A Likely Geometry of Generative Models

Frederik Möbius Rygaard; Shen Zhu; Yinzhu Jin; Søren Hauberg; Tom Fletcher

cs.LGOct 30, 2025

A Likely Geometry of Generative Models

Authors: Frederik Möbius Rygaard, Shen Zhu, Yinzhu Jin, Søren Hauberg, Tom Fletcher

Organizations: DTU Compute, Technical University of Denmark, Kongens Lyngby, Denmark · Department of Computer Science, University of Virginia, Charlottesville (VA), USA · Department of Electrical and Computer Engineering, University of Virginia, Charlottesville (VA), USA

Abstract

The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data dimension. In this paper, we construct a general geometry compatible with different metrics and probability distributions to analyze generative models that do not require additional training. We consider curves analogous to geodesics constrained to a suitable data distribution aimed at targeting high-density regions learned by generative models. We formulate this as a (pseudo)-metric and prove that this corresponds to a Newtonian system on a Riemannian manifold. We show that shortest paths in our framework can be characterized by a system of ordinary differential equations, which locally corresponds to geodesics under a suitable Riemannian metric. Numerically, we derive a novel algorithm to efficiently compute shortest paths and generalized Fréchet means. Quantitatively, we show that curves using our metric traverse regions of higher density than baselines across a range of models and datasets.

Figures & tables

Figure 1: A conceptual illustration of our method, where we consider the geometry corresponding to a Newtonian system on a Riemannian manifold. The gradient vector field of the density “pushes” the Riemannian geodesic to areas of high likelihood.

Figure 2: Examples of the three fundamental concepts derived from the (pseudo-)metric in Eq. 7 with $\lambda=20.0$ : boundary-value curves (row 1), initial value curves (row 2), and a generalized version of the Fréchet mean (row 3). The concepts are applied to an energy-based model ( ebm ) ( LeCun et al., 2006 ) for a checkerboard, normalizing flow ( nf ) ( Papamakarios et al., 2021 ) for the two-moon distributions, autoregressive model ( ar ) ( Uria et al., 2016 ) for a sinus curve, and a vae ( Kingma and Welling, 2014 ) for circle data. Red denotes the connecting curves between the green boundary points. For the generalized Fréchet mean, blue denotes the mean, while the white points are $100$ sampled data points. The black arrows denote the initial directions of the curves. The regularization function for each model is described in Section 4 .

Figure 3: Interpolation curves computed for different values of $\lambda$ in Eq. 5 with an Euclidean background metric and $S({\bm{z}})=-\log p({\bm{z}})$ . The data density, $p$ , is approximated by a gmm and kde shown in the background color for synthetic data (black). The mean log-likelihoods of the estimated curves are denoted $\mathcal{L}$ , where the first number is for gmm and the second for kde .

Method	Complexity
Gradient descent	$\mathcal{O}\left(N_{\mathrm{grid}}d\right)$
Quasi-Newton method	$\mathcal{O}\left(N_{\mathrm{grid}}^{2}d^{2}\right)$
(Sparse) Newton method	$\mathcal{O}\left(N_{\mathrm{grid}}d^{3}\right)$
ProbGEORCE	$\mathcal{O}\left(N_{\mathrm{grid}}d^{3}\right)$

Table 1: Complexity of different optimization algorithms.

Figure 4: The Fréchet mean (blue) for the dino-dataset using a ddpm computed in data space as well as 10 curves (red) in ten different directions (green arrows) for the corresponding ivp ( 8 ).

Figure 5: The regularized energy and runtime for different methods minimizing Eq. 7 for

\lambda=1.0

for the

n

-sphere and

S=-\log p

, where

p

is the density of three Gaussians randomly weighted with random mean values and identity as the covariance matrix. Note that the regularized energy for ProbGEORCE with line-search (LS) and adaptive update scheme (Adaptive) are very similar.

	EBM		NF		AR		VAE
Method	NLL $\downarrow$	Runtime $\downarrow$	NLL $\downarrow$	Runtime $\downarrow$	NLL $\downarrow$	Runtime $\downarrow$	NLL $\downarrow$	Runtime $\downarrow$
IVP
Ours ( $\lambda=20.0$ )	8.483	4.90 $\pm$ 0.09	$1.20\times 10^{3}$	101.08 $\pm$ 2.17	$1.37\times 10^{4}$	7.66 $\pm$ 0.08	-144.673	64.01 $\pm$ 0.49
Euclidean	30.010	0.00 $\pm$ 0.00	$4.01\times 10^{3}$	0.00 $\pm$ 0.00	$1.06\times 10^{5}$	0.00 $\pm$ 0.00	-105.053	0.00 $\pm$ 0.00
Spherical	16.536	0.00 $\pm$ 0.00	$4.10\times 10^{3}$	0.00 $\pm$ 0.00	$8.25\times 10^{4}$	0.00 $\pm$ 0.00	-85.552	0.00 $\pm$ 0.00
Fisher-Rao	29.682	12.61 $\pm$ 0.09	-	-	-	-	-	-

Table 2: Negative log-likelihood (NLL) and runtime results for the energy-based model, normalizing flow, autoregressive model and variational autoencoder used in Fig. 2 . For the vae , we report the elbo . We write

-

if the Riemannian metric is not invertible or if the generative model returns NaNs for the likelihood. In Appendix E , we provide details on the benchmarks and any hyper-parameters.

Figure 6: The distance between grid points for the normalizing flow model in Fig. 2 for varying values of

\lambda

Method	FID $\downarrow$	KID $\downarrow$
${\bm{G}}_{1/p_{\theta}}$ (conformal)	187.25	0.0800 $\pm$ 0.0028
${\bm{G}}_{\log p_{\theta}}$ (conformal)	189.47	0.0819 $\pm$ 0.0033
${\bm{G}}_{1/p_{\theta}}$ (diagonal)	198.06	0.0828 $\pm$ 0.0032
${\bm{G}}_{\log p_{\theta}}$ (diagonal)	189.32	0.0745 $\pm$ 0.0031
${\bm{G}}_{\mathrm{LAND}}$	186.87	0.0726 $\pm$ 0.0028
${\bm{G}}_{1/p_{\theta}}$ (full)	179.62	0.0676 $\pm$ 0.0024

Table 3: Comparison of methods using FID and KID metrics (lower is better).

Method	FID $\downarrow$	KID $\downarrow$
Ours (Noise)	182.73	0.12 $\pm$ 0.04
Ours (Data)	172.70	0.11 $\pm$ 0.04
Linear	200.62	0.08 $\pm$ 0.04
Spherical	184.21	0.12 $\pm$ 0.04
NoiseDiffusion ( Zheng et al., 2024 )	174.46	0.11 $\pm$ 0.04

Table 4: Comparison of methods using FID and KID metrics (lower is better) for the AFHQ dataset ( Choi et al., 2020 ) for the bvp connecting two images.

Figure 7: Interpolated images for different methods.

Figure 8: Left: Computed mean value using our method in data space, where the image shows the transition from the mean to the data points in the last row. Right: the estimated mean using our method in noise and data space, respectively, compared to the Fréchet mean using a linear and spherical geometry in noise space.

Figure 9: Three principal geodesics shown on the left. On the right, samples drawn within the principal subspace are shown.

Appendix figures & tables33 assets

Supplementary material from the paper’s appendix.

Appendix

\begin{split}&\left\{{\bm{z}}_{s}^{(k+1)}\right\}_{s=0}^{N_{\mathrm{grid}}-1},\left\{{\bm{u}}_{s}^{(k+1)}\right\}_{s=0}^{N_{\mathrm{grid}}-1}\\ &=\mathrm{ProbGEORCE}\left(\left\{\tilde{{\bm{G}}}_{s}^{(k)}\right\}_{s=0}^{N_{\mathrm{grid}}-1},\left\{\left(\tilde{{\bm{G}}}^{(k)}\right)^{-1}\right\}_{s=0}^{N_{\mathrm{grid}}-1},\left\{\tilde{\bm{\nu}}_{s}^{(k)}\right\}_{s=0}^{N_{\mathrm{grid}}-1},\left(\sum_{s=0}^{N_{\mathrm{grid}}-1}\left(\tilde{{\bm{G}}}_{s}^{(k)}\right)^{-1}\right)^{-1}\right)\end{split}

Appendix

Algorithm 1 Adaptive update scheme for ProbGEORCE

\begin{split}\min_{\alpha}\quad&E\left({\bm{z}}_{0:N_{\mathrm{grid}},1:N_{\mathrm{data}}}\right)\quad\text{(exact line search)}\\ \text{s.t.}\quad&{\bm{z}}_{s+1,i}={\bm{z}}_{s,i}+\alpha\tilde{{\bm{u}}}_{s,i}+(1-\alpha){\bm{u}}_{s,i}^{(k)},\quad s=0,\dots,N_{\mathrm{grid}}-1,\,i=1,\dots,N_{\mathrm{data}}.\\ &\tilde{{\bm{u}}}_{s,i}=\alpha{\bm{u}}_{s,i}+(1-\alpha){\bm{u}}_{s,i}^{(k)},\quad s=0,\dots,N_{\mathrm{grid}}-1,\,i=1,\dots,N_{\mathrm{data}}.\\ &x_{0,i}={\bm{a}}_{i}.\end{split}

Appendix

Algorithm 2 ProbGEORCE for means

Figure 10: From left to right: The Fréchet mean (blue) for two data points (red) on the circle with uniqueness of the Fréchet mean, two points on the circle, where there are multiple Fréchet mean, the area (blue) for the data support around the north pole (green) of $\mathbb{S}^{2}$ for which the existence and uniqueness is guaranteed and similarly for the torus embedded in $\mathbb{R}^{3}$ .

Figure 11: The negative loss of mean, where we for each grid point computes the energy from all other data points using Eq. 7 for different values of $\lambda$ . The right-most figure correspond to $\lambda=0$ . Note that each loss is normalized for each plot to be between 0 and 1, and therefore the absolute scale between the plots can not be compared.

Figure 12: The first row shows interpolation in noise space (red) using the isotropic Gaussian limiting distribution and transports the curve to data space using a ddpm -model for a dino-dataset. The second row computes the curve directly in data space using the score of the ddpm , which, for illustrative purposes, is repeated in each step of the sampling process.

Figure 13: Computed interpolations for ControlNet ( Zhang et al., 2023 ) for eagle images of size $768\times 768\times 3$ similar to the experiment by Zheng et al. (2024) . We set the regularizer function in Eq. 7 as the density of the $\chi^{2}$ -distribution on the squared norm of the grid points ${\bm{z}}_{0:N}$ . We plot the using our method incorporated into NoiseDiffusion using Eq. 46 and compare to NoiseDiffusion ( Zheng et al., 2024 ) .

Method	Metric	Hyper-parameters
Linear	$G({\bm{z}})=I$	-
Spherical	Riemannian geometry of a sphere embedded in Euclidean space	-
Fisher-Rao	$G({\bm{z}})=\nabla_{{\bm{z}}}\log p({\bm{z}})\nabla_{{\bm{z}}}\log p({\bm{z}})^{\top}$	-
Jacobian	$G({\bm{z}})=J_{\nabla_{{\bm{z}}}\log p}^{\top}({\bm{z}})J_{\nabla_{x}\log p}({\bm{z}})$	-
Inverse Density	$G({\bm{z}})=\frac{1}{p({\bm{z}})^{2}}I$	-
Generative	$G({\bm{z}})=\left(\frac{p({\bm{z}})+\lambda}{p_{0}+\lambda}\right)^{2}I$	$p_{0}=\lambda=1$

Appendix

Table 5: Benchmark methods and hyper-parameters

Metric type	Kinetic Energy
Conformal	$E_{\text{kin}}(x,\dot{x})=\left(\lambda+h(x)\right)\\|\dot{x}\\|^{2}$ , where $h$ is a scalar function.
Diagonal	$E_{\text{kin}}(x,\dot{x})=\left(\lambda+h(x)\right)\\|\dot{x}\\|^{2}$ , where $h$ is not a scalar function.
Full	$E_{\text{kin}}(x,\dot{x})=\lambda\\|\dot{x}\\|^{2}+\left(\langle\dot{x},\nabla h(x)\rangle\right)^{2}$

Appendix

Table 6: Kinetic energy for different metric types.

Model	Architecture	Training
Energy-Based model	We use a MLP with one hidden layer with 128 neurons and ReLU activations using the package Ghaderi and Contributors (2025) .	Trained for $1,000$ epochs with a batch size of $128$ for fixed $50,000$ samples.
Normalizing flow	We use 32 affine coupling blocks of MLPs with two hidden layers of 64 neurons using the package ( Stimper et al., 2023 ) .	Trained for $4,000$ epochs with a batch size of $2^{9}$ .
AR	We use a MLP with two hidden layers of 64 neurons and TanH activations using the package.	Trained for $10,000$ epochs with a batch size of $512$ for fixed $50,000$ samples.
VAE	We use the same architecture for a noisy circle embedded in $\mathbb{R}^{3}$ as in Shao et al. (2018) with minor modifications.	Trained for $5,000$ epochs with a batch size of $512$ for fixed $50,000$ samples.
DDPM dinosaur	We use the architecture from github.com/tanelp/tiny-diffusion .	Trained identical to github.com/tanelp/tiny-diffusion .

Appendix

Table 7: Summary of models, their architectures, and training details.

Figure 14: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{1/p_{\theta}}$ (conformal).

Figure 15: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{\log p_{\theta}}$ (conformal).

Figure 16: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{1/p_{\theta}}$ (diagonal).

Figure 17: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{\log p_{\theta}}$ (diagonal).

Figure 18: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{\mathrm{LAND}}$ .

Figure 19: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{1/p_{\theta}}$ (full).

Figure 20: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with ${\bm{G}}_{\log p_{\theta}}$ (full).

Figure 21: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with our method ( $S=-1/p_{\theta}$ ).

Figure 22: Image transitions on the AFHQ dataset ( Choi et al., 2018 ) with our method ( $S=-\log p_{\theta}$ ).

Figure 23: A transition corresponding to an initial value problem using our method in noise space for ControlNet, where we state the target prompt under the images. The initial prompt was "A photo of a cat".

Figure 24: A transition corresponding to an initial value problem using our method in noise space for ControlNet, where we state the target prompt under the images. The initial prompt was "A photo of a house".

Figure 25: A display of different boundary value methods for ControlNet. The prompt was "A photo of a mountain".

Figure 26: A display of different boundary value methods for ControlNet. The prompt was "A photo of a cat".

Figure 27: A display of different boundary value methods for ControlNet. The prompt was "A photo of an aircraft".

Figure 28: The estimated mean using spherical geometry for the data in Fig. 8 . The prompt was "A photo of a cat".

Figure 29: The estimated mean using linear geometry for the data in Fig. 8 . The prompt was "A photo of a cat".

Figure 30: The estimated mean using our method in noise space for the data in Fig. 8 . The prompt was "A photo of a cat".

Interpolation	OASIS3	CIFAR10 (SGM)	CelebAHQ (SGM)	CIFAR10 (LSGM)	CelebAHQ (LSGM)
Linear	27.83 / 85.94	0.33 / 391.64	-1.61 / 285.32	26.49 / 254.39	1.51 / 149.53
Spherical	28.46 / 68.97	4.30 / 237.64	-1.56 / 142.14	30.00 / 217.10	1.70 / 145.02
ProbGEORCE	28.41 / 66.54	4.12 / 233.84	-0.46 / 138.69	29.99 / 217.52	1.70 / 143.79

Appendix

Table 8: Mean log-likelihood / FID for interpolation in different diffusion models and datasets. SGM is score-based generative models ( Kingma and Ba, 2014 ) , while LSGM is latent score-based generative model ( Vahdat et al., 2021 ) . OASIS3 ( LaMontagne et al., 2019 ) was trained using latent diffusion model ( Rombach et al., 2022 ) used by Pinaya et al. (2022) .

Figure 31: Interpolation using a latent diffusion model similar to ( Pinaya et al., 2022 ) with our proposed method. Each row represents different slices of the brain. The interpolation shows the generative transition between a healthy brain (left) and a brain with Alzheimer disease (right).

Figure 32: Comparisons of interpolation curves for CelebAHQ with SGM.

Figure 33: Comparisons of interpolation curves for CIFAR10 with SGM.

Figure 34: The regularized energy and runtime for different optimizers for the regularized energy in Eq. 7 solving the bvp for the models in Fig. 2 .

$\lambda=1.0$
	EBM		NF		AR		VAE
Optimizer	Reg. Energy	Runtime	Reg. Energy	Runtime	Reg. Energy	Runtime	Reg. Energy	Runtime
ProbGEORCE (Adaptive)	0.124	7.45 $\pm$ 0.01	0.087	81.03 $\pm$ 0.39	0.115	6.83 $\pm$ 0.01	0.043	145.86 $\pm$ 0.53
ProbGEORCE (LS)	0.124	73.62 $\pm$ 0.29	0.087	6.42 $\pm$ 0.01	0.115	0.58 $\pm$ 0.00	0.037	514.56 $\pm$ 10.30
Adam	0.124	6.60 $\pm$ 0.00	0.087	181.43 $\pm$ 3.04	0.115	12.65 $\pm$ 0.03	0.045	194.05 $\pm$ 5.60
SGD	0.206	5.70 $\pm$ 0.01	0.111	166.12 $\pm$ 2.02	0.175	11.99 $\pm$ 0.02	0.150	135.86 $\pm$ 9.14

Appendix

Table 9: Runtime and regularized energy for different methods for different values of

\lambda

for the methods in Fig. 2 . For ProbGEORCE (LS) for nf , the neural networks estimate of the likelihood returned nans during the iterations, and therefore we set

-

Figure 35: The regularized energy and runtimes for different optimizers minimizing Eq. 7 . We consider the

n

-sphere with

n=10

for different values of

\lambda

and

S=-\log p

, where

p

is the density of a mixture of three randomly weighted Gaussians with random means and the identity as covariance matrix. We report the regularized energy and runtime (mean runtime

\pm

standard deviation runtime over five runs repeated five times).

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Modern generative models based on diffusion or flow matching transport noise toward the data manifold, without representing the data manifold explicitly. We propose Distance Marching that explores an alternative design by explicitly modeling a distance field of the data manifold. A distance field offers a geometric view of generation as a descent process toward the data manifold using only the current position. The negative gradient of the distance field provides the direction, and the distance value can optionally determine the step size. However, sparse training data in high-dimensional space make it difficult to identify the closest points on the data manifold needed to train the distance field. To address this challenge, we design new loss functions that emphasize denoising targets from closer data points. Our analysis shows that, without time input, emphasizing closer targets reduces denoising ambiguity and the bias toward the data mean. This distance-field perspective naturally motivates two principled deterministic samplers, showing fast convergence in inference. As a result, our model is the only one without time input that remains competitive with time-conditioned models across datasets and architectures. Moreover, our distance prediction is also helpful for early stopping during sampling and for out-of-distribution detection. We hope distance-field modeling can open a promising direction for exploring generative modeling from a geometric perspective.

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