing a geometrical structure. By leveraging the fact that the denoising distributions form exponential family, showed that can tractably estimate geodesics even for high- nigh-dimensional image diffusion models. We visualized our methods for image interpolations and demonstrated their utility in molecular transition path sampling.
This work deepens our understanding of the latent space in diffusion models and has the potential to inspire further research, including the development of novel applications of the spacetime geometric framework, such as enhanced sampling techniques.
REPRODUCIBILITY STATEMENT
We include the source code our submission, which allows for reproducing the results. Our claims made in the main text are proven in the appendices. Experiment details can be found in Appendix G.
ETHICS STATEMENT
The use generative models, especially those capable of producing images and videos, poses considerable risks for misuse. Such technologies have the potential to produce harmful societal

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effects, primarily through the spread of disinformation, but also by reinforcing harmful stereotypes and implicit biases. In this work, we contribute to a deeper understanding of diffusion models, which currently represent the leading methodology in generative modeling. While this insight may eventually support improvements to these models, thereby increasing the risk of misuse, it is important note that research does not introduce any new ethical risks beyond those already associated with generative AL.
We have used Large Language Models polish writing on sentence level.
ACKNOWLEDGMENTS
This work was supported by the Finnish Center for Artificial Intelligence (FCAI) under Flagship R5 (award 1501 1052). SH was supported by research grants from VILLUM FONDEN (42062), the Novo Nordisk Foundation through the Center for Basic Research in Life Science (NNF200C00666066) and the European Research Council (ERC) under the European Union's 's Horizon Programme (grant agreement 101 125003). VG acknowledges the support from Saab- WASP (grant 41 1025), Academy of Finland (grant 342077), and the Jane and Aatos Erkko Foundation (grant 7001703).
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A NOTATION
point in D-dimensional Euclidean space (a column vector), We denote x = (x"....,,"*" x D RDp
Tr(A) = E, Au - the trace operator of a square matrix A R
For scalar function f: PRD Differential operators.
R. f(z) RR we denote
0f Or
0f
gradient:
[=)( =
E RR.
Dr
j
Hessian: V=f(T)
E RR Dx
Ozi
x=
Σ 8 Ar²)²
Laplacian:
fz. Tr (V)(())
R.
==
For a curve y [0, + k + E k we denote
time derivative: 7s- = d ds 7s E R
For vector valued function / R Rm + (a) ....., (x)) E R we denote
0f(ā)
Offi
Jacobian:
m
dx
Oxi
by l==
When & m, we define
(af(i))
k Σ
05
divergence:
f = Tr
E
IR
dx
Or
i=1
==
Functions with two arguments. For f: RR x R2 + R, (I11 x22) + f(01, F2) E R we define (analogously w.r.t. second argument)
( as os
gradient w.r.t. first argument:
Va, f x222
E R
Ox
For f: R x Ri + R (æ1, + (f) x2), ..... pm (x) 22
E 1 we define (analo-
gously w.r.t. second argument)
0f )f(
0f
Jacobian w.r.t. first argument:
E ggmnk
dz
dr!
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B PULLBACK GEOMETRY IN DIFFUSION MODELS
Lemma B.1. Let be a latent space, = TRd a data space, and f: 2 - X a decoder. Then the length and energy of curve Y [0, + X under the pullback geometry given by
(pb(T) { |1 f(y)))) ds
(24)
EpB(7) = 2 / f(x))))l ds,
(25)
where ||
the Euclidean norm.
Proof. For a general Riemannian metric G, the length, and energy are given by
la(>) !.
l
(26)
Ea(7) = 2///" |HallEds,
(27)
where 1 =4 G(ys)ts. For G Gpn induced by the decoder f, we have
05 (z)
a
GpB(z) =
8z (z)
(28)
Oz
which leads
(sS 0z (76)
) (7s)
T (0s
4/- -05
T (76)9s)
)
(
11 =4
Oz
0z
(29)
d ((ts)
d ((ro) = [1ddf(7>))1
where (4) follows from the chain rule.
Proposition B.1 (Pullback geodesies decode straight lines). Let be latent space, X = Rd
data space, and f: Z + X bijective decoder. Fix z" ,²,=² zd E Z and write xx = f(z"), x² = f(z"). Then any shortest path between z" and z" in the pullback geometry decodes the straight segment from x" to xh.
Proof. Let Wg = (1- x + ssx for s E [0, 1]. Because f is bijective, define its latent preimage
7 - (xx),
E [0, 1].
The pullback length of a latent curve is the Euclidean length of its image (Lemma B.1):
(PB(T) [-1 5(7))) 1(7))) ds.
For Yo [();) = v. has constant velocity d;, = x²6 - x hence
(PB(r") S ||00 xz1l ds - x²1|
For any other smooth latent curve from =" to 2b, using the triangle inequality:
(PB(T) [' s(7.) ds > 1 dh f(x ds -|==='==((>))) = 1|0"~>011 =011=1, (r*).
Therefore (Pa(>) 2 (rB(Y*). Hence, any pullhack geodesic decodes to the straight segment:
f(xt) = f(f-((x.)) = (1 3) x= + 5
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In this proposition, emphasize that any minimizing path in the latent space 2, when measured with the pullback metric, will always decode to a straight line in the data space X. The reason is that the bijective decoder acts only on the ambient coordinates of X' = R regardless of whether the actual data lie on a lower-dimensional submanifold. In the denoising diffusion setting, this situation unavoidable, since the model enforces dim( ( dim(X). This stands in contrast with prior works using variational autoencoders (Arvanitidis et al., 2018), where latent geodesics live in wwer-dimensiioal space and can decode to curved trajectories in the ambient space.
Unless the dimension of the latent space is reduced to the intrinsic dimension of the data, the pullback metric carries no meaningful geometric information the standard denoising diffusion setting, where dim 2) = (X'.
C PROOF OF PROPOSITION 5.1
In this section, we prove Proposition 5.1. The proof consits of
1. Showing that curve energy & in exponential families simplifies (Appendix C.I).
2. Showing that the family of denoising distributions forms exponential family (Ap- pendix C.2).
3. Putting it together (Appendix C.4).
C.1 INFORMATION GEOMETRY IN EXPONENTIAL FAMILIES
We begin by defining an exponential family of distributions.
Definition C.1 (Exponential Family). A parametric family of probability distributions {p(-|z)} is called an exponential family if it can be expressed in the form
pxx)2) h(x) exp( n²2) ((x)))((z))
(30)
with x a random variable modelling the data and z the parameter of the distribution. In addition, T(x) is called a sufficient statistic, nz()) natural (canonical) parameter, v/(z) the log-partition (cumulant) function and h(x)) a base measure independent of z, and
mxz)= EE((<)) z]]
(31)
the expectation parameter.
In exponential families, the Riemannian metric tensor takes specific form, which show now. Propasition C.1 (Fisher-Rac Rao metric for an exponential family). Let {p(-|z)} be an exponential family. We denote TT2) the natural parametrisation, T(x) the sufficient statistic and μ(z) E[T(x)) 2] the expectation parameters. The Fisher-Rao metric is given by:
0m(())
@u((z)
Gıa(z)
(32)
0z
Oz
Proof. For p(x|2) h(x:) exp (n(2)TT(<>) +1(2)), we have
V. Σ (=) (x) v=v(2) (Ōm(z))
V V= Vologp(a
- -V(()))
(33)
Oz
Note that
V logp(x/>) = /
-f p(x|2) logp(x/2)dæ = / />p(a/2)dx== = V, / p(z)2))dx = 0.
E
(34)
Therefore, by taking the expectation of both sides of Eq. 33, we get
x=) = @m(2) T
1122),
(35)
Oz
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where (=)) [x) we get
Now we differentiate j-th component of both sides of Eq. 34 want 2", and
0
0
i log (x))())
2 Olog p(x]))
0 =
d2
0
0²
E
02
2
=
dzi
p(zlzz)
Ozd
dx
Öp(z x) a log p(x|z)
@²
log p( ( x))
dx
p (()z))
dx
(36)
Oz
Ozd
Oz'O21
① log p(a 2) θ log p(x
O8 log p( z)
E
Oz
02)
2
+
E
Oz'
Therefore
log p(
loog p(z]22
00 log p(a: x
(z)l = E
Oz
Ozd
= -E
Ozt
2
(37)
Now using Eq. 33, we have
0 [logp(x(>)
(z)
0b((z)
Oij (z)
S+((z))
Σ
T (xx)
T (xz)
(38)
0z'
Oz
O2i
Oz'
Oz'
Therefore, from Eq. 37:
GGG((z)] {z)((z)
(1((2))
Ozi 11 (2).
(39)
02
Now using Eq. 35, we have
{))())
(2)
= Σ (1)x(2)
01)(2)
02)
Ozi Σ
(z)
(z) + Σ
(40)
Oz
Oz
Ozi
Ozt
Ozd
k
b
Combining (Eq. 39) with (Eq. 40) yields:
(z)
0p
( 0z
[0n(z)
(00(zz)
[p((z)
[Gia(z)li
Σ
(41)
Ozl Oz
Oz
Corollary C.1 (Energy function for an exponential family). Let y 10, 1] + 2 be smooth curve in the parameter space Z of an exponential Then
EiG(7) f'' ( (()()))) (d. ds
(42)
GG(T) (/² (()((T))) (d.u())))
(43)
For a curve discretized uniformly into N points, we have 8n = N Yn = (8n). and pm =
p(7x), To. := (7()),, we have
N-2
N
EIG(7)
Σ nn++1 nam (Mut1 mm)
(44)
2
N-2
(i((7) 2
Σ
V (Ma+1 - nm) (fen+1 - phn)
(45)
0=
Proof. The energy of y is given by Eio(7) = fo 117, ds. We replace the Riemannian metric
GIG with the previously obtained expression of the Fisher-Rao metric (Fq. 32).
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Using Eq. (32)), we have
Hellera =1 G1G(T))9. = =/g- (on((z))
Oz
Oz
- (Op(Ts))
0z 98)
m (2cH()))))
0z
Therefore
EiG(7) = 2/ / | ds = 2 //' (mm(()))) ((H()))) ds
2
(I((7) = I. 1 1/ J. (())))) (2)
For discretized curve, the reasoning is similar to the proof of Proposition A.2 in Arvanitidis et al. (2022). We have ds = Snt1 8n N-I and we can approximate de (7>))) 2 Mot1- and de
4
d
()(o))))
Ph Mn
and
de
de
N-2
EiG(Y)
RM
M+ Un
Mn++
pn
ds
N
7o+1 - mm) (pat1 - pa)
ds
ds
n=0
2-2
N-2
(10(7) Rd
Mh+1 nm
Mn++
Mn
ds
Σ
V (M nn) (ku+11 pm))
ds
ds
C.2 DIFFUSION DENOISING DISTRIBUTIONS ARE EXPONENTIAL
A key observation is that the family of denoising distributions p(zolaz;) indexed by both space and time (a,, t) is exponential, which we prove now.
Proposition C.2 (Exponential family of denoising). Let an be noisy observation corresponding to diffusion time as introduced in Eq. 1. Then the denoising distribution be written
p(zu xx) = h(xo) exp (n(x,, T(xo) - ((<x;,t)
(46)
with h q the data distribution density, vp the log-partition function, and
oz
n)(x,,t) = o xit
(natural parameter)
(47)
20
T(wo)) = =(xo, || 1|
(sufficient statistic)
(48)
p( 1) F [ [z:]]
of
diva, BBaeo [ac]] + || [2:) 2
(49)
O
'space'
'time' B ||:00|| [mm]
which means that the family of denoising distributions {p(xo xx)) indexed by (æe, t) is exponential (Definition C.I).
Proof. Step 1: denoising is exponential. The denoising distribution is given by
p(zolxx) p(z))(za)g(z))
(x()))
where q is the data distribution, p(æ, (x)) (Nx of of is the forward density (Eq. 1), and pe(x,) = p(x; [x (20)q(20) dxy is the marginal distribution at time Therefore
[ 04,00
p( (ra) = (2m exp
20
|| ||
a4
a
D/1 exp
(50)
20
20
= exp
(-2
D
log( (2To²)
[
exp
a
12
+
ai
E2
20
20
20
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By substituting into the denoising density, get
p(zo)æ+)) = (xx)) exp {
aj
2 +
at
Tao 20 - (logp((a)) + log(2mo?) + ||xx112
D
} 20
2
= h(oo) exp (n(x,, t) (((x))
(xr, t)),
(51)
where
ai
(1x,,1) =
Dt,
E [R D+1
(52)
of
207
T T (zu, ||00o1
R D+1
(53)
M((o)) - q(v0))
(54)
D
|
( t) = log pt (xx) + log((no)) +
E R
(55)
2
20?
which proves that the denoising distributions form an exponential family.
Step deriving the expectation parameter μ.. The expectation parameter μa is given by p( ()
E (#)o)) mx]] (E[aro are), [| m.]). The denoising covariance is known (Meng al.,
2021):
of (I (I+ lo:))(z:)))
Cov 2o mx
(56)
of
Therefore, from the definition of conditional variance, we can deduce the second denoising moment:
[x]]] |
E
æt
E
xo Elwo
20=]
'Ir (Cov[æo xx))) + [B(200 [xe]
(56)
of
(D of A log px + [8(20o
ama]] ||
(57)
02 diva:
xt + of vx, x.
log Pr (x0e)
+ ||:00
xx]]
at
a4
of divan
E(2ra
Elaro ara]
ax.]
at
where we used the fact that (Efron, 2011)
De + of V log (xx))
(58)
Together, we have
o² div B|x0 xx] + ((xx,t)= [5[zo xz],
[[z
(59)
an
C.3 BOLTZMANN DENOISING DISTRIBUTIONS
Note that, if the data distribution is Boltzmann, i.e. q(æo) x exp(-U(Zo)) for some energy function U. have:
p( (2)e) x ((tt))((2)) x exp(-(U(2o)) exp [_1[z | - -2x0oll
20
exp (xu( |N 22)
This implies that p(zolzx;) also a Boltzmann distribution with p(zo xxe) x exx((UU(a) xx)) for
U(Eo (xx)) = (wo) + + {SNR()) -000 ar/ =00/00
(60)
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C.4 PUTTING TOGETHER: PROPOSITION 5.1
The claim of Proposition 5.1 follows from Proposition C.2 and Corollary C.I.
D KULLBACK-LEIBLER DIVERGENCE IN EXPONENTIAL FAMILIES
For any distribution family, the Fisher-Rao metric is the local approximation of the KI. divergence, ie (Arvanitidis et al., 2022):
2 (2) =22) GIG (21 -2)).
In the case of exponential families, we have GIG(z) =
(
[Oniz)
(H(z)
and thus we can write
8x
8x
(L(p(-I=))ll((1 ~ (z1 - 22) (0m((1)
(Op(zz)
(z1 - z2)
2
Oz
Oz
(n(21)
m(22)) (1(21) p(z2)).
turns that the RHS always corresponds notion of distribution divergence (not only when ø and 22 are close together), namely the symmetrized Kullback-Leibler divergence:
KL (xllg) = (KI((pllg) KT(glpp)).
(61)
Lemma D.1 (KL in exponential families). Let p {p(. 12) 2 E 2} be exponential family with p(z]z)) = h(xx) exp(n(=) T(x: - (=))), and sz(2) = Exep(a x) (x=). Then
<T(zll|z)) = (17(2) mz22) 12(z)) - (=11) + (=2)),
(62)
where we abuse notation and write (L(211|z2) instead of KL(p(- =p) x2)).
Proof.
(T.(21l/22) = Rewppl $1) (0)pp())22))) - log log:((2222)
Rxepp( $1) [n(21) T(x) n722) (m()) (22)) + (1(2))
= (m)(1) - n)22)) Em (2)) - ((21) + (222)
((1(2)) 17(22)) (221) ( (22))-
Lemma D.2 (Symmetrized KL in exponential families). With assumptions of Lemma D.I, we have
KL (zıllzz)) = (n(21) - m(Z2)) (f(21) - pl(22))
(63)
Proof.
2 2KL x2) (LL(21|12)) (L
=(7(x1) - (7(22)) T p(z1)= - Vo(TT + exta)+ + (n(22) n(x1)) p(zz) Axa) + 16(27)
(n7(2) ( (M(21) - p(zz)))
The formula for KL in Lemma D.1 is not useful in practice, because it requires knowing (2), which can be unknown or expensive to evaluate. However, the gradients with respect to both arguments depend only y and μ..
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Lemma D.3 (KL gradients). With assumptions Lemma D.I, we have for any 21, 22
0u(zz)
21, KL(21||z2)
(7(21) (((21)-((22)))
Oz
(64)
Ön(zz)
V KL((i||z>>
(H((22) (o(=1))
Oz
Proof. The proof is a straightforward calculation using Lemma D.I and Eq. 35. We have
v1 KL(zi||z2) vx, (17(z)) - (z22)) u(z1) (z1) (12))
1(11)) (m(21)
0p((21)
(m(z1) n)(2)) - (2(((2)
Oz
Oz
(35)
@m(z))
1((2)) + 01(21)
(m(z)) - m)22)) On(z1)
/el (1(21)
Oz
Oz
0z
@u(z))
(1(21) -1(22))
Oz
and
V 22
(T((2]|=2) = - ((n(z)) ()(2)) 11(21) - (122) + xb(22)
(35)
(2))
pl(z1) + @m(22)
Oz m(Z2) @m((2)
(H(22) - m(21))
Oz
8z
Knowing the gradients allows for estimating the KL divergence along curve without knowing tb. Proposition D.1 (KI. along curve). Let y : [0, 1] -> Z be a smooth denoising curve, and 2" 2.
Then:
KT(xx1l=*) KT(Toll=*)) + L. (
du ( (m1(Tw) m7(2))) du
(65)
KL(=*1l>> KL(=*1170) ( + ( (u(Tu) ul22)) du
Proof.
KT(Tall=*) <TT.>ro|l=)) l/¹ du
d (KL(Trul) )² du
// Fundamental theorem of calculus
/ V., (T((Tul|2") T dud
// Chain rule
(Qu(Ta) tm (m(()w) 17(x²) du
(zzz
Lemma D.3
Oz
/0/ (LuM(Y))) T (n(r))) (2²2)) du
Chain rule.
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Using the same reasoning have
KL - KI(==111o) = L." d
(KI(2"Ilxx)) du du //"0 2. KI(z*11Ta) T Yadu
''" (on(tui) ti.) (u(<)u) - [x(")) du
//- (dn(())) (u(TW)) (=(²))) du
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E GEODESIC AND DIFFED PSEUDOCODE
Algorithm Evaluate Curve
Require: Curve [0, 1]
IR D+11, number of points N,
{8n N₂ -1 +- linspace(O, 1, N,) 1 n=0
Uniformly discretized curve
2: for 0 to N₂ - do
3: In f- (sp))
4: Decompose 2n (a. (n) tn)
5: Th +- ,(( (n) using Eq. (15)
6: Mn + ulax In) using Eq. (16)
7: end for
8: return {nr}, {pm]
Algorithm Spacetime Geodesic Estimation
Require: Ta, 2t, E RR, N₂ > O discretization points
Require: Nier > 0, fmin > learning rate 1) >
1: <- (Ia; &min).
(2z, tmin)
Embed points into Spacetime
2: Initialize cubic spline Yeo with Yo (0) = Za. (1 = 2b
3: for k == 0 to Niir do
Optimization loop
4: {nm). [mn f- EVALUATECURVE((T0 N2.
5: E(7e) N.
(m mn) (Mh+1 pt)
Energy estimate Eq. (14)
6:
(770)
7: + 0 119
Gradient descent update of curve's parameters
8: end for
9: return Ya
Algorithm 4 Diffusion Edit Distance Estimation
Require: Ia Tb. E R.,,N Nr, > O, Nier O
Require: mmin > O, learning rate y) > 0
1: 7o < SPACETIMEGEODESSCCC (#., Lb, No, Nuers bminn 7)
D Use Algorithm
2: {7m) [ < EVALUATECCUVE((TO Ng,)
3: dDE(Fa: .36) + En (nn+1 mn) (pn+1 Mx)
b Length estimate Eq. (45)
4: return (DE 2.,)
F QUALITATIVE EXAMPLE OF DIFFED
In Fig. 8, we compare the Diffusion Edit Distance (DiffED) (Section 6.2) with LPIPS (Zhang et al., 2018), SSIM (Wang et al., 2004), and the Euclidean distance between images. Specifically, for ImageNet class "Space bar", generate 20 random image pairs, estimate the similarity of each pair with each method, and rank the pairs according to each method, from the most similar to the most dissimilar.
G EXPERIMENTAL DETAILS
G. TOY GAUSSIAN MIXTURE
For the experiments with ID Gaussian mixture (Fig. 1, and Fig. 3 left), we define the data
distribution Po 3 T,N(,, with μ1 225, #2 0.5. M4 2.5, π 0.275, n2 0.45, T3. = 0.275, and = 0.75. We specify the forward process (Eq. 1) as Sariance-Preserts (Song al., 2021), i.e. satisfying of of and assume log-SNR linear noise schedule,
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LPPS
SSIM
Figure 8: Comparison of DifED with other image similarity metries. Each row corresponds to different image similarity measure, and images and sorted by their similarity, from most similar (left) most dissimilar (right). Images shown 20 random image pairs from class "Space bar"..
i.c. log SNR(t) Amax + (Amin Amax)t for Amin -10, Amax 10. Which implies: at = ()g) sigmoid(-Xr).
Since Po is a Gaussian mixture, all marginals pt are also Gaussian mixtures, and training diffusion model is unnecessary, as the score function V log, (x) is known analytically. In this example, the data ID. and the spacetime is 2D.
To generate Fig. we estimate the geodesic between = (-2.3, 0.35), and و = 0.4) by parametrizing 7 with a cubic spline (Arvanitidis et al., 2022) with two nodes, and discretizing it into N = 128 points and taking 1000 optimization steps with Adam optimizer and learning rate 17 = 0.1. which takes a few seconds on an MI CPU.
To generate Fig. 3 left, we generate 3 PF-ODE sampling trajectories starting from 1, 0, -1 using an Fuler solver with 512 solver steps. We solve only until / mmin = 0.1 (as opposed to / = 0), because for t 0, the denoising distributions p(zo xx) become closer to Dirac delta distributions Se,, which makes the energies very large. For each sampling trajectory, we take the endpoints (z1, 1). (Ztmmn (min) and estimate the geodesic between them using Proposition 5.1 with cubic spline with 10 nodes, discretizing into 512 points, and taking 2000 gradient steps of AdamW optimizer with learning rate 17 = 0.01. This takes roughly 10 seconds on M1 CPU.
G.2 IMAGE DATA
For all experiments on image data, we use the pretrained EDM2 model trained on ImageNet5 12 (Karras et al., 2024) (specifically, the 1112--mmm522xx1--- Exploding model, i.c. a, 1, and using the noise schedule o, using a fixed Stability VAE (Rombach et al., 2022) as the encoder/decoder.
checkpoint), which is a Variance-
latent diffusion model,
Image interpolations. To interpolate between to images, we encode them with Stability VAE to obtain two latent codes 2/ 2 and encode them both with PF-ODE (Eq. 3) from t 0 to == Imin = 0.368, corresponding to log SNR(4min) 2. This is to avoid very high values of energy for Rz We then optimize the geodesic between (mr (mmn)) and ( by parametrizing with cubic spline with 8 nodes, and minimizing the energy defined in Proposition 5.1 using AdamW optimizer with learning rate 7 = 0. The curve is discretized into 16 points, and optimized for 200 gradient steps, which takes roughly 6 minutes on A100 NVIDIA GPU per interpolation image pair.
Note that in our experiments, used the largest release model 1e22-imm512--0 image interpolation time can be reduced to roughly a minute by considering the smallest model version edm2- 1dm2-img511 mgg512 -
The
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the 2nd order Heun solver (Karras et al., 2022) with 64 steps, and solve from = 80 to Emin = 0.135 corresponding to log (tmi) with the entire sampling trajectory "Yt = (#t, 1) for = ......,,iin where the schedule corresponds to EDM2 model's s sampling schedule.
PF-ODE sampling trajectories. To generate PF-ODE sampling trajectories,
4. This is to avoid instabilities for small t. We parametrize the geodesic directly
We then fix the endpoints of the trajectory, and optimize the intermediate points using Adam optimizer with learning rate 7/ = 0.0001 (larger learning rates lead to NaN values) and take 600 optimization steps. This procedure took roughly 2 hours on an A100 NVIDIA GPU per a single sampling trajectory.
To visualize intermediate noisy images at diffusion time t, we rescale them with Adata before
Vo to?
decoding with the VAE deocoder, to avoid unrealistic color values, where we set data 0.5 in Karras et al. (2022).
G.3 MOLECULAR DATA
Approximating the base energy function with neural network. We follow Holdijk al. (2023) and represent the energy function of Alanine Dipeptide in the space of two dihedral angles é, dh [-T,,), R). We use the code provided by the authors at github com/LarsiBl which estimates the energy U(Ø, v). However, even though the values of the energy U looked reasonably, we found that the provided implementation of BU Be Op
jk/
SoCTransitionPate
and a0
yielded
unstable results due to discontinuities.
Instead, we trained an auxiliary feedforward neural network Up approximate U. We parametrized with two hidden layers of size 64 with SiLU activation functions, and trained it on uniformly discretized grid [-T, x] [- π into 16384 points. We trained the model with mean squared error for 8192 steps using Adam optimizer with learning rate 0.001 until the model converged to an average loss of 1.5. This took approximately two and a half minutes on an MI CPU. In the subsequent experiments, we estimate (xx) with automatic differentiation on the trained auxiliary model.
Generating samples from the energy landscape. To generate samples from the data distribution Po()o) x exp(-U((x)),, we initialize the samples uniformly on the [- π x [-T,, π] grid, and use Langevin dynamies
dæ V.U(() + V2dW, with the Euler-Maruyama solver for dt 0.001 and N = 1000 steps.
(66)
Training a diffusion model on the energy landscape. estimate the spacetime geodesies, we need a denoiser network approximating the denoising mean 20(0,, t) = E[ We parametrize the denoiser network with
from ddpm import MLP
model = MLP {
hidden_layers=-
emb_size=128, input_emb==siinsson
using the TinyDiffusion implementation github the model using the weighted denoising loss: httpa///.ooo/eeeeeeoeoooooooeaae
We trained zull2 with weight function w(A)) / (A, + 2) and an adaptive noise schedule (Kingma & Gao, 2023). We train the model for 4000 steps using the AdamW optimizer with learning rate 77 = 0.001, which took roughly I minute on an MI CPU,
Spacetime geodesics. With a trained denoiser 2,(2,, t). we can estimate the expectation parameter fe (Eq. 16) and thus curves energies in the spacetime geometry (Proposition 5.1).
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In Section 6.3, want to interpolate between two low-energy states: z (-2.55,2.7 and z = (0.95, -0.4). To avoid instabilities for = O, we represent them on the spacetime manifold as 21 (-2.55, fmin), and 22 (0.95, -0.4, fmin), where log (fR(f = 4. We then approximate the geodesie between them by parametrizing 7 as a cubic spline with 10 nodes and fixed endpoints Tho 21, and 7 22 and discretize it into 128 points. We then optimize it by minimizing Proposition 5.1 with the Adam optimizer with learning rate 0.1 and take 10000 optimization steps, which takes roughly 4 minutes on an MI CPU.
Annealed Langevin dynamics. To generate transition paths, we use Annealed Langevin dynamics (Algorithm 1) with the geodesic discretized into N 128 points, K 128 Langevin steps for each point on the geodesic y, and use dt 0.0004, i.e., requiring 16385 evaluations of the gradient of the auxiliary energy function per path. Generating 1000 independent paths in parallel takes 32 seconds on an MI CPU and requires a total of 16,385,000 energy function evaluations.
Constrained transition paths. Constrained transition paths were also parametrized with cubic splines with 10 nodes, but discretized into 1024 points.
For the low-variance transition paths, we chose the threshold o = 3, and = 0 for the first 1200 optimization steps, and À linearly increasing from 0 to 100 for the last 3800 optimization steps, for the total of 5000 optimization steps with the Adam optimizer with a learning 1 = 0.01. This took just under 6 minutes on an MI CPU.
For the aeviiinavoiiing transition paths, we encode the restricted region with =' = (-0.8.
with log SNR((") 4, and combine two penalty functions: h1 is the low-variance penalty described above, but with P1 3.75 threshold, and h2 is the KL penalty with P2 = -4350 threshold. We define a as in the low-variance transitions, and fix X2 1. The optimization was performed with Adam optimizer, learning rate 1) = 0.1, and ran for 4000 steps for runtime of just under 5 minutes on an MI CPU.
The reason we include the low-variance penalty in the region- avoiding experiment is because (L(p(+Iz*) Il p(-1xs)) can trivially be increased by simply increasing entropy of p(-1%s) which would not result in avoiding the region defined by p(-)2").
H NOTE ON TRANSITION PATH SAMPLING BASELINES
For transition path experiments performed in Section 6.3, we considered Holdijk et al. (2023); Du et al. (2024); Raja et al. (2025) as baselines. However, we encountered reproducibility issues. Specifically
Holdijk et al. (2023) released the implementation: github, com/LarsHolle Idii jk/ sOCTransit onn ss. However, does not appear be supported. Several issues in the repository highlight failures to reproduce results, which have remained unresolved for more than year.
Raja et al. (2025) released the implementation: github. com/ Bae///KK-BBeeeeeee OM-TPS. However, it does not contain the code for the alanine dipeptide experiments, and the authors did respond request to release it.
* Du et al. (2024) released the implementation: that we were able to use. However, we obtained results signif-
github. com/plal plla.... iin rman/
Variati Marrattional- icantly worse than those reported in the original publication. We have contacted the authors, who acknowledged our question but did not provide guidance on how to resolve the issue.
For Doob's Lagrangian (Du et al., 2024), experimented with: different numbers of epochs, different numbers of Gaussians, first vs second order ODE, MLP vs spline, and internal vs external coordinates. We reported the results of the configuration that was the best. Many configurations either diverged completely (returned NeN values) or collapsed to completely straight transition paths, oblivious the underlying energy landscape. These issues persisted even after switching double precision (as advised in the official code repository).
I EXPECTATION PARAMETER ESTIMATION CODE
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import jax
import rax.om as jr
3 Import jax.n as jnp
def (x,, key): Implemenation of the expected denoising pass
def sigma_and anpha # Depands on the choice of SDE and noise schedule pass
11 def mu (x, t, key):
12
model_key, eps_key jr.sp (key, 2) eps = rr.rademacherrere (eps key, (x.size,), dtype=jap. f2) def fred_fn(x
15
return fxx,, t, key=nodel key)
16
f_pred, f_grad jax jvp (pred_f, (x,), (cps,)) div = jnp. sup.sum(f..) sf_graaad eps)
17
sigma, alpha aa__ma__nd__aphaaaa return sigma=>2/alpi
19
div jnp. (f_pred 2), f_pred
Listing 1: JAX Implementation of estimation
J LICENCES
EDM2 model (Karras et al., 2024): Creative Commons BY-NC-SA 4.0 license
ImageNet dataset (Deng et al., 2009): Custom non-commercial license
SDVAE model (Rombach al., 2022): CreativeML Open RAIL++MM license
OpenM++ (OpenMP Architecture Review Board, 2008): MIT License
25


Reviews:
[{"review_id": "SvndFg8AZM", "reviewer": "Reviewer_zLiS", "summary": "This paper revisits the geometric structure underlying diffusion models and argues that meaningful geometry does not reside in the noisy latent space x_t alone, but rather in the full spacetime domain formed by pairs (x_t, t) combining state and noise level. The authors show that when considering only the latent variable x_t at a fixed noise level, the denoising distribution p(x_0 \\mid x_t) becomes increasingly isotropic as t increases, causing the geometry (e.g., tangent and normal decomposition, pullback metrics) to collapse. To recover a non-trivial Riemannian structure, the paper proposes defining the metric using the Fisher–Rao information geometry over spacetime, resulting in the GIG metric that varies jointly with state and time.\n\nUsing this metric, the authors define the Diffusion Edit Distance (DiffED); the length of the shortest geodesic path in spacetime connecting two clean samples (x,0) and (y,0). This yields a principled notion of semantic distance that measures “how naturally one data point can be transformed into another” under the diffusion generative process. Moreover, the paper introduces a transition-path sampling method based on Annealed Langevin Dynamics (ALD) that follows the computed spacetime geodesic to generate actual interpolation trajectories. Empirical results in Section 6 demonstrate that these spacetime geodesics produce smooth, consistent, and semantically meaningful transformations between examples.", "questions": "1. Comparing image pairs with different visual attributes:\nFor two cases (i) two images with similar color tone but completely different content, and (ii) two images with identical structure/shape but very different color, which pair does the Diffusion Edit Distance assign a larger distance to? In other words, does the metric primarily reflect semantic structure, appearance, or a combination of both?\n\n1-1. Potential analytical uses of DiffED.\nSince Diffusion Edit Distance provides a meaningful notion of distance in the spacetime geometry, it seems possible to use it for analyzing model behavior (e.g., structure of learned manifolds, semantic neighborhood relationships, mode connectivity). Have the authors explored such analytical applications, or do they have ideas for experiments where DiffED could be used as a diagnostic tool?\n\n2. Extension to video data:\nDo the authors believe that the proposed spacetime geodesic framework can be extended to video sequences? If so, what would be a reasonable way to handle the temporal consistency constraint, for example, by treating time as an additional dimension alongside the diffusion timestep, or by defining geodesics directly in a trajectory space?\n\n3. Effect of different latent representations.\nThe behavior of DiffED may depend on the type of latent space in which the diffusion model operates (e.g., VAE latent space, pixel space, or non-image modalities). Have the authors observed different geometric patterns or performance characteristics when applying the method to models operating in these different latent representations?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. Clear and compelling reframing of the latent representation in diffusion models.\nWhile prior work typically analyzes individual x_t states or only the fully-noised x_T, this paper emphasizes that the entire trajectory \\{x_t\\} constitutes the latent representation. Defining geometry on the spacetime manifold (x_t, t) using the Fisher–Rao metric is both conceptually straightforward and surprisingly underexplored, making the contribution feel natural yet novel.\n2. Diffusion Edit Distance offers meaningful semantic comparisons.\nThe proposed DiffED provides a principled notion of distance based on how naturally one data point can transform into another through the diffusion process. Although currently computationally expensive, the definition itself seems robust and opens opportunities for future research in efficient approximations and downstream applications such as morphing, retrieval, and generative editing.\n3. Transition-path sampling via Annealed Langevin Dynamics adds practical value.\nThe use of ALD to follow the spacetime geodesic demonstrates that the metric is not only theoretically motivated but also operationally useful. The method enables the generation of smooth and interpretable interpolation paths, giving the framework concrete applicability rather than remaining purely abstract.", "weaknesses": "1. High computational cost.\nThe method requires computing geodesics in spacetime and then sampling along those paths with ALD. This is quite expensive in practice, and the paper does not propose any way to reduce this cost. As a result, it may be difficult to use the method in large-scale or time-sensitive settings.\n2. Limited variety of data types in experiments.\nThe experiments are mostly on datasets where the structure is relatively simple and smooth (e.g., faces, digits). It is unclear how well this approach works on more complex images (e.g., multiple objects, cluttered scenes) or on stylized data (e.g., cartoons, anime). Testing or discussing these cases would make the evaluation stronger.\n3. Unclear stability of geodesic optimization.\nThe final interpolation path comes from optimizing an energy function, which may have multiple local minima. The paper does not analyze how sensitive the result is to initialization or parameter choices. Because of this, it is not yet clear how stable or reproducible the method is.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4tiPaf7L1A", "reviewer": "Reviewer_ixrc", "summary": "The authors extend a geometric perspective on diffusion models by defining the Fisher–Rao metric not only over the latent space but across the latent spacetime $(z = (x_t, t))$. This structure allows them to define the length, or energy, of curves within the latent spacetime, and to define geodesics, or shortest path, over latents and timesteps. The authors show that geodesics from noise $(t = T)$ to denoised states $(t = 0)$ closely follow PF-ODE sampling trajectories. This also leads to a new measure of distance, the Diffusion Edit Distance, expanding upon known interpolation distances such as LPIPS.", "questions": "The paper could benefit from discussion on image editing approaches such as DDIM and Null-text inversion. \n\nThe authors mention that their method is numerically slower and suggest potential distillation. What are their preliminary thoughts on the architecture and training objective for a separate model intended to predict DiffED? Specifically, should the distillation target focus on replicating the absolute geodesic length, or rather on the manifold’s tangent structure defined by the Fisher–Rao metric, to ensure geometrically meaningful similarity estimation?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "The paper is clearly written, and the proposed approach is well presented.\n\nThe work introduces a novel and interesting notion of latent spacetime. While previous studies usually considered interpolation between samples as transitions within a slice of latent space fixed in timestep, this paper generalizes that perspective. It may have a broad impact on the image editing domain, effectively connecting geometric ideas with techniques such as DDIM inversion.", "weaknesses": "There are concerns regarding the computational complexity. Although the paper mentions that the method may be slower, it would be beneficial to include exact runtime comparisons and additional evaluations.\n\n\nThe authors demonstrate that PF-ODE sampling trajectories and geodesics are similar but do not provide any quantitative metrics. It would be interesting to see how this approach compares with LPIPS, for example by fixing the starting and final points to the same timestep.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LbYEVCUEXS", "reviewer": "Reviewer_r6Rp", "summary": "The paper proposes a new geometric framework for diffusion models based on information geometry, introducing a spacetime manifold that combines both the noisy sample and the diffusion time (x_t, t). The authors show that the conditional denoising distributions, $p(x_0|x_t)$, form an exponential family, allowing them to endow the diffusion process with a Fisher–Rao metric (a principled way to measure distances between nearby denoising distributions)\n\nUsing this metric, they define spacetime geodesics (shortest paths) between data points and introduce the Diffusion Edit Distance (DiffED), a new notion of distance induced by the geometry of the diffusion process. They compute these geodesics by discretizing the trajectory in spacetime and minimizing a Fisher–Rao energy functional using gradients derived from the trained denoiser.\n\nEmpirically, the paper illustrates the framework on image diffusion models and molecular simulation tasks, showing that the resulting geodesics correspond to meaningful transition paths and that the proposed distance can characterize structure in data manifolds.", "questions": "- What is the motivation for comparing to the ODE baseline? It is not clear what specific insight this comparison is intended to convey.\n\n- The pullback geodesic experiment seems somewhat like a strawman comparison—does it add significant insight? It may not warrant an entire page of the paper.\n\n- Is the time discretization used to compute the geodesic linear, or is it adapted in some way along the path?\n\n- How do the proposed results compare to stochastic interpolants? A direct comparison or discussion would be helpful for positioning this work relative to recent diffusion-based interpolation methods.\n\n- What theoretical or empirical guarantees can you provide that the Diffusion Edit Distance (DiffED) indeed corresponds to a true geodesic under your defined Fisher–Rao metric?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper addresses an interesting theoretical question: how to define a meaningful geometric structure for diffusion models and connects it to information geometry.\n\n- The idea of modeling denoising distributions as an exponential family and equipping the resulting spacetime with a Fisher–Rao metric is conceptually sound and mathematically motivated.\n\n- The second application (transition path sampling in molecular systems) is original and can lead to generating more research in that direction (with the caveat that I'm not very familiar with this particular area of applications of diffusion models)", "weaknesses": "- The paper does not clearly describe the algorithm used to compute the Diffusion Edit Distance (DiffED). Including pseudocode or a concise algorithmic summary would greatly improve clarity and reproducibility.\n\n- The computational cost of finding geodesics in spacetime using DiffED is unclear. Since Equation (16) must be evaluated at many noise levels, the method appears potentially expensive. How does its efficiency compare to related approaches? For instance, What’s Inside Your Diffusion Model? For example in [1]  the authors propose a similar framework for computing geodesics with diffusion models. Some discussion or comparison would help. This paper also contains some comparisons with other methods which the authors can use. \n\n*A Score-Based Riemannian Metric to Explore the Data Manifold, Azeglio & Di Bernardo, arXiv:2505.11128\n\n- The paper notes that the proposed distance correlates with SSIM but not with LPIPS. Could the authors elaborate on why this occurs, given that LPIPS is generally regarded as a stronger measure of perceptual or semantic similarity than SSIM?\n\nMinor comments: \n\n- Some terminology seems unnecessarily reinvented. For example, using “denoising” or “decoding distribution” instead of the standard term posterior may reduce clarity for readers familiar with Bayesian or diffusion-model terminology.\n\n- Correlations are reported as percentages in line 281. should these not be expressed as standard correlation coefficients?\n\n- The caption of Figure 1 describes the geodesic in spacetime as the “shortest path between two distributions.” Shouldn’t it instead be the shortest path between two images (or corresponding distributions over clean images)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Or7SQdRm6M", "reviewer": "Reviewer_XMm1", "summary": "This paper presents an information-geometric perspective on the latent space of diffusion models. The authors first show that the conventional pullback metric induced by the Probability Flow ODE (PF-ODE) is degenerate: geodesics under this metric always decode to straight lines in data space, thus failing to capture any intrinsic geometry. To address this limitation, the authors propose a new Fisher–Rao metric defined on the denoising distributions $p(x_0 | x_t)$. They further introduce a latent spacetime representation $z = (x_t, t)$, where both the sample and its diffusion timestep jointly parameterize the geometry. They prove that these denoising distributions form an exponential family, which enables a tractable estimator for geodesic energy and distance. Based on this framework, they define the Diffusion Edit Distance (DiffED) and demonstrate applications in image interpolation and transition path sampling for molecular systems, showing competitive performance with specialized baselines.", "questions": "- Could the proposed Diffusion Edit Distance be leveraged for practical downstream applications, such as by combining with SDEdit?\n- How would the transition path differ if one were to fix a specific diffusion timestep $t$ (as in Line 169) instead of employing the current spacetime modeling? What are the conceptual or empirical benefits of jointly modeling both space and time in the proposed spacetime representation? The current discussion in Lines 175–181 mainly explains how the framework enables spacetime geodesic modeling, but it does not clearly state why this joint modeling is beneficial or what additional insights it provides.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper provides a conceptually novel and rigorous framework.\n- This paper shows the inherent limitations of the pullback metric defined by the PF-ODE sampler from $x\\_{T}$ to $x\\_{0}$ and motivates the introduction of the information geometry using the denoising distribution $p(x\\_0 | x\\_{T})$.\n- The transition-path sampling experiments on the Alanine Dipeptide system demonstrate that the framework has potential beyond visualization or interpolation tasks.", "weaknesses": "- The claim that “memorylessness” causes $p(x\\_0 | x\\_T) \\approx q(x\\_0​)$ deserves clearer probabilistic justification. While the forward process is memoryless, it seems like this does not directly imply conditional independence in the reverse direction. Isn't it $p(x_0 | x_{T}) \\propto p(x_{T}) q(x_0)$?\n- Table 1 reports the number of energy evaluations but omits the actual trajectory computation time or runtime comparison with baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)