Probing the Geometry of Diffusion Models with the String Method

Elio Moreau; Florentin Coeurdoux; Grégoire Ferre; Eric Vanden-Eijnden

stat.MLFeb 25, 2026

Probing the Geometry of Diffusion Models with the String Method

Authors: Elio Moreau, Florentin Coeurdoux, Grégoire Ferre, Eric Vanden-Eijnden

Organizations: Capital Fund Management, 23 Rue de l’Université, 75007 Paris · ML Lab, Capital Fund Management, 23 Rue de l’Université, 75007 Paris · Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

Abstract

Understanding the geometry of learned distributions is fundamental to improving and interpreting diffusion models, yet systematic tools for exploring their landscape remain limited. Standard latent-space interpolations fail to respect the structure of the learned distribution, often traversing low-density regions. We introduce a framework based on the string method that computes continuous paths between samples by evolving curves under the learned score function. Operating on pretrained models without retraining, our approach interpolates between three regimes: pure generative transport, which yields continuous sample paths; gradient-dominated dynamics, which recover minimum energy paths (MEPs); and finite-temperature string dynamics, which compute principal curves -- self-consistent paths that balance energy and entropy. We demonstrate that the choice of regime matters in practice. For image diffusion models, MEPs contain high-likelihood but unrealistic ''cartoon'' images, confirming prior observations that likelihood maxima appear unrealistic; principal curves instead yield realistic morphing sequences despite lower likelihood. For protein structure prediction, our method computes transition pathways between metastable conformers directly from models trained on static structures, yielding paths with physically plausible intermediates. Together, these results establish the string method as a principled tool for probing the modal structure of diffusion models -- identifying modes, characterizing barriers, and mapping connectivity in complex learned distributions.

Figures & tables

Figure 1: The likelihood-realism paradox. Top: schematic showing the MEP (green) passing through the high-likelihood region (yellow), while the principal curve (red) stays within the typical set where data concentrates (blue points). Dashed lines indicate Voronoi cells—regions of points closest to each image along the string. Bottom: actual images at numbered locations. Endpoints (1, 2) are identical for both paths; the principal curve intermediate (3) is realistic; the MEP intermediate (4) is cartoonish. Full pathways computed by our method are shown in Figures 4 and 5 .

Panel of Figure 2 — Figure 2: The string method. Grey dashed arrows show Step 1 (evolution): each image moves according to $v_{t}$ , landing at positions marked with $\times$ . Blue dotted arrows show Step 2 (reparametrization): images are redistributed to restore equal arc-length spacing along the string.

Figure 3: Relative score estimation error:

\mathbb{E}_{\text{model}}[|s_{t}-\hat{s}_{t}|/|s_{t}|]

as a function of

t

for a mixture of Gaussians in various dimensions. The error increases sharply near

t=1

, motivating the quenching of

\gamma_{t}

t\to 1

. For details see Appendix C

	Transport	MEP	Principal curve
	$\gamma_{t}=0$	$\gamma_{t}\gg 1$	$\gamma_{t}\gg 1$
		$T=0$	$T>0$
Geometric	No	Yes	Yes
Entropic	No	No	Yes
Realistic	Yes	No	Yes

Table 1: Three regimes of string dynamics.

Figure 4: Effect of score weight $\gamma$ . Left: string realizations for $\gamma$ ranging from 15 ( top ) to 2 ( middle ) to $10^{-2}$ ( bottom ) in logarithmic steps (factor of $\sqrt{10}$ ). Higher $\gamma$ drives paths through abstract, high-likelihood modes; lower $\gamma$ preserves realism. Right: log-likelihood of images along each string (colored curves), overlaid on the likelihood distribution of ImageNet validation images (heatmap; see Figure 6 for details). The intermediate images along the MEP reach likelihoods far exceeding typical images.

Figure 5: Effect of temperature $T$ . Left: string realizations for $T$ ranging from 0.1 ( top ) to 0.5 ( middle ) to 0.9 ( bottom ). Lower $T$ drives paths through cartoon-like, high-likelihood regions; higher $T$ produces realistic samples. Right: log-likelihood of images along each string (colored curves), overlaid on the likelihood distribution of ImageNet validation images (heatmap; see Figure 6 ). As $T$ increases, the likelihood of the intermediates images decreases toward typical values, and images become more realistic.

Figure 6: Likelihood distributions. Histogram of log-likelihoods for ImageNet

256\times 256

validation images (blue), compared with images along MEPs (orange) and finite-temperature strings at

T=0.1

(green),

T=0.5

(red), and

T=0.9

(purple), aggregated across multiple strings. MEP intermediates have significantly higher likelihood than real images, confirming they lie outside the typical set. As temperature increases, string images approach the likelihood distribution of real data. Inset: same data on log scale.

Panel of Figure 7 — Figure 7: Principal curve for images from the goose class. Top row: images along the principal curve, computed as the EMA of associated walkers ( $T=0.9$ , $\gamma=7$ ). Bottom three rows: individual walkers at $t=1$ . The walkers exhibit subtle variations due to entropic effects (best seen when zoomed in); averaging produces the smoother images in the principal curve.

Figure 8: Adenylate Kinase transition pathway. Pathway between the open (4AKE) and closed (1AKE) conformations computed using DiG. Intermediate structures maintain physical plausibility (secondary structure preservation, no steric clashes).

Panel of Figure 9 — Figure 9: BBA folding pathway. Top: structures along the initial string obtained by pure transport ( $\gamma=0$ ). Bottom: structures along the converged MEP. Both show progressive formation of secondary structure. Left panel: initial string (purple) and MEP (green) projected onto the first two TIC components, overlaid on a free-energy landscape estimated from ScoreMD samples; darker regions indicate lower free energy. Asterisks mark equal arc-length intervals. Right panel: energy profile along each pathway. The MEP achieves significantly lower energy than the initial string.

Panel of Figure 10 — Figure 10: Chignolin folding pathway. Top: structures along the initial string obtained by pure transport ( $\gamma=0$ ). Bottom: structures along the converged MEP. Both show progressive formation of secondary structure. Left panel: initial string (purple) and MEP (green) projected onto the first two TIC components, overlaid on a free-energy landscape estimated from ScoreMD samples; darker regions indicate lower free energy. Asterisks mark equal arc-length intervals. Right panel: energy profile along each pathway. The MEP achieves significantly lower energy than the initial string.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameters	Values
Neural network
Depth	28
Hidden Size	1152
Patch Size	2
Number of Attention Heads	16
MLP ratio	4.0

Appendix

Table 2: Hyperparameter choices for the SiT Model

Hyperparameter	Initialization	PIDP	Data Training
Model depth	12
Hidden dim (Single)	768
Hidden dim (Pair)	256
Hidden dim (Feed Forward)	1024
Number of Heads	32

Appendix

Table 3: Hyperparameters of the Distributional Graphormer Protein Model.

Model name	hidden_nf	feature_embedding_dim	n_layers	potential	dropout
transformer_large_score	128	16	3	false	0.0
transformer_large_potential	128	16	3	true	0.0

Appendix

Table 4: Graph transformer model definitions extracted from the configuration file.

Entry	Model reference	Range
1	transformer_large_score	$[1.0,\;0.6]$
2	transformer_large_score	$[0.6,\;0.1]$
3	transformer_large_potential	$[0.1,\;0.0]$

Appendix

Table 5: Ranged models configuration extracted from the configuration file of the model, it should be kept in mind that for this model

t=0

is actually the interpolant time

t=1

and vice versa.

Figure 11: Pathway from extended to folded conformation computed using ConfDiff. Intermediates show progressive formation of secondary structure.

Hyperparameters	Values
Neural network
Number of IPA blocks	4
Dimension of single repr.	256
Dimension of pairwise Repr.	128
Dimension of hidden	256
Number of IPA attention heads	4

Appendix

Table 6: Hyperparameter choices for ConfDiff Model

Figure 12: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 13: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 14: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 15: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 16: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 17: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 18: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 19: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 20: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 21: Effect of temperature $T$ on images principal curve. Lower $T$ drives paths through cartoonish high-likelihood modes (top); higher $T$ preserves realism (bottom).

Figure 22: A principal curves with three realizations coming from this curve

Figure 23: A principal curves with three realizations coming from this curve

Figure 24: A principal curves with three realizations coming from this curve

Figure 25: A principal curves with three realizations coming from this curve

Figure 26: A principal curves with three realizations coming from this curve

Figure 27: A principal curves with three realizations coming from this curve

Figure 28: A principal curves with three realizations coming from this curve

Figure 29: A principal curves with three realizations coming from this curve

Figure 30: A principal curves with three realizations coming from this curve

Figure 31: A principal curves with three realizations coming from this curve

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