Intrinsic Geometry Matching: learning observation-equivariant process geometry

Manuel Lecha; Michael Bronstein; Iolo Jones

cs.LGOct 12, 2026

Intrinsic Geometry Matching: learning observation-equivariant process geometry

Authors: Manuel Lecha, Michael Bronstein, Iolo Jones

Organizations: AITHYRA · University of Oxford

Abstract

Real data are often arbitrary, indirect observations of an unknown underlying process, and those observations can distort its geometry. Existing geometric methods equip data with the induced geometry of the ambient space, but this entangles the true geometry of the process with the distortion of the observation. To address this, we introduce Intrinsic Geometry Matching (IGM): a scalable approach to learn the intrinsic geometry of the underlying process. Specifically, we learn the carré du champ operator of the process, allowing us to recover its diffusion geometry. Our method uses the fact that data often come with natural perturbations, like steps in a time series, to recover the intrinsic carré du champ while transforming naturally with changes in observation: it is observation-equivariant. Furthermore, since this equivariance relates the geometry of different spaces, we can train a model using additional sources of data, even when that data lies in another space. We evaluate IGM on molecular dynamics, image, and video data and find better performance and scaling than existing neural and graph-based geometric tools.

Figures & tables

Figure 1: Observation equivariance. Standard geometric tools give data the geometry of the observation space (left) , so depend on the observation map. This affects the carré du champ, which measures local covariance and describes the data geometry, like the distances. These quantities cannot be reconciled across different observations. We learn the intrinsic geometry of the underlying process (right) . The carré du champ measures intrinsic covariance and is observation-equivariant: it transforms naturally when we change observation. Distances are intrinsic and consistent.

Figure 2: Observation invariance of the LogDet divergence.

Panel of Figure 3 — Figure 3: IGM recovers process geometry with intrinsic-dimensional scaling. Orange : IGM; teal : LD-RMM; grey : $k$ NN. (a) Geodesics in latent coordinates (left) and on the observed torus (right): IGM recovers the true geometry, while LD-RMM induces the observation geometry; markers ( $\bullet$ , $\blacksquare$ ) identify endpoint pairs. (b) Latent-cometric and tangent-projector errors across intrinsic dimension. (c) Tangent-projector error versus sample size, varying intrinsic (left) and observation dimension (right).

Figure 4: Error vs number of views.

Panel of Figure 5 — Figure 5: Process geometry from image transitions. (a) Leading CDC eigenvectors: metamorphic augmentations induce semantically meaningful tangent vectors. (b) Interpolating paths with LD-RMM (observation-space geometry). (c) Interpolating paths with IGM (intrinsic metamorphic geometry). Models trained on MNIST (top row) are transferred to EMNIST (bottom row) without fine-tuning. Additional eigenvectors across scales and interpolation paths are shown in figs. 11 and 12 , respectively.

Panel of Figure 6 — Figure 6: Natural image geometry from video time steps. (a) NLL (bits/dim) of intrinsic IGM and observation-space LD-RMM, across model rank. (b) Explained variance of CDC eigenvectors.

Figure 7: Tangent vectors from video time steps. Leading CDC eigenvectors reveal the tangent structure to the image ‘manifold’, according to the intrinsic process observed through videos.

Figure 8: Held-out transition fit. NLL decreases with CDC rank (left), while leading directions capture increasing residual variability (right).

Panel of Figure 9 — Figure 9: Process geometry in molecular representations. (a) Held-out MD transition, IGM bridge, and linear interpolation (LERP) between fixed endpoints in latent PCA. (b) Decoded paths (top) and predicted per-residue mobility (bottom). Orange : IGM; teal : LERP; grey : held-out MD. Results for all $20$ test trajectories appear in fig. 18 ; additional decoded paths and mobility ellipsoids appear in figs. 16 and 17 .

Fréchet distance
LERP	$2.00\pm 0.14\,\text{\AA}$
IGM	$\mathbf{1.40\pm 0.16}\,\text{\AA}$

Table 10

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 10: Sampled image transitions. Samples at increasing transition scales $\tau\in\{0.5,1,2\}$ . Orange : metamorphosis; teal : isotropic Gaussian perturbations.

Panel of Figure 11 — Figure 11: IGM eigenvectors across transition scales. Leading CDC eigenvectors of the MNIST-trained model, evaluated on unseen EMNIST letters at (a) $\tau=0.05$ , (b) $\tau=0.1$ , and (c) $\tau=0.2$ . Larger scales capture progressively larger, morphology-aligned variations.

Panel of Figure 12 — Figure 12: Additional IGM interpolation paths. Paths between randomly sampled source–target pairs from held-out images in (a) MNIST and (b) EMNIST. The MNIST-trained model produces smooth, structured transformations on both datasets, without fine-tuning on EMNIST.

Panel of Figure 13 — Figure 13: Training-smoothing ablation on CelebV-HQ. Left: validation NLL, varying training smoothing $\lambda$ at rank $128$ . Right: explained transition variance in the leading $k$ CDC directions. Orange : IGM; teal : LD-RMM. Isotropic NLL references are defined in eq. 27 .

Figure 14: Intrinsic IGM geometry of images, with unregularised LogDet. Leading CDC eigenvectors on unseen validation images, with no training smoothing.

Figure 15: Intrinsic IGM geometry of images, with LogDet regularisation $\lambda=0.1$ . Leading CDC eigenvectors on unseen validation images, with training smoothing $\lambda=0.1$ .

Panel of Figure 16 — Figure 16: Decoded endpoint-conditioned paths. Two held-out MD transitions (trajectories 111 and 145; top and bottom) compared with the decoded IGM bridge and latent linear interpolation (LERP) at matched path progress. Bridges use rank $r=32$ and lag $\tau=16\,\mathrm{ps}$ . Orange : IGM; teal : LERP; grey : held-out MD.

Panel of Figure 17 — Figure 17: Learned process geometry in molecular coordinates. Per-residue mobility along held-out trajectories $111$ (top) and $145$ (bottom), shown as one-standard-deviation ellipsoids. The LID and NMP domains exhibit greater predicted mobility than the CORE .

Figure 18: Endpoint-conditioned paths on held-out MD trajectories. Each panel compares a held-out trajectory ( grey ), its IGM bridge ( orange ), and latent linear interpolation (LERP; teal ) between the same closed ( $\bigstar$ ) and open ( $\bigstar$ ) endpoints. Paths are projected onto the first two principal components of the training states. Headings report trajectory IDs and decoded discrete Fréchet distances in Å, using C $\alpha$ Kabsch-RMSD as the ground distance. IGM improves on LERP for all $20$ test trajectories, with mean distances of $1.418$ and $2.039$ Å, respectively. Rank $r=32$ and lag $\tau=16\,\mathrm{ps}$ were selected on the validation trajectories.

Geometric Representation Learning

Geometric representation learning encodes data to preserve its intrinsic shape, distances, symmetries, and relationships.

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Geometric Representation Learning Metric Learning Riemannian Metric Learning Riemannian Geometry

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High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of

k

-NN-based diffusion geometry estimators, while enabling amortized inference that is up to

400\times

faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

Abstract

Geometric Representation Learning

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