Geodesic

Activation steering provides a lightweight way to control large language models (LLMs) by modifying their hidden activations at inference time. Among these approaches, norm-preserving steering aims to change model behavior without altering the activation norm, reducing the risk of representation collapse and degradation. However, existing norm-preserving methods are limited by predefined steering trajectories and by their reliance on one-step updates, which may fail to capture the complex structure of activation distributions. We propose GeoSteer, an optimization-based method for norm-preserving activation steering. GeoSteer formulates steering as a Riemannian optimization problem and updates activations through a sequence of small geodesic steps on the representation manifold. To avoid fixed steering directions, GeoSteer learns a nonlinear activation-space objective that distinguishes desired from undesired activations, and uses this function to adaptively guide each steering step. This multistep formulation yields smoother, more stable, and more consistent steering behavior while preserving the activation norm. Across TruthfulQA, RealToxicityPrompts, and UltraFeedback benchmarks, GeoSteer consistently improves over state-of-the-art activation steering baselines. These results suggest that norm-preserving steering can be made more effective by replacing predefined one-step edits with adaptive, geometry-aware optimization.

Generative diffusion models have emerged as a class of powerful techniques for various imaging applications, including but not limited to synthesis, reconstruction, and segmentation. Despite their success, current generative models pose two key limitations. First, they primarily rely on image intensity and texture information, with limited attention to underlying object geometry. As a result, they do not guarantee geometric or topological consistency during the generation process, which is a crucial requirement for high-stakes domains such as computational anatomy, biology, and robotics, where preserving object structure is critical. Second, existing models fail to explicitly learn or represent shape changes in the generative process. Such deformation dynamics remain occluded within network parameters; hence leaving the transformation process uninterpretable and physically uninformed. To address these challenges, we introduce IGG (Image Generation informed by Geodesic dynamics), a novel framework that integrates topology-preserving geodesic principles into the diffusion-based generative process. In contrast to conventional methods that operate in image intensity space, IGG learns and synthesizes diverse samples within geodesic deformation spaces, where geometric object changes are learned as smooth and invertible smooth mappings from a given template/source image. Our code is publicly available at https://github.com/nellie689/IGG.

3D Gaussian Splatting has recently revolutionised novel view synthesis as well as many other 3D vision methods and applications. Drawing inspiration from this representation, we now rethink textures to overcome the main issues of UV mapping while considerably lowering their memory footprint. Heat Kernel Textures (HKTex) eliminate UV unwrapping as well as their persistent issues of wasted UV space, seams, distortions, vertex-duplication, and varying resolution. Grounded in discrete Riemannian geometry and intrinsically defined on any manifold surface discretised as a triangular mesh, HKTex uses anisotropic heat kernels as geodesic equivalents to Gaussians. Like our kernels, also the optimisation of their position and the adaptive densification strategies were redefined to operate on the surface of the object to be textureised. Our novel representation is also fully integrated with a physically based renderer and can be optimised either from existing textures or multi-view images. Our project page and code are available at circle-group.github.io/research/HeatKernelTextures.

Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example, $k$-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the $k$-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.

We introduce cut-cell skinning, a geometric prior designed to augment data-driven skinning weight generation. While data-driven methods show promise in producing high-quality skinning weights, they often lack the generalizability of classic geometric approaches. To bridge this gap, we propose a geometric prior that can be robustly computed for in-the-wild meshes and is efficient for large-scale machine learning workflows. The key idea of our cut-cell skinning is a fast graph-based approximation of the volumetric geodesics distances, motivated by their importance in classic skinning weight computation. Our method achieves orders of magnitude speedup compared to optimization-based solvers and remains resilient to topological artifacts common in cage- or voxel-based alternatives. We demonstrate the efficacy of the cut-cell skinning prior by integrating it into recent neural skinning models, showing consistent improvements across existing methods and achieving state-of-the-art results. Project page: https://wenchao-m.github.io/CutCell.github.io/

This study introduces an advanced lateral control strategy for autonomous vehicles using a clothoid-based approach integrated with an adaptive lookahead mechanism. The primary focus is on enhancing lateral stability and path-tracking accuracy through the application of Euler spirals for smooth curvature transitions, thereby reducing passenger discomfort and the risk of vehicle rollover. An innovative aspect of our work is the adaptive adjustment of lookahead distance based on real-time vehicle dynamics and road geometry, which ensures optimal path following under varying conditions. A quasi-feedback control algorithm constructs optimal clothoids at each time step, generating the appropriate steering input. A lead filter compensates for the vehicle's lateral dynamics lag, improving control responsiveness and stability. The effectiveness of the proposed controller is validated through a comprehensive co-simulation using TruckSim and Simulink, demonstrating significant improvements in lateral control performance across diverse driving scenarios. Future directions include scaling the controller for higher-speed applications and further optimization to minimize off-track errors, particularly for articulated vehicles.

Predicting how a subcortical structure's shape will evolve from a few prior scans could support prognosis and clinical-trial enrichment. Existing longitudinal mesh predictors either extrapolate shape trajectories via high-dimensional embeddings or regress vertex deformations directly. We instead predict the surface's intrinsic geometry in continuous time: a single per-structure graph network predicts the future per-vertex first fundamental form (metric tensor) for an arbitrary causal multiple-visit history and an arbitrary prediction horizon, conditioned on a Fourier encoding of the lead time. The predicted metric is decoded into a surface by a differentiable As-Rigid-As-Possible solver, and the model is trained end-to-end on the rigid-aligned vertex error. Training through the reconstruction keeps the decoded prediction a valid surface and consistently improves it. On 14 subcortical structures from the ADNI dataset, the proposed mesh evolution model (MT-GNN) predicts best among the evaluated methods at every horizon ($-2.29\%$ mean vertex error vs. the temporal mean, $p{=}6.1{\times}10^{-5}$, beating it on 14/14 structures), ahead of geodesic shape regression (DCM, $-0.19\%$) and a mesh transformer (TransforMesh, $-0.45\%$; $p{=}1.2{\times}10^{-4}$), with the lead widening as the horizon grows.

Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential map. Every hidden state thus keeps a constant $\ell_{2}$-norm, confining the residual stream to a hypersphere. The exponential map, however, is only one member of a broad family of retraction maps. We show that on the hypersphere this entire family collapses to a single scalar design choice. What distinguishes one retraction from another is only how the magnitude of an update is converted into a rotation angle within the plane spanned by the hidden state and the update. This view places Euclidean residual connections and GeoNorm in one framework. Instantiating it with the metric projection retraction and the Cayley retraction yields Proj-SpheretNorm and Cay-SpheretNorm, which are exactly norm-preserving yet require only algebraic operations. Both prove to be members of a one-parameter family of angular retractions, $p$-SpheretNorm, whose rotation angle saturates rather than growing without bound. The two methods above are recovered exactly at $p = 1$ and $p = 2$, while the identity map and GeoNorm arise only as limits at either end. On nanoGPT, all three methods outperform existing lightweight deep connection schemes, and the best validation loss is attained at finite $p$, indicating that the exponential map is not the preferred retraction for spherical residual streams but merely one end of a spectrum.

The Word Problem has been a subject of intensive mathematical study for over a century, initially driving advances in combinatorial group theory and more recently emerging as a foundational hardness assumption in post-quantum cryptography (PQC). While generally undecidable, several families of infinite non-abelian groups exhibit solvable or algorithmically fast word problems, making them attractive platforms for cryptographic design. This paper introduces WPNet, a novel Graph Neural Network architecture capable of solving the Word Problem heuristically, which is demonstrated on the Baumslag-Solitar group $BS(1,2)$ and on an Artin group. By mapping unreduced words to dynamic graph structures, the model learns to cluster algebraically equivalent elements in a continuous embedding space, effectively identifying the geodesic representative of a word without executing discrete reduction steps. As an application, a model variant is developed that can predict the geodesic length of an unreduced word in both groups. To demonstrate the cryptographic severity of this structural leakage, WPNet is successfully deployed against the Wagner-Magyarik public-key cryptosystem.

Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.

Hierarchical Implicit Q-Learning (HIQL), an offline goal-conditioned RL method, selects subgoals by value-function advantages alone. This rule has two coupled failure modes. Optimistic bias treats lucky stochastic outcomes as skillful choices, and mode collapse reduces a multi-modal subgoal distribution to a single Gaussian mean that often falls in unreachable regions. We propose NFTR (Normalizing Flows subgoal policies with Triangle-slack Reweighting). A conditional Normalizing Flow replaces the Gaussian policy, and a closed-form mode-averaging result identifies NFs as the minimal generative class for AWR-based subgoal selection. A triangle slack score, built on the architectural triangle inequality without relying on distance accuracy, multiplicatively corrects the AWR weight to downweight subgoals whose detour cost exceeds average reachability. Triangle-slack vanishes on geodesics in deterministic MDPs and remains a conservative upper bound on composability violation under stochastic dynamics. The RWDR objective preserves AWR's population-level monotonic improvement and admits a three-term suboptimality decomposition. Together, these two ingredients yield subgoal selection that provably avoids the Gaussian collapse described above and remains stable under stochastic dynamics. GitHub page: https://github.com/erdemtbao/NFTR

Nonlinear least-squares optimization is central to regression, physics-informed neural networks, and other machine-learning tasks. Such problems have a natural geometric interpretation, model predictions form a manifold in data space, while the chosen parameterization can introduce parameter-effects curvature that becomes a dominant source of nonlinearity. This exposes a limitation of the Levenberg-Marquardt (LM) method, its tangent-space step is applied as a straight update in parameter coordinates. Geodesic acceleration gives a second-order correction, but its removal of parameter-effect curvature is exact only in the infinitesimal-step limit. We propose a Riemann-normal-coordinate Levenberg-Marquardt method (RNC-LM) to improve this consistency for finite optimization steps. By reformulating the geodesic equation, RNC-LM extends geodesic acceleration to arbitrary-order corrections and constructs finite-step updates with progressively higher reparameterization consistency. A line search along the resulting RNC curve controls the traveled distance while keeping the cost close to standard LM. The method eliminates the tangential component of residual acceleration order by order in a moving tangent frame, making the actual objective reduction more consistent with the linear model prediction of LM. On classical nonlinear least-squares benchmarks, RNC-LM improves convergence and robustness in curved valleys and rank-deficient problems. On a reaction-diffusion PINN failure-mode benchmark, it reduces the relative L2 error to the order of 1e-3 and recovers a physically meaningful solution. On a large-scale machine-learning potential-energy-surface fitting task, it achieves a 34-fold speedup over standard LM.

We introduce EntroPath, a manifold learning method that recovers geodesic geometry from data graphs through ensembles of diffusion paths. Many existing graph-based embeddings rely either on locally normalised random walks or on shortest-path distances. The former can concentrate diffusion in densely sampled regions, while the latter are sensitive to spurious shortcut edges in the graph. EntroPath instead builds its dissimilarities from the maximum entropy random walk (MERW), which aggregates the full ensemble of k-step paths between points rather than relying on any single trajectory. We show that the resulting free-energy dissimilarity converges to squared geodesic distance in the short-time limit, via Varadhan's heat-kernel formula. The diffusion depth k interpolates smoothly between local neighbourhood structure and global manifold geometry, and the symmetrised kernel admits an exact Gram factorisation connecting EntroPath to kernel methods. We further provide scalable extensions via landmark projection and diffusion-potential pseudotime. Across synthetic manifolds and single-cell benchmarks, EntroPath consistently matches or outperforms diffusion- and shortest-path-based methods, while remaining competitive with neighbourhood-preserving embeddings (UMAP, t-SNE) on local-structure metrics. Its gains are most pronounced on manifolds with non-uniform sampling density and well-separated branching trajectories, where path-ensemble diffusion more faithfully preserves the underlying geodesic geometry.

When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We observe that more generally, when $\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ for all $p \in \mathcal{M}$, there exists $K > 0$ such that $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with $D$ and discuss examples where $D$ is given by the Sinkhorn divergence on a family of probability measures parametrized by a manifold.

Conformal prediction provides distribution-free reliability guarantees for vision systems, but these guarantees depend on how prediction errors are measured in the output space. Many vision tasks produce outputs on curved spaces (e.g. gaze directions on the sphere or 3D head rotations), yet intermediate prediction heads, residuals, uncertainty estimates, or conformal scores are often defined in flat coordinate charts such as yaw-pitch or Euler angles. We show that this scoring choice introduces systematic geometric distortion near coordinate singularities (large pitch angles on the sphere and poses approaching gimbal lock in 3D rotations). Across four datasets (ETH-XGaze, Gaze360, BIWI, AFLW2000-3D), slice-conditional coverage at a nominal 90% target drops by 30-50 percentage points in these regions, falling to 38.9% on ETH-XGaze and 42.0% on Gaze360 at gaze pitch above 70 degrees, and to 57.5% on BIWI and 55.2% on AFLW2000-3D at head pose pitch above 60 degrees near gimbal lock, despite marginal coverage remaining near 90%. We prove that this is structural. Scalar thresholding changes the size of chart-coordinate prediction sets but leaves their distorted axis ratios unchanged. To diagnose this hidden failure mode, we show that a simple geometric quantity, the Riemannian volume density, strongly correlates with where coverage collapse occurs. Finally, we show that coordinate-free geodesic scoring removes this distortion. It requires no retraining and adds negligible computational cost.

The space $\mathcal{P}_2(\mathbb{R}^d$) of probability measures with finite second moment carries a natural geometry: the quadratic Wasserstein distance W_2 makes it a complete metric space and, following Otto, a (formal) Riemannian manifold whose geodesics are the optimal-transport interpolations. On this manifold, the gradient flow of the free energy F(rho) = KL(rho || π) is exactly the Fokker-Planck equation, and its implicit-Euler discretization is the JKO scheme. This is the geometry underlying diffusion models: the forward process descends the free energy, and each denoising step realizes one JKO step, which recovers DDPM, DDIM, NCSN/SMLD, and Energy Matching; this is one scheme, not separate theories. The same manifold supports a second variational principle. Its geodesics - the minimum-action curves of the Benamou-Brenier formula - are precisely the optimal-transport paths that Flow Matching learns. Fixing both endpoints and following the geodesic, generation becomes a deterministic ODE along a straight line, hence far fewer sampling steps. Placing both families of models on one manifold makes their relationship exact: diffusion follows a free-energy gradient flow, an initial-value problem; optimal-transport Flow Matching follows a Wasserstein geodesic, a boundary-value problem. The two reach the same endpoints along different paths.

We present an iterative algorithm to compute an arc-length parameterized spline interpolating a set of points. This differs from other methods where the computed spline either does not interpolate the original points or the parameterization is not the arc-length of the returned curves. Our method is applicable in any dimension $D \ge 2$, and we illustrate it with numerical results for plane curves.

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

Despite great advances, finding accurate segmentation remains a challenging task, especially in scenarios with cluttered backgrounds, complex intensity variations and topology appearance. Minimal path models have exhibited their strong ability in addressing image segmentation tasks. However, the performance of minimal paths-based segmentation approaches is heavily influenced by model initialization, hence limiting their application scope in practice. In this work, we propose a novel mask proposal voting framework that overcomes the major drawback of classical approaches, allowing robust segmentation even in complicated scenarios. Firstly, we introduce an efficient method for constructing adaptive domain cuts as a constraint for initializing the region-based min-cut evolution, by which diverse and reliable mask proposal candidates can be generated, substantially increasing the possibility of accurately covering the objective region by these proposals. Secondly, we propose a new mask voting scheme to build a voting score map encoding the final segmentation information. In contrast to classical path voting methods, our model allows incorporating priors to assign different importance to each individual mask. As a consequence, the proposed segmentation model is capable of accurately delineating object boundaries under complex scenarios, and is insensitive to initialization. Experiments demonstrate that our method consistently outperforms state-of-the-art minimal path-based approaches in both accuracy and robustness.

Curvature of planar curves serves as a key regularization term for computing second-order minimal paths, due to its tight relevance to desirable geometric properties such as smoothness, rigidity, and elasticity. In this paper, we tackle a more challenging problem in computational physics and geometry problem: tracking minimal paths whose curvature is constrained by arbitrary upper and lower bounds. For that purpose, we propose a new curvature-bounded geodesic model, developed under the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) framework. It provides strong geometric control over minimal paths by enforcing curvature range constraints, whose paths are smooth and of bounded curvature limitation. We also present a discretization scheme for the Hamiltonian and the HJB PDE incorporating curvature bounds, allowing efficient solver for estimating numerical solutions to the model. Finally, we illustrate the capability of the proposed curvature-bounded geodesic model in applications of robot path planning and curvilinear structures tracking from images. Numerical experiments demonstrate that the proposed curvature-bounded geodesic model serves as a powerful and robust tool for finding satisfactory paths.

We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.

Traditional change point detection in dynamic networks assumes abrupt transitions between stationary states, overlooking scenarios of continuous evolution which arise in most real-world applications, such as social networks or physical systems. We address this gap by formally defining regimes as periods of coherent dynamics in temporal graphs, which we characterize as trajectories along geodesics in a suitably defined graph space. This original perspective allows us to define regime changes as significant drifts in dynamics, either toward new trajectories or with pace changes. We leverage graph regression methods to measure the cumulative distance of sequences of observed graphs from the estimated geodesics between their endpoints, in the relevant graph space, which we can combine with change point detection algorithms. We present experiments on dynamic networks, with changing trajectories and varying speeds, in which we outperform state of the art change point detection models. Then, we analyse mobility data during the Covid-19 pandemic, and show that our assumptions on regular network evolution lead to change points that are more aligned to external events compared to the outcomes of baseline methods. Our work is the first to model and detect changes between evolving regimes in graph space, providing a realistic and powerful tool for analyzing complex temporal graph data.

We present the knowledge manifold: a Riemannian geometric space in which a corpus of documents is arranged according to semantic positional relationships derived from character n-gram TF-IDF representations. The framework proceeds in five tightly coupled stages. First, each document is converted to a character-level n-gram TF-IDF vector (4-7 grams, up to 250,000 features, L2-normalized) and embedded in a two-dimensional knowledge map via constrained stress minimization with repulsion, variance, and centering regularizers. Second, knowledge at an arbitrary query point is estimated through Smoothed Particle Hydrodynamics (SPH) interpolation using a cubic-spline kernel, yielding an interpolated TF-IDF feature vector that can be linguistically characterized. Third, directional knowledge gradients at 0, 45, and 90 degrees are computed from the SPH interpolation map, and pairwise directional similarity is quantified via inner product and cosine similarity. Fourth, a Gaussian Process Regression (GPR) model, with a Constant x RBF + White kernel fitted on a 10-dimensional SVD projection, provides a Bayesian posterior mean, uncertainty estimate, and per-document contribution rate at the query point. Fifth, geodesics in the knowledge space are obtained by minimizing a discrete Riemannian path energy derived from the SPH-induced metric tensor, using L-BFGS-B with seven deterministic initial-path candidates. We apply the formulation to a corpus of 20 papers in fiber-reinforced composite materials and aerospace structural mechanics, showing that the semantic map recovers meaningful research clusters, geodesic paths reveal natural conceptual bridges between distant topics, and SPH/GPR interpolation enables the generation of virtual knowledge: hypothetical paper abstracts describing unstudied but geometrically predicted research directions.

Learning from demonstratins (LfD) is usually performed over Euclidean spaces, while the robot state, e.g. orientation, naturally evolves over curved spaces. Therefore, to ensure natural, complex motion generation, we investigate learning from demonstrations over Riemannian manifolds that are capable of encoding both position and orientation data. Here, geodesic paths provide for natural motion between two arbitrary points within the manifold. We propose to numerically estimate geodesics via neural ordinary differential equations, mitigating large computational overhead of existing approaches. Finally, these geodesics can be decoded back into the original task space before deploying on the robot. In this extended abstract, we discuss the architecture of our framework, provide some initial insights from our simulation experiments, including comparison to other geodesic computation mechanisms, and discuss the challenges and prospects for future work.

Open-vocabulary 3D scene segmentation usually assumes RGB-D video, calibrated multi-view imagery, or a reconstructed mesh. GeoSAM-3D studies a lighter setting: a user uploads a short monocular video, clicks or names an object in one frame, and receives a propagated 3D mask over a Gaussian scene. The implementation combines frozen image and video foundation models with a monocular 3D Gaussian Splatting reconstruction and a differentiable graph-geodesic propagation kernel over Gaussian centroids. The central design choice is to propagate prompts by heat-kernel distance on the reconstructed scene graph, rather than by Euclidean nearest neighbors in 3D. This preserves continuity around curved surfaces and reduces leakage across nearby but disconnected objects. This paper describes the repository state, the mathematical kernel implemented in geosam3d.propagate, the feature head trained from Segment Anything masks, and the validation already present in the codebase. The evaluation protocol separates implementation validation, graph propagation quality, leakage control, and interactive latency.

Curvature-penalized geodesic models have proven their effectiveness in image segmentation by computing globally optimal curves. Unfortunately, these models remain susceptible to shortcuts when delineating objects with complex shapes and image intensity distributions, as they lack mechanisms to enforce shape-aware tangent constraints. To address this limitation, we propose a unified geodesic framework that integrates tangent-constrained priors with curvature penalization. The key idea is to formulate tangent admissibility directly within the orientation-lifted space, where path tangents are restricted to spatially varying angular sectors derived from intrinsic shape representatives (ISR) such as skeletons or interior landmarks. This formulation gives rise to a family of tangent-constrained Finslerian metrics, extending the classical curvature-penalized geodesic models while enforcing mandatory tangent constraints. The resulting Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) admit efficient numerical solutions via variants of the fast marching method, preserving the single-pass computational complexity. Experiments on synthetic, natural, and medical images demonstrate that the proposed geodesic framework indeed improves robustness against weak boundaries and topological shortcuts, yielding segmentation results with enhanced shape fidelity compared to existing geodesic models.

Reported retrieval scores for training-free shape descriptors conflate local signal design, normalization, aggregation, codebook fitting, and metric choices, making isolated component evaluation difficult. This paper reframes descriptor evaluation as a {\em protocol audit}. We introduce Diffused Geodesic Moments (DGM), a seed-conditioned descriptor that computes sparse implicit heat responses, converts them to distance-like fields, and summarizes each vertex by low-order moments across seeds and scales. DGM is used both as a practical non-spectral baseline and as an instrument for isolating protocol effects. On the registered FAUST benchmark split (FAUST-Reg) and the TOSCA shape collection, aggregation-matched experiments show that an independent Geometric Moment Shape Descriptor baseline built on Heat Kernel Signature features (GMSD-HKS) obtains the highest scores in this implementation ($0.621/0.820$ and $0.865/0.963$ mean average precision (mAP)/top-1), Wave Kernel Signature (WKS) remains a strong classical signal, and DGM is useful mainly when sparse solves, non-spectral deployment, or symmetry-informative seed frames are priorities. The broader finding is methodological: the input field and aggregation protocol can dominate the moment formula. The paper contributes a reproducible protocol-cascade analysis, a cross-shape alignment diagnostic for functional-map compatibility, and concrete recommendations for designing and reporting training-free shape descriptors.

Steering a language model - intervening on its internal activations to change downstream behaviour - has recently expanded beyond linear interpolation to nonlinear methods such as angular and kernelized steering, which define intervention transformations without learning an explicit geometry over paths in activation space. Freshly introduced geometry-aware manifold methods do learn such a geometry, but require labelled class centroids together with prescribed cyclic or sequential structure. These assumptions restrict where manifold steering can be applied, since existing constructions require labelled centroids and compatible boundary conditions. We recast manifold steering more broadly as \textbf{Riemannian geodesic computation} on activation space, recovering linear and labelled-spline steering as geodesics under particular choices of metric. A principled metric within this framework is the output-space Hellinger distance pulled back to activations; we approximate this with a learned encoder trained on output distances over a small concept-token schema - no per-prompt labels, no topology prior, and no per-task curve fitting. Empirically, the method reliably drives the model onto the target class across all tasks in a standard four-task language-model arithmetic benchmark, while following more behaviourally natural trajectories than baselines on smaller output spaces. We thereby provide a unified Riemannian framework for manifold steering together with a schema-supervised, label-free instantiation that operates without labelled centroids or prescribed boundary conditions.

Inferring time-varying graph structures from high-dimensional nodal observations is a fundamental problem arising in neuroscience, finance, climatology, and beyond. Two intrinsic challenges govern this problem: maintaining the \emph{temporal coherence} of the latent graph across successive observation windows, and respecting the \emph{intrinsic Riemannian geometry} of the symmetric positive definite manifold on which precision matrices naturally reside, a curved space whose geodesic structure departs fundamentally from that of the ambient Euclidean space. In this paper we propose dynamic estimation on the Grassmann manifold with a factor model (\textsc{Degfm}), a novel algorithm that jointly addresses both challenges. We model the time-varying precision matrix sequence as a low-rank-plus-diagonal structure governed by a latent elliptical graph factor model, which drastically reduces the effective parameter count and enables reliable estimation in the challenging small-sample regime. Temporal coherence is enforced through a Riemannian geodesic penalty defined on the Grassmann manifold, ensuring that the estimated graph trajectory is smooth with respect to the intrinsic geometry rather than the ambient Euclidean space. To solve the resulting non-convex optimization problem over Grassmann-manifold-valued sequences subject to the LRaD constraint, we derive an efficient Riemannian gradient descent algorithm that respects the manifold structure at every iterate and rigorously establish its convergence to a stationary point. Extensive experiments on both synthetic benchmarks and real-world datasets demonstrate that \textsc{Degfm} consistently outperforms state-of-the-art baselines across all evaluation metrics, confirming the practical effectiveness of the proposed framework.

Few-Shot Medical Image Segmentation (FSMIS) aims to delineate novel anatomical targets from one or a few annotated support images, addressing the annotation scarcity in medical imaging. Notwithstanding recent advancements, current prototype-based methods are bottlenecked by two coupled limitations: 1) cue entanglement, where a single spatial-domain prototype is forced to summarise organ silhouette, parenchymal texture and boundary appearance simultaneously, so any support-query mismatch on one cue propagates indiscriminately to the others; and 2) topology-blind matching, where cosine similarity measures distance in the ambient Euclidean space and ignores the connectivity of the underlying feature manifold, causing fragmented activations inside low-contrast organs and leakage into neighbouring tissues. To this end, we propose Spectral-Geodesic Prototype Network (SGP-Net), built around a Spectral-Geodesic Prototype Module with two coupled components. A Spectral Prototype Bank (SPB) decomposes support and query features into low-, mid- and high-frequency bands via learnable radial Fourier filters, yielding three disentangled prototypes per class that separately encode shape, texture and boundary cues. A Geodesic Matcher (GM) then replaces cosine similarity with a differentiable heat-diffusion approximation of geodesic distance, propagating matching signals along a feature affinity graph so that on-manifold pixels accumulate consistent responses while off-manifold look-alikes are suppressed. Experiments on three public FSMIS benchmarks demonstrate that SGP-Net achieves competitive performance against recent state-of-the-art methods.

Vision-language models like CLIP have demonstrated remarkable zero-shot transfer capabilities. However, their susceptibility to imperceptible adversarial perturbations remains a critical security concern. While test-time defenses offer a pragmatic solution for deployed models, existing approaches typically rely on gradient-based optimization during inference, incurring significant computational overhead. In this paper, we revisit the role of data augmentation in CLIP robustness and observe that augmentations are not equally effective: specific augmentations consistently provide robust geometric cues that align with correct class semantics in the hyperspherical feature space. Based on this, we propose Adaptive Geodesic Correction (AGC), a training-free defense mechanism that requires no parameter updates. AGC identifies a reliable augmentation as a geometric anchor and corrects the input feature towards it, utilizing an adaptive step size to balance robustness against clean accuracy preservation. AGC achieves superior performance across eight fine-grained datasets and three CLIP backbones, improving average robust accuracy by 44.4% over state-of-the-art baseline while delivering a 10$\times$ reduction in inference latency. Our findings reveal a fundamental geometric property of CLIP features, offering a highly efficient and effective paradigm for robust multimodal deployment.

Recent advances have markedly improved the cross-scene generalization of relative depth estimation, yet its practical applicability remains limited by the absence of metric scale, local inconsistencies, and low computational efficiency. To address these issues, we present \emph{\textbf{M}idas \textbf{T}ouch for \textbf{D}epth} (MTD), a mathematically interpretable approach that converts relative depth into metric depth using only extremely sparse 3D data. To eliminate local scale inconsistencies, it applies a segment-wise recovery strategy via sparse graph optimization, followed by a pixel-wise refinement strategy using a discontinuity-aware geodesic cost. MTD exhibits strong generalization and achieves substantial accuracy improvements over previous depth completion and depth estimation methods. Moreover, its lightweight, plug-and-play design facilitates deployment and integration on diverse downstream 3D tasks. Project page is available at https://mias.group/MTD.

Video anomaly detection (VAD) aims to automatically identify events that deviate from normal patterns in untrimmed surveillance videos. Existing methods universally depend on large-scale annotations or task-specific training procedures, severely limiting their rapid deployment to novel scenes. We observe that intermediate-layer features of pre-trained multimodal large language models (MLLMs) already encode rich anomaly semantics, yet existing approaches rely on the language output pathway and fail to exploit the geometric discriminability latent in these representations. Based on this finding, we propose SphereVAD, a fully training-free, zero-shot VAD framework that recasts anomaly discrimination as von Mises-Fisher (vMF) likelihood-ratio geodesic inference on the unit hypersphere, unleashing latent discriminability through principled geometric reasoning rather than learning new representations. Specifically, SphereVAD first applies Frechet mean centering to unfold feature distributions and eliminate domain biases, then employs Holistic Scene Attention (HSA) to reinforce feature consistency using cross-video priors, and finally performs vMF-guided Spherical Geodesic Pulling (SGP) to align ambiguous segments with directional prototypes on the spherical manifold. This training-free pipeline requires only minimal synthetic images for calibration. SphereVAD establishes new state-of-the-art results among training-free approaches on three major benchmarks and remains competitive with fully supervised baselines. Code will be available upon acceptance.

Text-to-image diffusion models can generate visually stunning images, yet, controlling what appears and how it appears, remains surprisingly difficult, especially when operating solely within the constraints of the text-conditioning space. For example, changing a subject or adjusting an attribute often leads to unintended side effects, such as altered backgrounds or distorted details. This is because most existing text-based control methods treat the embedding space as Euclidean and apply simple linear transformations, which do not reflect how semantic concepts are actually organized. In this work, we take a step back and ask: what is the true geometry of these embeddings? We find that text encoder representations lie on a hypersphere, where concepts are not linear directions but structured, anisotropic distributions better captured by Kent distributions. Building on this insight, we propose HEART, a training-free framework that performs Kent-aware geodesic transformations directly on the hypersphere. By respecting the underlying geometry, HEART enables intuitive and precise edits, such as consistent subject replacement and fine-grained attribute control, while preserving the original scene. Importantly, HEART requires no finetuning, inversion, or optimization, and generalizes across diffusion model architectures. Our results show that a simple shift in perspective, from linear to spherical, can unlock fast, and controllable image generation.

We present the E$Δ$-MHC-Geo Transformer, a novel architecture that unifies Manifold-Constrained Hyper-Connections (mHC), Deep Delta Learning (DDL), and the Cayley transform to obtain input-adaptive, unconditionally orthogonal residual connections. Unlike DDL, whose Householder operator is orthogonal only at $β\in \{0,2\}$, our Data-Dependent Cayley rotation $Q(x)=(I+(β/2)A(x))^{-1}(I-(β/2)A(x))$ preserves orthogonality for all $β$ and all inputs. To handle negation, an eigenvalue $-1$ case that Cayley provably excludes, we introduce the E$Δ$-MHC-Geo Hybrid, which combines Cayley rotation with Householder reflection via a learned operator-selection gate $X'=γ(X)Q(X)X+(1-γ(X))H_2(X)X$. A midpoint-collapse regularizer, $4γ(1-γ)$, encourages boundary gate decisions, where each selected component is orthogonal. In matched-parameter comparisons, with approximately 1.79M parameters per model and mean +/- standard deviation over 3 seeds, against four baselines including the concurrent JPmHC, E$Δ$-MHC-Geo achieves the best long-horizon stability, 1.9x over JPmHC and 3.8x over GPT; the best near-$π$ rotation loss, 4.5x over JPmHC on single-plane; strong norm preservation, with 0.001 mean deviation; and 0.96 negation cosine alignment in a diagnostic reflection probe, all with 33% fewer layers. While JPmHC's wider representation excels on pure rotation, its finite Cayley residual mixer excludes an exact $λ=-1$ operator and has no reflection branch, motivating our hybrid approach for accessing both connected components of $O(n)$.

Vector search systems built on frozen vision encoders face queries from unseen classes at deployment, yet existing adapter training collapses under this shift: high-capacity adapters with global contrastive losses silently reassign unseen-class samples to wrong seen-class clusters, dropping worst-case Label Precision by over 40 points below the frozen baseline in our tests. We propose Euclidean Geodesic Alignment (EGA), a residual adapter that couples three principles: zero initialization, local triplet loss, and hypersphere projection. These collectively induce a self-limiting dynamic: triplets that already satisfy a small margin stop producing gradients, so the adapter automatically stops updating where the local geometry is already correct. Our experiments show that at convergence $96.5\%$ of triplets are gradient-free, leaving unseen-class regions largely untouched while still enabling full-capacity refinement of seen classes. Across five diverse out-of-distribution (OOD) benchmarks, EGA achieves the highest worst-case Label Precision on the four primary splits and a consistent improvement on the fifth. The design also transfers to stronger backbones in addition to CLIP, and we provide an analytical justification linking gradient sparsity to bounded OOD perturbation.

\We introduce the horospherical depth, an intrinsic notion of statistical depth on Hadamard manifolds, and define the Busemann median as the set of its maximizers. The construction exploits the fact that the linear functionals appearing in Tukey's half-space depth are themselves limits of renormalized distance functions; on a Hadamard manifold the same limiting procedure produces Busemann functions, whose sublevel sets are horoballs, the intrinsic replacements for halfspaces. The resulting depth is parametrized by the visual boundary, is isometry-equivariant, and requires neither tangent-space linearization nor a chosen base point. For arbitrary Hadamard manifolds, we prove that the depth regions are nested and geodesically convex, that a centerpoint of depth at least $1/(d+1)$ exists, and hence that the Busemann median exists for every Borel probability measure. Under strictly negative sectional curvature and mild regularity assumptions, the depth is strictly quasi-concave and the median is unique. We also establish robustness: the depth is stable under total-variation perturbations, and under contamination escaping to infinity the limiting median depends on the escape direction but not on how far the contaminating mass has moved along the geodesic ray, in contrast with the Fréchet mean. Finally, we establish uniform consistency of the sample depth and convergence of sample depth regions and sample Busemann medians; on symmetric spaces of noncompact type, the argument proceeds through a VC analysis of upper horospherical halfspaces, while on general Hadamard manifolds it follows from a compactness argument under a mild non-atomicity assumption.

Composed Image Retrieval (CIR) aims to retrieve target images by integrating a reference image with a corresponding modification text. CIR requires jointly considering the explicit semantics specified in the query and the implicit semantics embedded within its bi-modal composition. Recent training-free Zero-Shot CIR (ZS-CIR) methods leverage Multimodal Large Language Models (MLLMs) to generate detailed target descriptions, converting the implicit information into explicit textual expressions. However, these methods rely heavily on the textual modality and fail to capture the fuzzy retrieval nature that requires considering diverse combinations of candidates. This leads to reduced diversity and accuracy in retrieval results. To address this limitation, we propose a novel training-free method, Geodesic Mixup-based Implicit semantic eXpansion and Explicit semantic Re-ranking for ZS-CIR (G-MIXER). G-MIXER constructs composed query features that reflect the implicit semantics of reference image-text pairs through geodesic mixup over a range of mixup ratios, and builds a diverse candidate set. The generated candidates are then re-ranked using explicit semantics derived from MLLMs, improving both retrieval diversity and accuracy. Our proposed G-MIXER achieves state-of-the-art performance across multiple ZS-CIR benchmarks, effectively handling both implicit and explicit semantics without additional training. Our code will be available at https://github.com/maya0395/gmixer.

Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein space, induced by the optimal transport map. Compared with plain networks, ResNets can better approximate the geodesic curve, which explains why ResNets can be optimized and generalize better. Numerical experiments show that the data tracks of a ResNet tend to be line-shaped in terms of the line-shape score, and the map learned by a ResNet is closer to the optimal transport map in terms of the optimal transport score. In a word, we conclude that ResNets learn the geodesic curve in the Wasserstein space and discretely engineer the data transformation in high-dimensional spaces.