Latent Cluster Geometry

Residual Vector Quantization turns continuous representations into discrete, multi-level token sequences. Yet most methods operate in Euclidean space, despite the coarse-to-fine structure of the resulting codes and the latent hierarchies present in many data domains. Hyperbolic geometry offers a natural alternative for hierarchical representations, but naive hyperbolic extensions introduce geometric inconsistencies: non-associative hyperbolic addition prevents consistent residual aggregation, while standard straight-through gradient estimation ignores the geometry of the latent space. We propose a geometry-aware hyperbolic residual quantization that addresses these issues in both the forward and backward passes. In the forward pass, Hyperbolic Residual Aggregation restores the telescoping behavior of residual quantization on the Poincare ball. In the backward pass, a discounted Hyperbolic Straight-Through Estimator routes the reconstruction gradient through the quantizer as a single geometric block, avoiding unstable recursive gradient transport across residual stages. Evaluations on hierarchical prediction, recommendation, image tokenization, and neural audio coding tasks show that our method improves the stability and structural organization of hyperbolic residual codes over naive hyperbolic baselines. At the same time, we observe a clear structure-compression trade-off: Euclidean residual quantization remains preferable for pure compression, while geometry-aware hyperbolic quantization is most useful for hierarchically organized discrete latent spaces.

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

Classical approaches to event-based egomotion estimation, including those adopted by the top-performing teams of the ELOPE challenge, rely on geometric optimization frameworks such as contrast maximization, homography estimation, or dense optical flow combined with analytic motion inversion. This work investigates the geometric structure that emerges inside a multi-modal network for egomotion estimation. Event tensors, inertial measurements, and range signals are fused through a cross-modal attention architecture and trained in a batch setting. We analyze the latent space geometry and attention dynamics, showing that (i) embeddings lie on low-dimensional manifolds aligned with motion variables, (ii) attention weights adapt with angular excitation and visual reliability, and (iii) the fused representation recovers classical observability cues. These results bridge analytical estimation theory and modern data-driven fusion.

We study sparse random geometric graphs generated by connecting pairs of high-dimensional vectors whose inner product exceeds a threshold. The latent vectors are sampled either uniformly from the sphere or from a standard Gaussian distribution. Although every edge appears with probability $p$, the edges are dependent through their shared latent vectors. For the spherical model, at the connectivity scale $np=Ω(\log n)$, we prove $\|A-\mathbb E A\|=O\left(\sqrt{np\log n}+npτ\right)$, with high probability, where $τ$ is the cap threshold. This sharpens the spectral norm bound of Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions. An analogous result holds for the Gaussian model after removing the fluctuations of the vector norms, yielding improved global synchronization guarantees for the homogeneous Kuramoto model. We then recover the latent geometry from the leading eigenspace. When $np\gg\log n$, both the latent vector and relative Gram matrix errors vanish provided $d\ll np\log(1/p)/\log n$. The required lower dimension is only $d\gg\log(1/p)$ for the spherical model and $d\gg\log^2(1/p)\log n$ for the Gaussian model, improving the recovery guarantees of Li and Schramm (2023). Finally, we prove the first exact recovery result for the Gaussian mixture block model of Li and Schramm (2023). At the optimal connectivity scale $np=Ω(\log n)$, a polynomial-time semidefinite program exactly recovers all labels in a moderate-separation regime, whereas larger separation makes exact recovery impossible because isolated vertices appear with high probability. Our proofs combine orthogonal polynomial expansions, decoupling, and matrix concentration, avoiding the trace-moment arguments used in previous work.

Online learning in non-stationary streams is often formulated as tracking a point estimate, but many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and adversarial corruption. Our approach represents each candidate law through a latent cluster geometry: a variable-size configuration of centers that organizes probability mass and induces a predictive distribution. A Gibbs quasi-posterior over these configurations yields an online predictor by posterior averaging, and the resulting variable-dimensional posterior can be sampled with reversible-jump MCMC. The method therefore avoids specifying a parametric streaming law while retaining a structured latent space for uncertainty, regularization, and comparison. We evaluate performance by cumulative Wasserstein-1 regret against the time-varying true law. The analysis separates two effects: corruption perturbs the loss-based posterior update, whereas drift makes long-horizon posterior memory stale. We address the latter with a restarted variant that temporally localizes the same quasi-Bayesian update. The resulting high-probability bounds decompose into a PAC-Bayesian complexity term, a corruption-sensitive posterior perturbation term, and a dynamic optimal-transport term driven by $A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)$. Under bounded support, stable latent geometry, predictive-map regularity, oracle realizability, localized restart windows, sublinear transport action, and sublinear corruption budget, the restarted predictor achieves sublinear cumulative Wasserstein regret. These guarantees require no parametric model for the stream, drift mechanism, or corruption process.

Contrastive learning has become a leading paradigm for self-supervised representation learning, yet the conditions under which it recovers meaningful latent geometry remain incompletely understood. We develop a measure-theoretic framework formalizing the diversity condition, a support requirement on positive-pair sampling that is necessary for isometric latent recovery. We show that the standard full-support von Mises-Fisher setting implies the satisfaction of the diversity condition and as a consequence global contrastive loss minimizers recover latent geometry up to orthogonal transformation, while restricted conditionals can make non-orthogonal maps attain strictly lower asymptotic contrastive loss. We introduce a support-corrected Information Noise Contrastive Estimation (InfoNCE) variant as a theoretical fix: this correction makes orthogonal latent space recovery achievable but does not uniquely select it. Experiments on synthetic benchmarks validate the identifiability predictions, and CIFAR-10 experiments are consistent with the qualitative prediction that architectural inductive bias becomes more important when sampling diversity is limited. Together, our results clarify how sampling mechanisms and encoder inductive bias interact in contrastive representation learning.

Real-world dynamics shifts pose a critical challenge for reinforcement learning in robotics, as policies tightly coupled to nominal environments often fail catastrophically when physical conditions change. Most existing methods rely on encoding explicitly identified physical parameters into a latent context, a parameter-centric paradigm that depends on pre-specified axes of variation and becomes brittle under unmodeled or compound dynamics changes. We revisit dynamics adaptation from an outcome-centric perspective: rather than telling policies what the dynamics are, we enable them to learn how dynamics affect interaction outcomes. Theoretically, this is grounded in a monotonic relationship between target-domain regret and the Lipschitz constant of a trajectory dynamics encoder. Practically, this constant can be upper-bounded through contrastive learning, yielding a smooth, task-relevant latent topology without privileged dynamics information. On MuJoCo benchmarks, our method consistently outperforms parameter-centric baselines under severe dynamics shifts, including unmodeled and time-varying parameters, while also improving in-distribution stability and latent interpretability. Overall, these results validate that controlling latent geometry is a principled mechanism for robust adaptation.

Compact Vision Transformers are attractive for medical imaging in low-data and resource-constrained settings, but most existing variants assume that Euclidean latent geometry is sufficient for organizing image representations. We introduce hZACH-ViT, a family of curved-geometry extensions of ZACH-ViT, a compact zero-token Vision Transformer that removes positional embeddings and the class token and relies on global average pooling over patch representations. To isolate the role of geometry, we preserve the verified ZACH-ViT backbone and modify only the final representation space and prototype-based classifier head, enabling a controlled comparison between Euclidean, hyperbolic, and spherical latent geometries. We evaluate Poincaré, Klein, and spherical hZACH-ViT heads on seven MedMNIST datasets under an identical few-shot protocol with 50 samples per class and five random seeds. The completed benchmark contains 770 training runs spanning seven datasets, three non-Euclidean geometries, seven curvature magnitudes, and a Euclidean baseline. Across all seven datasets, the best non-Euclidean hZACH-ViT configuration improves over Euclidean ZACH-ViT, with an average gain of +0.021 in the dataset-specific primary metric and the largest improvement on OCTMNIST (+0.055 MacroF1). Fixed low-curvature configurations retain positive gains on the majority of datasets, and low curvature values (c = 0.1 or 0.2) account for six of the seven dataset-level winners. Rather than identifying a universally optimal manifold, our results establish geometry and curvature as dataset-dependent model-selection variables, with fixed low-curvature analyses confirming that gains persist beyond exhaustive per-dataset tuning.

Inference and control in engineered physical systems pay a heavy physics cost at deployment: state estimators, inverse-problem solvers, model-predictive controllers, schedulers, and observers are often not closed-form and must re-solve a numerical optimization per instance, with the operator re-supplied each time. Physics-informed learning moves this cost to training, but uses a single encoder pathway whose latent geometry de-learns under fine-tuning and admits no quantitative transfer guarantee. We propose an asymmetric two-pathway architecture that resolves both issues. A teacher encoder consumes privileged dense states from a high-fidelity simulator and represents the system through operator-polynomial features stable under spectral perturbation; a student encoder learns the same latent geometry from sparse field data and operator descriptors. At deployment the teacher is discarded, and the frozen student runs in a single forward pass with a transfer certificate. The design connects to privileged-information learning, knowledge distillation, and cross-modal distillation, but targets cross-instance transfer rather than fixed-instance prediction: topology and operator may change, while the latent task does not. We establish sufficient and near-necessary transfer conditions via Wasserstein proximity between latent laws, yielding a zero-shot error bound, and develop a finite-sample certification protocol with active expansion when coverage is incomplete. The framework applies wherever a system admits an operator with reportable spectrum. On power-system estimation, it achieves zero-shot transfer to 100 unseen topologies, a 95% certificate pass rate, accuracy competitive with topology-aware Newton--Raphson, and sub-millisecond inference. These results suggest asymmetric pathways plus operator-anchored latent geometry provide a foundation for certified zero-shot inference and control.

Geometry-aware generative models and novel view synthesis approaches have shown strong potential in visual fidelity and consistency. In parallel, equivariant representation learning has emerged as a powerful framework for constructing latent spaces where analytically known group transformations could act directly, capturing geometric structure in data and enhancing both interpretability and generalization in novel view synthesis. However, we identify that existing approaches often suffer from latent misalignment, a discrepancy between the intended group action and the actually required transformations in the latent space. Consequently, the learned latents often fail to consistently preserve the equivariant relations imposed by the underlying group symmetry. To address this, we propose Residual Latent Flow, a flow-based framework that corrects the misaligned latents, thereby improving compliance with the underlying equivariance relation. Our comprehensive experiments show that our method significantly reduces latent misalignment and improves novel view synthesis quality, under rotation groups SO(n).

Modern matrix completion problems often involve heterogeneous data whose rows simultaneously belong to many meta-categories, such as demographic and age groups in recommendation systems, or region and recording session labels in neural electrophysiological experiments. Standard low-rank estimators impose a single global latent geometry, which can recover average structure but may smooth away subgroup-specific variation, especially when observations are unevenly distributed across groups. We introduce Group-Aware Matrix Estimation (GAME), a convex estimator for overlapping subgroup-wise low-rank matrix estimation. GAME regularizes category-specific submatrices through overlapping nuclear-norm penalties, allowing related groups to borrow information while preserving local latent structure in a shared coordinate system. We provide finite-sample guarantees for both reconstruction error and subgroup-specific subspace recovery, showing how performance depends on sampling density, subgroup rank, and overlap structure. Experiments on synthetic, recommendation, ecological, and neuroscience datasets show that GAME is most beneficial in structured missingness regimes, where subgroup-aware regularization improves both reconstruction accuracy and latent subspace fidelity. Across these benchmarks, GAME is competitive or best among global low-rank, side-information, and modern imputation baselines, with the largest gains when subgroups exhibit distinct low-rank structure.

The Strong Platonic Representation Hypothesis suggests that representational convergence in artificial neural networks can be harnessed constructively: embeddings can be translated across models through a universal latent space without paired data. We ask whether an analogous geometry can be recovered across human brains. Using fMRI data from the Natural Scenes Dataset, we propose a self-supervised encoder that learns subject-specific embeddings from brain data alone by exploiting repeated stimulus presentations. We show that these independently learned spaces can be translated across subjects using unsupervised orthogonal rotations, without paired cross-subject samples or intermediate model representations. Synchronizing pairwise rotations into a single shared latent space further improves cross-subject retrieval, indicating that subject-specific spaces are mutually compatible with a common coordinate system. These results provide evidence for a shared neural geometry in the human visual cortex: subject-specific fMRI representations are approximately isometric across individuals and can be translated through purely geometric transformations.

Recent healthcare foundation models have achieved strong predictive performance through large scale self supervised learning, yet their latent representations frequently entangle physiologic severity, intervention intensity, observational structure, and institutional workflow into shared embedding directions. While effective for downstream prediction, such representations remain semantically opaque and unstable under contextual shift. We introduce AURORA, Adaptive Uncertainty aware Representations through Orthogonalized Relational Alignment, a new framework for healthcare representation learning based on contextual latent geometry. Rather than optimizing a single unified embedding manifold, AURORA decomposes representations into orthogonal semantic subspaces corresponding to distinct contextual factors and learns relational consistency objectives within each subspace. This induces latent spaces that are both semantically disentangled and geometrically interpretable. Across multiple clinical prediction and retrieval tasks, AURORA consistently outperforms reconstruction, contrastive, and self distillation baselines while substantially improving contextual disentanglement, neighborhood purity, and robustness under institutional distribution shift. Our results suggest that latent geometry itself constitutes an important axis of healthcare foundation model design and that explicitly structuring representation space according to contextual semantics provides a complementary direction beyond conventional predictive compression objectives.

Controllable multimodal generation is commonly formulated as an inference-time conditioning problem using prompts, guidance, or auxiliary modules. While effective, such approaches do not explicitly structure how semantic attributes evolve, which can lead to identity drift and inconsistent cross-modal behavior. We propose Controlla, a modular factorized-control framework that treats controllability as a property of structured latent geometry. Controlla learns identity and attribute factors from multimodal inputs and aligns them with graph priors using graph-constrained optimal transport, encouraging attributes to follow graph-consistent trajectories while preserving reference identity. To evaluate this setting, we construct AffectHuman-43K, a leakage-aware multimodal benchmark for reference-grounded affective control, and introduce geometry-aware metrics for trajectory consistency and latent disentanglement. Experiments show consistent improvements in controllability, identity preservation, and cross-modal alignment, with additional analyses on graph sensitivity, extensibility, and robustness.

Latent flow matching for image generation usually transports Gaussian noise to variational autoencoder latents along linear paths. Both endpoints, however, concentrate in thin spherical shells, and a Euclidean chord leaves those shells even when preprocessing aligns their radii. By decomposing each latent token into radial and angular components, we show through component-swap probes that decoded perceptual and semantic content is carried predominantly by direction, with radius contributing much less. We therefore project data latents onto a fixed token radius, use the radial projection of Gaussian noise as the spherical prior, finetune the decoder with the encoder frozen, and replace linear interpolation with spherical linear interpolation. The resulting geodesic paths stay on the sphere at every timestep, and their velocity targets are purely angular by construction. Under matched training, the method consistently improves class-conditional ImageNet-256 FID across different image tokenizers, leaves the diffusion architecture unchanged, and requires no auxiliary encoder or representation-alignment objective.

Dense 3D tracking from monocular video is fundamental to dynamic scene understanding. While recent 3D foundation models provide reliable per-frame geometry, recovering object motion in this geometry remains challenging and benefits from strong motion priors learned from real-world videos. Existing 3D trackers either follow iterative paradigms trained from scratch on synthetic data or fine-tune 3D reconstruction models learned from static multi-view images, both lacking real-world motion priors. Pre-trained video diffusion transformers (video DiTs) offer rich spatio-temporal priors from internet-scale videos, making them a promising foundation for 3D tracking. However, their frame-anchored formulation, which generates each frame's content, is fundamentally mismatched with reference-anchored dense 3D tracking, which must follow the same physical points from a reference frame across time. We present TrackCraft3R, the first method to repurpose a video DiT as a feed-forward dense 3D tracker. Given a monocular video and its frame-anchored reconstruction pointmap, TrackCraft3R predicts a reference-anchored tracking pointmap that follows every pixel of the first frame across time in a single forward pass, along with its visibility. We achieve this through two designs: (i) a dual-latent representation that uses per-frame geometry latents and reference-anchored track latents as dense queries, and (ii) temporal RoPE alignment, which specifies the target timestamp of each track latent. Together, these designs convert the per-frame generative paradigm of video DiTs into a reference-anchored tracking formulation with LoRA fine-tuning. TrackCraft3R achieves state-of-the-art performance on standard sparse and dense 3D tracking benchmarks, while running 1.3x faster and using 4.6x less peak memory than the strongest prior method. We further demonstrate robustness to large motions and long videos.

The paper studies the local geometry of embedding clouds induced by \emph{controlled local classes of semantically close sentences}. The central question is how controlled paraphrase-like semantic variation is organized in sentence embedding space and whether this local structure can be explicitly modeled by low-degree fitted carriers. We introduce a local geometric modeling scheme based on affine, quadratic, and cubic fitted models. We also use a surface-based latent probing procedure that constructs synthetic latent points in a reduced local PCA space with respect to the fitted carrier. The procedure is intended as an offline method for representation-space analysis, local manifold modeling, and geometry-aware latent probing. Generated latent points are evaluated using criteria that measure consistency with the fitted surface, preservation of neighborhood structure, agreement with the empirical distribution, stability of Hessian-based second-order shape descriptors, and stability of fitted-model coefficients. Experiments on controlled sets of semantically close sentences show that nonlinear local models describe embedding clouds more accurately than affine models. Surface-based generation provides strong fitted-geometry fidelity, including surface consistency, Hessian-based shape consistency, and coefficient consistency. Downstream experiments show that geometric validity of synthetic latent points does not automatically translate into improved classification performance. The results support explicit local geometric modeling of sentence embedding space and highlight the need to distinguish geometric validity from discriminative utility. As a resource contribution, we introduce \textbf{CoPaGE-300K}, a controlled template-based dataset of semantically close sentence variants with slot-level annotations and precomputed sentence embeddings.

Modern self-supervised representation learning methods often relies on empirical heuristics that are not theoretically grounded. In this study we propose HyDeS, a theoretically grounded method based on multi-view mutual information maximization within an hyperspherical space using Shannon differential entropy with a non-parametric von Mises-Fisher density estimator. We show that HyDeS bias the trained model towards focusing on foreground features of the images and perform well on segmentation tasks such as VOC PASCAL, while it lags in fine-grained classification. We provide a detailed analysis of the induced latent space geometry and learning dynamics, that can be used for designing other theoretically grounded self-supervised learning methods.

Stochastic dynamical systems with slow or metastable behavior evolve, on long time scales, on an unknown low-dimensional manifold in high-dimensional ambient space. Building a reduced simulator from short-burst ambient ensembles is a long-standing problem: local-chart methods like ATLAS suffer from exponential landmark scaling and per-step reprojection, while autoencoder alternatives leave tangent-bundle geometry poorly constrained, and the errors propagate into the learned drift and diffusion. We observe that the ambient covariance~$Λ$ already encodes coordinate-invariant tangent-space information, its range spanning the tangent bundle. Using this, we construct a tangent-bundle penalty and an inverse-consistency penalty for a three-stage pipeline (chart learning, latent drift, latent diffusion) that learns a single nonlinear chart and the latent SDE. The penalties induce a function-space metric, the $ρ$-metric, strictly weaker than the Sobolev $H^1$ norm yet achieving the same chart-quality generalization rate up to logarithmic factors. For the drift, we derive an encoder-pullback target via Itô's formula on the learned encoder and prove a bias decomposition showing the standard decoder-side formula carries systematic error for any imperfect chart. Under a $W^{2,\infty}$ chart-convergence assumption, chart-level error propagates controllably to weak convergence of the ambient dynamics and to convergence of radial mean first-passage times. Experiments on four surfaces embedded in up to $201$ ambient dimensions reduce radial MFPT error by $50$--$70\%$ under rotation dynamics and achieve the lowest inter-well MFPT error on most surface--transition pairs under metastable Müller--Brown Langevin dynamics, while reducing end-to-end ambient coefficient errors by up to an order of magnitude relative to an unregularized autoencoder.

We study neural architectures in which each hidden layer is defined by the stationary state of a dissipative Schrödinger-type dynamics on a learned latent graph. On stable branches, the local stationary problem defines a differentiable implicit graph layer. To learn the graph itself, we optimize over the stratified moduli space of weighted graphs and equip each stratum with a non-degenerate Kähler-Hessian metric that keeps natural-gradient descent and face crossing well posed. We then show that a multilayer stationary network is equivalent to an exact global stationary problem on a supra-graph, and that it admits a penalized global relaxation whose stationary states converge to the exact one as the penalty parameter tends to infinity. Reverse-mode differentiation is recovered as the adjoint of the exact global system, and the penalized adjoint converges to it in the same limit. Finally, under finite-dimensional strong-monotonicity and admissible-lift assumptions, the corresponding represented hypothesis classes coincide among resolvent feed-forward networks, graph-stationary networks, supra-graph stationary systems, and sheaf-based architectures with unitary connection. The resulting structural identifications yield complexity bounds controlled by sparse graph or supra-graph geometry rather than dense ambient connectivity.