Definition

Rather than claiming the representation manifold is some named shape (sphere, torus, paraboloid…), this node covers papers that measure how curved it is at a given layer/training point — a computed quantity, not a geometric object in its own right. Two genuinely different notions of “curvature” appear here, and they must not be merged:

• Extrinsic principal curvature (mean absolute principal curvature, MAPC/NMAPC): fit local quadratic surfaces (via local PCA on neighborhoods) to the point cloud and measure how much it bends away from its own local tangent plane, embedded in the ambient activation space.
• Intrinsic graph curvature (Ollivier-Ricci, Forman-Ricci): build a k-NN or similarity graph over activations and compute a discrete, combinatorial/optimal-transport curvature at each vertex or edge, entirely from the graph’s own connectivity and neighbor-mass transport cost — no ambient embedding or ambient ruler is referenced.

These measure different things: extrinsic curvature depends on how the manifold sits inside its ambient space, while intrinsic (graph) Ricci curvature is a property of the graph alone and would be unchanged by any embedding that preserves local neighbor relations. A paper reporting one does not thereby report the other.

Relative to intrinsic-dimension-profile

Intrinsic-dimension profile across depth tracks how many coordinates the manifold needs (its dimensionality) across depth or training; this node tracks how bent the manifold is, independent of its dimension. The two are orthogonal: a 1-dimensional circle embedded in a high-dimensional space has intrinsic dimension 1 at every radius, yet nonzero (and tunable) curvature — dimension and curvature are separate facts about the same manifold, and a flat, zero-curvature object can still have arbitrarily high dimension. Papers in this node should not be read as making any claim about dimensionality, and vice versa.

Key evidence: extrinsic curvature (MAPC)

Kaufman & Azencot (2023) estimate principal curvatures of the latent data manifold at hidden layers of real trained ResNet and VGG image classifiers, via a normalized mean-absolute principal curvature (NMAPC) computed from local-PCA neighborhoods. Trained networks show a characteristic curvature profile across layers — an initial steep increase, a long plateau, then a further increase — absent in untrained networks (which stay flat). The curvature gap between the last two layers correlates strongly with generalization capability, and the paper explicitly notes that intrinsic dimension of the same latent codes is “not necessarily indicative of curvature” — an independent confirmation, from the source paper itself, of this node’s orthogonality claim relative to Intrinsic-dimension profile across depth. See kaufman-azencot-2023-mapc-curvature-tracks-generalization-in-real-resnet-and-vgg-image-classifiers.

Rahman, Barrett & Last (2026) measure a third, simpler extrinsic instrument on Google AlphaEarth’s 64-dimensional satellite embeddings (~12.1 million real CONUS samples): the angle between locally-fit tangent spaces (local PCA) at neighboring points of the manifold. A flat (affine) manifold would show zero tangent-space rotation everywhere; instead, 84% of sampled locations show rotation exceeding 60 degrees between adjacent probe tangent spaces, with the largest instability in mountainous regions with steep environmental gradients. A separate local-versus-global comparison — how far each local tangent space tilts relative to the global principal axes — gives a mean |cos theta| of 0.169, a distinct metric from the adjacent-probe rotation. Both are direct, if cruder, extrinsic-bending signals in the same family as MAPC (both are local-PCA-based), reported alongside (not merged with) the same paper’s separate intrinsic-dimension-profile measurement (effective dimensionality 13.3, local ID ~10). See alphaearths-64-dimensional-embedding-manifold-has-effective-dimensionality-13-3-and-tangent-spaces-that-rotate-over-60-degrees-across-84-percent-of-locations.

Key evidence: intrinsic curvature and Ricci-flow-like evolution

Hehl, von Renesse & Weber (2025) build k-NN graphs from layer-wise activations of real feedforward MLPs (widths 15/25/50, depths 7/10/15, over 20,000 networks total) trained on binary MNIST/Fashion-MNIST/ CIFAR-10 subsets, computing discrete Ollivier-Ricci and augmented Forman-Ricci curvature per vertex, plus a derived local Ricci-evolution coefficient (the correlation, across layers, between curvature and neighborhood-distance change). A majority-negative coefficient pattern (88.7%-98.3% of vertices, architecture-dependent) indicates genuine Ricci-flow-like contraction dynamics in the real trained network’s feature geometry — a dynamical pattern measured on top of the curvature quantity, not a claim that the manifold itself is a dynamical object with attractors or bifurcations of its own (which would instead warrant a dynamical-object role). The coefficient’s trend reverses at the point coinciding with the onset of overfitting and with the depth that maximizes test accuracy. See hehl-etal-2025-discrete-ollivier-ricci-curvature-of-real-trained-mlp-activation-graphs-shows-flow-like-contraction-dynamics-that-reverse-at-the-onset-of-overfitting.

Key evidence: generator/decoder pullback metric (VAEs and GANs)

A third extrinsic instrument, distinct from both MAPC (local-PCA on a static point cloud) and tangent-space rotation, applies specifically to generative models with an explicit differentiable latent-to-data generator map: the Generator/decoder Jacobian pullback metric, $M(z) = J_g(z)^\top J_g(z)$. Arvanitidis, Hansen & Hauberg (2018) introduce a stochastic version for real VAEs trained on MNIST and video-frame data, showing geodesics under this metric substantially improve interpolation and density estimation over naive Euclidean latent distance. Shao, Kumar & Fletcher (2018) apply the deterministic version to real VAEs trained on CelebA and SVHN, finding the learned manifold is surprisingly close to zero curvature — linear interpolation nearly matches true geodesics, a notably different emphasis from Arvanitidis et al.’s finding that latent distortion is significant enough to warrant correction. Wang & Ponce (2021) eigen-decompose the same metric across several real pretrained GANs (BigGAN, PGGAN, StyleGAN2, DCGAN and others), finding image variation concentrates along surprisingly few, position-homogeneous major axes. Shukla, Bhagat, Anand & Turaga (2018) compare the metric’s curvature across a vanilla VAE and three disentangling VAE variants trained on MNIST, MultiPIE and 3D Chairs, finding class-distinguishing latent regions carry measurably higher curvature than a vanilla VAE’s latent space. All four are purely observational (no causal intervention on the manifold itself). See arvanitidis-hansen-hauberg-2018-a-real-vaes-decoder-induced-pullback-metric-and-its-geodesics-substantially-improve-interpolation-and-density-estimation-over-naive-euclidean-latent-distance, shao-etal-2018-real-celeba-and-svhn-trained-vae-manifolds-are-surprisingly-close-to-zero-curvature-so-linear-latent-interpolation-nearly-matches-true-geodesics, wang-ponce-2021-image-variation-in-several-real-pretrained-gans-concentrates-along-surprisingly-few-position-homogeneous-major-axes-of-the-pullback-metric, and shukla-etal-2018-class-distinguishing-latent-regions-of-real-disentangled-vae-variants-carry-measurably-higher-pullback-metric-curvature-than-a-vanilla-vaes-latent-space.

Key evidence: Fisher information metric and phase-transition structure

A fourth, intrinsic instrument reconstructs a Fisher information metric from a generative model’s implicit log-partition function (the Hessian of $\log Z$), rather than fitting a point cloud or pulling back a generator’s Jacobian — see Fisher information metric reconstruction from a log-partition function. Lobashev, Guskov, Larchenko & Tamm (2026) validate this reconstruction against exactly-solvable statistical-physics systems (2D Ising model, TASEP) before applying it to a real Stable Diffusion 1.5 (Dreamshaper8 checkpoint) latent space, finding a fractal structure of phase transitions: geodesic interpolation is locally linear within a phase, but breaks down at phase boundaries, where the local Lipschitz constant diverges — a genuinely different kind of finding from the other entries on this node (discontinuous phase structure, not a smooth curvature number or profile). See lobashev-etal-2026-a-real-stable-diffusion-1-5-latent-space-shows-a-fractal-structure-of-fisher-metric-phase-transitions-with-diverging-lipschitz-constants-at-phase-boundaries.

Key evidence: spin-glass phase diagnostics and attention-weighted graph curvature

Two further instruments extend this node’s intrinsic/regime-detecting side. Ascárate, Lebrat, Santa Cruz, Fookes & Salvado (2026) apply Spin-glass order-parameter diagnostics for latent phase structure (overlap distributions between sampled latent codes, order parameters, susceptibility — tools imported from spin-glass statistical physics) to real trained VAE/ autoencoder latent spaces on CIFAR-10, CelebA64, Mars Rover Mastcam, and Galaxy Zoo 64 image data, finding the latent space organizes into measurably distinct ordered, disordered, and glassy phase regimes; a hyperspherical-compression intervention that shifts the regime toward more order improves downstream anomaly-detection AUROC (Mars Rover: 0.66->0.76 kNN-AUROC; Galaxy Zoo 64: 0.74->0.79). This is a phase-regime finding in the same family as Lobashev et al.’s Fisher-metric phase transitions above, via a different (spin-glass rather than information-geometric) instrument. See ascarate-etal-2026-real-trained-vae-latent-spaces-organize-into-measurably-distinct-ordered-disordered-and-glassy-phases-and-shifting-toward-a-more-ordered-phase-improves-anomaly-detection.

Tori, Bini, Sorbi, Marchand-Maillet & Ginis (2026) apply Attention-weighted graph curvature analysis (Balanced Forman Curvature computed on a real trained graph transformer’s own attention-re-weighted input graph, rather than a k-NN graph built from activations) to real Graph Transformers on ZINC, Tox21, and Long-Range Graph Benchmark peptide datasets, finding a “Curvature Collapse”: the fraction of negatively-curved edges in the attention-weighted effective graph jumps from 57% to 84% (peptides-func) and 57% to 82% (peptides-struct) relative to the raw input graph’s own curvature — attention concentrates on and worsens existing topological bottlenecks rather than smoothing them, confirmed as functionally load-bearing (not an artifact) via causal edge-pruning that spikes loss by 22-27%. This is a genuinely different graph-curvature construction from Hehl et al.’s k-NN-of-activations Ollivier/Forman-Ricci above: it re-weights a pre-existing external graph rather than deriving a graph from the network’s own representation cloud. See tori-etal-2026-a-real-trained-graph-transformers-attention-weighted-effective-graph-undergoes-curvature-collapse-concentrating-on-and-worsening-bottleneck-edges-rather-than-smoothing-them.

Key evidence: hyperbolic-embedding distortion and Gromov delta-hyperbolicity fit

A fifth instrument, Hyperbolic embedding distortion and Gromov delta-hyperbolicity fit, asks a different question than the others on this node: not how curved an arbitrary representation manifold is, but whether a network architecturally given a curved (hyperbolic) embedding space actually achieves a low-distortion, metric-faithful embedding of the graph it is trained on — a genuinely measured, task-conditional property, not a confirmation that the imposed geometry works as designed. Naddeo, Linkerhägner, Toschi, Skenderi & Lachi (2026) train real HGCN (Poincare-ball, learnable curvature) and HyboNet (Lorentz-model) graph neural networks — versus Euclidean GCN/GAT/MLP baselines — on Cora, Citeseer, Pubmed, Disease and Airport (node classification and link prediction), plus synthetic tree/grid graphs, measuring embedding distortion $\delta(h) = d_c(h)/d_e(h)$ (contraction/expansion of learned latent distances vs. graph shortest-path distances), a normalized Stress Loss, and each dataset’s exact Gromov delta-hyperbolicity (Disease 0.0, Airport 1.0, Pubmed 3.5, Citeseer 4.5, Cora 11.0). The hyperbolic advantage is sharply task-conditional: on synthetic trees, HyboNet/HGCN show markedly lower stress than Euclidean models at low dimension, but the advantage vanishes by $d=128$ and reverses entirely on grid graphs; critically, on real-world node classification, “none of the models learn metric-preserving embeddings” regardless of architecture, while only link prediction (a geometry-aligned task) shows HGNNs achieving both lower distortion and higher ROC-AUC/AP, especially on low-hyperbolicity datasets (Disease, Airport) — confirmed as functionally load-bearing, not incidental, via a feature-corruption experiment where HGNNs retain higher performance and lower distortion under progressive noise than Euclidean models. See naddeo-etal-2026-hyperbolic-gnns-achieve-low-distortion-embeddings-only-on-tree-like-graphs-under-geometry-aligned-tasks.

Why one node, not two

Kaufman & Azencot’s extrinsic principal curvature and Hehl et al.’s intrinsic graph-Ricci curvature are hosted on the same node deliberately, mirroring Intrinsic-dimension profile across depth (which also hosts several distinct estimators — TwoNN, GRIDE, effective rank, MST-based — under one “what is the dimensionality trajectory” question). Both papers here answer the same higher-level question — “how much, and how, is the representation manifold curved, as a function of depth/training” — with different instruments. Splitting into two separate nodes (principal-curvature-profile / ricci-curvature-profile) is deferred until either notion is independently replicated by a second paper using that same instrument; until then, a single node with the extrinsic/intrinsic distinction carried explicitly in the text (never merged into one number) is the more honest representation of the current evidence base.

Independent contemporaneous discovery of the pullback metric, applied to robot motion and human mocap. Chen, Jiang, Klushyn, Kurle, Bayer & van der Smagt (2018) independently derive the same deterministic pullback metric as Arvanitidis et al. and Shao et al. above, applying an SVD smoothing regularizer and demonstrating it on a real IWAE trained on MNIST, a simulated 6-DOF robot-arm circular-motion dataset, and real CMU motion-capture walking data — geodesics under the pullback metric consistently beat naive Euclidean latent interpolation at producing natural, class-consistent, or physically plausible interpolation paths in all three settings. See chen-etal-2018-a-real-trained-importance-weighted-autoencoders-pullback-riemannian-geodesics-outperform-euclidean-latent-interpolation-on-mnist-a-simulated-robot-arm-and-real-human-motion-capture-data.

Generalizing the pullback metric to a Finsler norm. Pouplin, Eklund, Ek & Hauberg (2023) generalize the expected-pullback-metric approach to a Finsler metric that retains the full distribution of stochastic pullback metrics rather than collapsing it to a single expectation, proving both notions converge at rate $O(1/D)$ as ambient dimension grows, and confirming this convergence empirically on real GP-LVMs trained on a 256-dimensional font-contour dataset and a 48-dimensional qPCR single-cell dataset (with visible divergence only in low-dimensional synthetic settings). See pouplin-etal-2023-a-real-trained-gp-lvms-riemannian-and-finslerian-latent-geodesics-converge-on-high-dimensional-real-datasets-but-diverge-in-low-dimensional-synthetic-settings.

Score-Jacobian spectral gaps in diffusion models, a curvature-adjacent diagnostic along the noise axis. Ventura, Achilli, Silvestri, Lucibello & Ambrogioni (2025) apply a statistical-physics/random-matrix treatment to a diffusion model’s score-function Jacobian, predicting and then empirically confirming — on real diffusion models trained on MNIST, CIFAR-10, and CelebA — that the Jacobian’s eigenvalue spectrum develops gaps as sampling noise decreases, with gap locations tracking each dataset’s sub-manifold dimensionality and marking distinct geometric phase transitions in the generative dynamics. See ventura-etal-2025-real-diffusion-models-trained-on-mnist-cifar10-and-celeba-show-score-jacobian-spectral-gaps-matching-random-matrix-theory-predictions-for-sub-manifold-dimensionality.

Retargeting the score-Jacobian spectral gap from the true score to the empirical score reveals memorization as a dimension-loss process. Achilli, Ventura, Silvestri, Pham, Raya, Krotov, Lucibello & Ambrogioni (2024) reuse Ventura et al.’s score-Jacobian spectral-gap estimator, built above for the true (population) score, on the empirical score of real DDPMs trained from scratch on MNIST, Fashion-MNIST, CIFAR-10, CelebA-HQ and LSUN-Church across 38 training-set-size splits per dataset — new spectral gaps appear that only a theory of the empirical score, not the true score, predicts, and the resulting local intrinsic dimension declines smoothly as training-set size shrinks into a memorization regime before collapsing toward zero (see also Dimensional collapse). See achilli-etal-2024-real-ddpm-spectral-gap-dimension-shrinks-toward-zero-as-training-set-size-falls-tracing-memorization-induced-dimensional-collapse.

The same score-Jacobian curvature diagnostic, mapped onto a Dense Associative Memory energy landscape. Pham, Raya, Negri, Zaki, Ambrogioni & Krotov (2025) identify a diffusion model’s own energy function with a Hopfield-network (Dense Associative Memory) energy and reuse the identical score-Jacobian SVD/spectral-gap curvature diagnostic on real DDPMs trained from scratch on MNIST, Fashion-MNIST, CIFAR-10 and LSUN-Church, finding the resulting memorized/spurious/generalized transition point matches Achilli et al.’s dimensional-collapse critical points on the same checkpoints, and that this transition’s location scales with model capacity (a U-Net width ablation on CIFAR-10 and LSUN-Church) rather than dataset size alone. See pham-etal-2025-mapping-real-trained-ddpm-score-jacobian-curvature-onto-a-densam-energy-landscape-reveals-a-memorized-spurious-generalized-transition-that-shifts-with-model-capacity.

A local, trajectory-level Jacobian-norm diagnostic, complementary to the population-level score-Jacobian spectral gap. Sakamoto & Sakamoto (2026) define a Critical Boundary Detector (CBD), the Frobenius norm of the Jacobian of a diffusion model’s own normalized score/velocity field along a single generation trajectory, and show on a real pretrained DDPM-CIFAR10 checkpoint that this local instrument correlates strongly with LPIPS perceptual distance (mean Pearson r=0.928 across 5 seeds) and, when used to target classifier guidance to a detected “switching band,” matches full-guidance target-class accuracy (0.96-1.00) using only about 4% of the diffusion steps. Real experiments on Stable Diffusion 3.5 Medium, DiT-XL (ImageNet), EDM2-XS (CIFAR-10) and a CelebA-HQ DDPM show the same instability-band structure across architectures. The paper positions CBD explicitly as a complementary, single-trajectory geometric diagnostic to the population-level score-Jacobian spectral-gap technique above, not a replacement for it. See sakamoto-sakamoto-2026-a-real-ddpm-cifar10-checkpoints-critical-boundary-detector-correlates-with-lpips-and-matches-full-classifier-guidance-accuracy-using-only-4-percent-of-diffusion-steps.

Key papers

• Kaufman, I. & Azencot, O. (2023). Data Representations’ Study of Latent Image Manifolds. ICML 2023, arXiv:2305.19730 — origin of the MAPC/NMAPC extrinsic curvature-profile measurement.
• Hehl, M., von Renesse, M. & Weber, M. (2025). Neural Feature Geometry Evolves as Discrete Ricci Flow. arXiv:2509.22362 — origin of the discrete Ollivier/Forman-Ricci intrinsic curvature measurement and its Ricci-flow-like evolution across depth.
• Rahman, M., Barrett, S. J. & Last, C. (2026). Characterizing AlphaEarth Embedding Geometry for Agentic Environmental Reasoning. arXiv:2604.18715 — tangent-space-rotation extrinsic curvature signal on a real satellite-embedding manifold.
• Arvanitidis, G., Hansen, L. K. & Hauberg, S. (2018). Latent Space Oddity: On the Curvature of Deep Generative Models. ICLR 2018, arXiv:1710.11379 — stochastic generator-pullback-metric curvature on real VAEs.
• Shao, H., Kumar, A. & Fletcher, P. T. (2018). The Riemannian Geometry of Deep Generative Models. CVPR Workshop 2018, arXiv:1711.08014 — near-zero-curvature finding on real VAE manifolds.
• Wang, B. & Ponce, C. R. (2021). The Geometry of Deep Generative Image Models and its Applications. arXiv:2101.06006 — pullback-metric eigen-decomposition across several real pretrained GANs.
• Shukla, A., Bhagat, S., Anand, S. & Turaga, P. K. (2018). Geometry of Deep Generative Models for Disentangled Representations. ICVGIP 2018, arXiv:1902.06964 — comparative curvature across disentangled vs. vanilla real VAEs.
• Lobashev, A., Guskov, D., Larchenko, M. & Tamm, M. (2026). Hessian Geometry of Latent Space in Generative Models. arXiv:2506.10632 — Fisher-information-metric phase-transition structure in a real Stable Diffusion latent space.
• Ascárate, A., Lebrat, L., Santa Cruz, R., Fookes, C. & Salvado, O. (2026). High-Dimensional Latents Should Be Diagnosed Through Phase Structure. arXiv:2606.02600 — spin-glass overlap-diagnostic phase regimes in real trained VAE latent spaces.
• Tori, F., Bini, L., Sorbi, M., Marchand-Maillet, S. & Ginis, V. (2026). Probing Graph Neural Network Activation Patterns Through Graph Topology. arXiv:2602.21092 — attention-weighted graph curvature collapse in real trained graph transformers.
• Chen, N., Jiang, X., Klushyn, A., Kurle, R., Bayer, J. & van der Smagt, P. (2018). Metrics for Deep Generative Models. AISTATS 2018, arXiv:1711.01204 — independently-discovered pullback metric with SVD smoothing, on real IWAEs (MNIST, robot arm, motion capture).
• Pouplin, A., Eklund, D., Ek, C. H. & Hauberg, S. (2023). Identifying Latent Distances with Finslerian Geometry. TMLR 2023, arXiv:2212.10010 — Finsler generalization of the pullback metric, on real GP-LVMs (font contours, qPCR).
• Ventura, E., Achilli, B., Silvestri, G., Lucibello, C. & Ambrogioni, L. (2025). Manifolds, Random Matrices and Spectral Gaps: The Geometric Phases of Generative Diffusion. ICLR 2025, arXiv:2410.05898 — score-Jacobian spectral gaps on real diffusion models (MNIST, CIFAR-10, CelebA).
• Achilli, B., Ventura, E., Silvestri, G., Pham, B., Raya, G., Krotov, D., Lucibello, C. & Ambrogioni, L. (2024). Losing Dimensions: Geometric Memorization in Generative Diffusion. arXiv:2410.08727 — retargets the score-Jacobian spectral gap to the empirical score, revealing memorization-induced dimensional collapse.
• Pham, B., Raya, G., Negri, M., Zaki, M. J., Ambrogioni, L. & Krotov, D. (2025). Memorization to Generalization: Emergence of Diffusion Models from Associative Memory. arXiv:2505.21777 — maps the same score-Jacobian curvature diagnostic onto a Dense Associative Memory energy landscape, capacity-dependent memorization transition.
• Sakamoto, R. & Sakamoto, K. (2026). The Geometry of Phase Transitions in Generative Dynamics via Projection Caustics. arXiv:2606.13191 — Critical Boundary Detector, a local trajectory-level Jacobian-norm curvature diagnostic correlating with LPIPS on a real DDPM-CIFAR10 checkpoint.
• Naddeo, D., Linkerhägner, J., Toschi, N., Skenderi, G. & Lachi, V. (2026). Hyperbolic Graph Neural Networks Under the Microscope: The Role of Geometry-Task Alignment. arXiv:2602.01828 — embedding distortion and Gromov delta-hyperbolicity fit, showing the hyperbolic advantage is task-conditional in real trained HGCN/HyboNet models.