Riemannian Manifolds

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations -- a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

Scaling deep learning faces critical bottlenecks: data exhaustion, exponential training costs, and resource concentration. Model merging combines pre-trained checkpoints without gradient descent, offering orders-of-magnitude savings versus retraining. Combining independently trained vision models is difficult when their architectures and parameter shapes differ. Existing weight-space merging methods generally assume aligned, shape-compatible checkpoints, whereas a Vision Transformer (ViT) and a state-space model (SSM) implement token mixing with different operators. We study a hybrid Heterogeneous merging setting that retains both architectures while aligning parameter groups by semantic role. Our proposed Riemannian--Lorentz Parameter Fusion (RLPF) method projects aligned groups to common coordinates, lifts selected coordinates to the Lorentz hyperboloid model of hyperbolic space, computes a regularized geodesic barycenter, and decodes the result into the two branches. A learned gate then combines branch logits for each input. Component groups use fixed curvature values, with normalization parameters treated as Euclidean. In the results available in this manuscript, the fine-tuned system obtains 82.37% on CIFAR-10, 75.04% on Oxford-IIIT Pet, and 78.58% top-1 accuracy on ImageNet-1K; the corresponding best-parent accuracies are 76.54%, 71.42%, and 76.42%. On ImageNet-1K, the reported pre-fine-tuning initialization reaches 77.80%. These results support further study of geometry-aware heterogeneous fusion, but not a training-free single-checkpoint merge: RLPF is a two-branch hybrid whose gate and reported final models are trained.

The query and key projections $\WQ,\WK$ in attention are almost always trained by Euclidean optimizers with no constraint on their geometry. We constrain them to the Stiefel manifold and optimize them there with a Riemannian Adam that carries one scalar second moment per frame, caps its step by a trust region, and retracts polarly. Four propositions prove this update is steepest descent in the embedded metric, independent of gradient scale, well conditioned, and exactly $\mathrm{O}(d)$-equivariant, each certified numerically in \texttt{float64}. A fifth supplies the mechanism: weight decay has \emph{identically zero} Riemannian gradient on $\St(d,r)$, since $W = W I_r$ lies in the normal space, so the learned attention geometry survives the collapse cycles that decay drives through the rest of the model. On modular arithmetic grokking, a single run holds $97.0\%$ validation accuracy at epoch 20,000 against the baseline's $61.1\%$---an unstable endpoint we report as evidence for the mechanism rather than as an effect size. On CIFAR-10 patches the same rule gains $\mathbf{+8.98}$,pp over 12 paired starts ($t{=}60.6$, $12/12$), and the gap widens with data rather than eroding. The step rule earns this: a fixed-step Riemannian update is degree one in the gradient, so it moves $24$--$40\times$ less per step than an identically shaped AdamW matrix---its frames barely leave their initialization, and freezing them outright costs only $0.28$,pp. An ablation credits the whole gain to making the step scale free, and nothing measurable to the projector or to equivariance. A negative result sharpens the account: gauge removal cannot motivate the method, because a direction along which the loss is invariant carries no gradient at all.

The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on $(p,q)$ only through the exponent $r=p+q$. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member $p=q=r/2$ attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune $r$ for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning $r$. A task covariance example shows a further gain from tuning the shape.

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the $\mathbb{SL}(n)$ space, a representation geometry defined by the simple $\det(A)=1$ constraint and a left invariant Schatten-$p$ Finsler structure. Despite this minimal construction, $\mathbb{SL}(n)$ exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically, $\mathbb{SL}(n)$ consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by $44.3\%$ on KEGG and $40.5\%$ on HumanCyc, and improves Hits@20 by $42.8\%$ on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

Recently, researchers have proposed a graph pooling operation, akin to the pooling process in conventional convolutional neural networks (CNN), aimed at reducing the computation cost of Graph convolutional neural networks (GCNNs). While most GCNN-based methods treat graph pooling as a node clustering problem and propose learning a cluster assignment matrix, existing clustering-based pooling methods tend to focus solely on the rough topology information of graphs, neglecting the exploitation of higher-order mutual connections among neighbors. In terms of message passing on graph, the ease of information passing on edges reflects the closeness between neighboring nodes, which significantly relies on the interconnectivity among neighbors. In this study, we address this gap by considering such local connection information and introducing a novel graph pooling method named RicciPool. We introduce discrete graph curvature, particularly Ollivier-Ricci curvature, as a measure of higher-order connectivity around an edge. Subsequently, we construct an Ollivier-Ricci flow formula to reweigh edge weights, leveraging the crucial information provided by Ricci curvature, particularly vital for extracting clusters in graphs. Building upon this foundation, we utilize the spectral clustering technique to learn a new cluster assignment matrix. Experimental results on multiple bioinformatics protein datasets and social networks underscore the effectiveness of our proposed method.

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.

This paper introduces rlaopt, a PyTorch-based package for large-scale optimization and scientific computing using randomized numerical linear algebra (RandNLA). Despite substantial progress in RandNLA-based algorithms, few implementations combine GPU acceleration with a simple interface for specifying optimization problems. rlaopt addresses this gap by providing GPU-enabled solvers for positive-definite linear systems and convex empirical risk minimization with constraints and regularizers. These solvers use RandNLA to accelerate conjugate gradient (NystromPCG), operator splitting (NysADMM), and stochastic gradient methods (SAPPHIRE). Moreover, rlaopt includes a modeling language that lets users specify problems using natural mathematical syntax. rlaopt automatically checks compatibility with the selected solver and performs the required problem decomposition. The solvers also support differentiation through their iterations, enabling applications such as hyperparameter tuning. Experiments on ridge regression, bounded multinomial logistic regression, and bounded elastic net identify when randomized preconditioning improves performance and demonstrate substantial speedups from GPU execution. The package is open-source under an Apache license, with source code at https://github.com/udellgroup/rlaopt and version 0.1.0 available on PyPI.

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.

Quantum game theory is an extension of classical game theory that uses quantum principles in game theory. The Eisert-Wilkens-Lewenstein (EWL) quantum game is an early example of the two-player classical Prisoner's Dilemma transformed into a quantum Prisoner's Dilemma. In the EWL game, the players choose pure quantum strategies represented by unitary matrices. This extension can resolve the classical dilemma by enabling cooperative equilibrium with higher payoff. In this paper, we first discuss the Extended EWL (EEWL) for multiplayer quantum games with mixed strategies. In EEWL, each player controls a set of unitary operators as quantum actions and uses a classical mixed strategy over these actions. The payoffs are defined as expectation values of Hermitian reward operators acting on a shared quantum state, which is generated and measured according to the EEWL protocol. We then propose the Unitary Strategy Matrix Exponential Algorithm (USMEA), a geometry-aware sequential algorithm for the EEWL mixed-strategy setting, in which each player jointly learns a trainable set of local unitary actions and the associated classical mixing probabilities. Thereby it acts as a learning-and-control layer for multi-agent quantum decision systems. We analyze the convergence properties of USMEA under standard smoothness and step-size conditions and validate the theory with numerical experiments. These results show how classical optimization methods can be systematically integrated into the design and analysis of engineered quantum strategic interactions.

Tensor networks, originally developed for quantum many-body physics, are promising models for machine learning. We derive stochastic Riemannian optimizers for tree tensor networks (TTNs) on both their parameter and quotient manifolds, including adaptive and learning-rate-free schemes suitable for minibatch training. Using a hybrid CNN-TTN architecture, we evaluate the methods on Fashion-MNIST, CIFAR10, and Imagenette. The proposed optimizers achieve predictive performance comparable to unconstrained optimization while enabling numerically stable downstream compression.

Activation steering has emerged as a lightweight approach for controlling model refusal at inference time. A growing line of research explores trainable rotations of activations to develop geometrically principled intervention mechanisms. However, existing techniques rely on auxiliary constructs, such as refusal vectors, to define these rotations. In our work, we develop a self-contained methodology for learning parameter-efficient rotational transformations based on Riemannian optimization. We empirically validate the proposed scheme, demonstrating its superiority in intervention efficiency. An extensive ablation study highlights the importance of key design choices in our method. Our results identify the proposed rotation-based steering scheme as a promising direction for more reliable control over the behavior of LLMs.

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/

Cross-subject electroencephalogram (EEG)-based emotion recognition remains challenging due to substantial inter-individual variability and discrete formulation that overlooks affective continuity. Existing methods operate in Euclidean space and focus on marginal distribution alignment, failing to preserve the semantic structure of emotions across subjects. This article proposes MGMCL, reconceptualizing emotion recognition as learning continuous representations on symmetric positive definite (SPD) Riemannian manifolds. The frame?work introduces multi-granularity manifold contrastive learning at instance, emotion, and trajectory levels while preserving semantic ordering. Neural ordinary differential equations on manifolds model continuous emotion dynamics. Cross-subject generalization employs Gromov-Wasserstein manifold alignment. Weakly-supervised learning enables continuous valence-arousal-dominance prediction from discrete labels. Extensive experiments on three public datasets demonstrate state-of-the-art performance: 91.23% accuracy on SEED, 73.82% on SEED-IV, and 76.38% on DEAP, achieving consistent improvements of 1.89%, 1.66%, and 1.28% over previous best methods, respectively.

Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.

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.

Temporal graph learning has become essential for analyzing real-world systems whose interactions continuously evolve over time, including financial transaction networks, communication systems, and online social platforms. However, learning from large-scale temporal graphs remains computationally challenging when networks are dense and rapidly changing. To address this limitation, we propose a network-curvature-inspired edge sparsification framework for dynamic graph learning. Our proposed method, TRicci, extends classical Forman-Ricci curvature to directed weighted temporal graphs by capturing structural support, temporal recency, and local interaction competition. Experiments on 9 transaction networks and 3 temporal graph benchmark datasets demonstrate that the proposed framework preserves predictive performance across multiple graph-level prediction tasks. The results show that TRicci sparsifies temporal graphs by approximately 80% while reducing end-to-end downstream training and inference time by an average of 55.94%, without substantial degradation in predictive performance. Our findings suggest that temporal curvature can serve as a principled basis for scalable temporal graph learning by preserving predictive temporal-structural information under substantial sparsification.

This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA's effectiveness for both 2D contour and 3D shape analysis.

We study Muon, a recently proposed matrix-aware optimization method, in the context of the Stiefel manifold. This manifold consists of matrices with orthonormal columns and is ubiquitous in machine learning and scientific computing. Existing extensions of Muon to this manifold rely on heuristic, approximate, or iterative updates with varying computational efficiency. We show that the corresponding Stiefel Muon update admits an exact closed-form solution and use this result to develop Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation. We further establish first-order convergence guarantees for Skewon in the smooth non-convex setting.

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.

Optimal trajectory generation for rigid-body motions on Lie groups can be formulated as a variational problem that minimizes energy functionals defined by Riemannian metrics. While closed-form solutions exist for special cases such as product metrics and rest-to-rest boundary conditions, solving the general problem with arbitrary boundary states and coupled rotational-translational metrics often requires computationally expensive numerical boundary value solvers. These limitations restrict the use of geometrically consistent trajectory generation in real-time robotic planning and control. This paper presents a learning-based framework for approximating higher-order smooth trajectories on SE(3) under general left-invariant Riemannian metrics. The method parameterizes body-twist trajectories using high-order polynomials and relies on a neural network to learn a subset of the polynomial coefficients and the trajectory duration. The remaining coefficients are analytically determined to enforce the boundary conditions. The training of the network is guided by losses derived from Euler-Lagrange optimality conditions, metric-weighted smoothness objectives, and feasibility constraints. The metric-conditioned framework enables generalization across diverse metric structures and motion conditions. Extensive numerical experiments demonstrate that the proposed approach generates smooth trajectories that closely approximate solutions from numerical optimization while achieving millisecond-level inference times. We demonstrate two practical applications of the proposed framework: real-time generation of diverse motion primitives with waypoint traversal, and refinement for quadrotor flight under dynamic conditions. These results suggest that learning-based motions with geometric structure can provide an efficient alternative to conventional optimization-based methods for trajectory generation on SE(3).

All Transformer-based large language models compute attention via the Euclidean inner product, an architectural choice that Dong et al. (2021) proved causes representational rank to decay doubly exponentially with depth in pure self-attention stacks. We develop a theoretical framework that targets this structural limitation at the mathematical level by replacing the flat Euclidean metric with learned per-token Riemannian metrics. Our contributions are threefold. (1) We prove that Riemannian attention scores with heterogeneous per-token metrics are non-Gram---they cannot be factorized as QK^T with factorization dimension O(d). We are explicit that this is a structural observation, not a proof of rank preservation. (2) We establish that low-rank metric factors render all geometric operations tractable: geodesic distance in O(dr) per token and metric inversion in O(dr^2) via the Woodbury identity---both far below the O(d^3) cost of a general matrix---making Riemannian attention feasible at billion-parameter scale with negligible overhead. (3) We present the Fiber Bundle Transformer, a complete architecture specification in which each token position carries its own Riemannian metric, attention is geodesic distance computation, feed-forward updates use metric-preconditioned steps, and the connection carries explicit curvature and torsion proxies. We derive formal predictions about correctly implemented geometric architectures and identify the central open problem: proving or disproving that heterogeneous Riemannian metrics prevent the rank collapse that row-stochastic attention matrices otherwise cause. This paper presents theoretical analysis and architectural design; empirical validation is the subject of future work.

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.

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.

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.

Optimization over the Stiefel manifold plays a significant role in various machine learning tasks. Existing methods either use the retraction operators, requiring costly orthonormalization for large-scale matrices, or employ landing methods that rely on careful step size selection and penalty parameter tuning. To address these challenges, we propose a retraction-free and penalty parameter-free algorithm that directly lands on the manifold. By leveraging the strongly-convex-like property of the quadratic penalty function and the proximal smoothness of the Stiefel manifold, we establish global convergence guarantees with the best-known iteration complexities under both constant and diminishing step sizes. Then, we reformulate the low-rank adaptation (LoRA) fine-tuning problem for large language models as a manifold optimization problem, introducing Manifold-LoRA for geometry-accelerated adaptation. This approach employs the proposed landing technique and a carefully designed step size strategy to accelerate the training process. Numerical experiments on benchmark datasets demonstrate the efficiency and strong downstream performance of the proposed method.

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.

Point cloud denoising is essentially a geometric recovery task that aims to reconstruct the intrinsic structure of a smooth 2D Riemannian manifold embedded in R^3 from noisy, discrete ambient-space samples. Despite the remarkable progress of modern manifold-aware encoders and generative transport models in geometric representation learning, a fundamental objective-geometry mismatch remains underexplored. Theoretically, we identified that this mismatched coupling leads to geometric gradient interference, where conflicting optimization objectives result in structural degradation and point clustering. We introduce Riemannian Orthogonally Decoupled Regularization (RODR) to reformulate the optimization trajectory by disentangling the normal (fitting) and tangential (distribution) components. Guided by a vector-attention and entropy-aware adaptive strategy, RODR effectively preserves high-fidelity geometric details while maintaining sampling uniformity. Experiments demonstrate that RODR reaches performance comparable to state-of-the-art baselines and suggests improved distribution regularity and reduced local aggregation effectively. Our work establishes a generic and interpretable framework for disentangled geometric optimization in point cloud processing.

This paper develops an online, off-policy policy-iteration framework for reinforcement learning (RL), based on sparse Gaussian-mixture-model Q-functions (S-GMM-QFs). The framework reconciles streaming, non-stationary data with the Riemannian structure of the parameter space while handling distributional mismatch through experience replay. S-GMM-QFs are introduced via Hadamard overparametrization, enabling interpretable sparsification through smooth regularization that facilitates Riemannian-based optimization. Overparametrization allows the framework to adaptively identify meaningful components from a large initial pool, yielding sparse models where interpretability emerges naturally from geometry: each component's parameters (means and covariances) explicitly encode its geometric role in the ambient state-action space. These geometric roles are learned through online gradient descent on a smooth objective over a (Cartesian-product) Riemannian manifold. Numerical tests demonstrate that S-GMM-QFs match or exceed deep RL methods while using substantially fewer parameters and achieving faster improvement per observed transition. Notably, parameter efficiency and interpretability combine to maintain strong generalization in low-parameter regimes where sparsified deep RL approaches degrade.

Energy natural gradient descent (ENGD) aligns parameter updates with the curvature of an underlying function-space energy, but existing formulations assume an unconstrained Euclidean parameter domain. We introduce \EMNGDfull{}, a manifold optimization framework for physics-informed and variational neural PDE solvers whose parameters lie on a Riemannian manifold. EMNGD restricts the energy-induced quadratic model to feasible tangent directions and uses retractions to preserve parameter constraints throughout optimization. Under coercivity, we prove that the push-forward of the undamped EMNGD direction is the best feasible approximation to the function-space Newton vector in the energy metric. We establish coordinate invariance, exact reduction to ENGD in Euclidean space, global first-order convergence with Armijo backtracking, and robustness to inexact tangent solves. For quadratic residual energies and generalized Gauss--Newton pullbacks, the Woodbury identity transfers the tangent system to sample space without changing the direction. Nyström approximation provides scalable sample-space solves with controlled direction error and recovers the exact direction after iterative convergence. On the evaluated neural PDE benchmarks, EMNGD achieves higher accuracy and faster convergence than the compared state-of-the-art baselines. Woodbury preserves the EMNGD direction, while scalable-solver diagnostics quantify the accuracy--cost trade-off of preconditioning and residual subsampling.

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.

Protein fold classification can be approached via sequence-based representations or structural descriptors, but direct comparisons between lightweight handcrafted descriptors and pretrained protein language model embeddings remain limited. We investigate discrete Ricci curvature on Calpha contact graphs as a lightweight structural descriptor for fold classification. Each protein domain is represented by a 22-dimensional fixed-length feature derived from summary statistics and quantiles of Ollivier-Ricci and Forman-Ricci edge curvature distributions. We evaluate on CATH top-10 Topology classification and on the ASTRAL 40%-identity SCOPe top-10 Fold benchmark, comparing against geometry, contact-graph statistics, persistent homology, and mean-pooled ESM-2 (150M) baselines. On both datasets, lightweight structural descriptors substantially outperform mean-pooled ESM-2 embeddings, with a larger performance gap on the ASTRAL 40% SCOPe benchmark. Ricci alone uses 22 dimensions, or 3.4% of the ESM-2 baseline dimensionality, and already outperforms mean-pooled ESM-2 on both datasets. Combining Ricci with persistent homology yields the strongest performance, achieving macro-F1 of 0.71 on CATH and 0.68 on SCOPe with a 112-dimensional feature vector. These results identify a regime where lightweight interpretable graph descriptors offer a practical alternative to pretrained protein language model embeddings.

Reinforcement learning (RL) has become a dominant paradigm for enhancing LLMs' reasoning capabilities. However, RL algorithms with PPO-Clip are inherently limited by exploration collapse. Subsequent works remain primarily heuristic and fail to identify the essential cause of PPO-Clip's failure. This work reveals the fundamental flaw of PPO-Clip: it implicitly measures policy discrepancy using Euclidean metric, which is theoretically inconsistent with the intrinsic geometry on the policy Riemannian manifold. This geometric mismatch results in overly conservative updates in low-probability regions while aggressive in high-probability regions, ultimately collapsing exploration. To correct this geometric flaw, we propose Riemannian Isometric Policy Optimization (RIPO), which guarantees isometric policy updates on the Riemannian manifold, effectively balancing exploration and exploitation. We further show that RIPO achieves a favorable bias-variance trade-off, which stabilizes optimization. Extensive experiments demonstrate that RIPO significantly surpasses existing LLM RL algorithms across seven competition-level benchmarks (up to 60% improvement over GRPO on AIME24).

Modern robotic perception increasingly involves large-scale geometric optimization problems distributed across multiple robots or sessions. However, existing distributed solvers often depend on brittle hand tuning and primarily target rigid body pose graphs. To address this, we present DeepCORD, a learning-augmented framework for distributed factor graph optimization on general matrix Lie groups. By unfolding a parallel and accelerated Riemannian optimizer into differentiable iterations, DeepCORD learns a self-supervised feedback policy that dynamically adapts solver parameters according to the optimization phase and communication status. The resulting method enables adaptive distributed optimization over matrix Lie groups under both synchronous and asynchronous communication regimes. Extensive experiments on real-world $\mathrm{SE}$(3) pose graph optimization and $\mathrm{SL}$(4) projective submap alignment show that our method achieves lower objective values than existing distributed baselines on most benchmarks across realistic operating scenarios.

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.

Understanding the geometric structure of pre-trained language model embeddings matters for interpretability and safety. We ask whether sentence-level classification signal lives in the Riemannian geometry of contextual token embeddings, and probe it by extracting per-token pullback metrics from a learned encoder's analytical Jacobian and aggregating them with the Fréchet mean on the symmetric positive definite (SPD) manifold; we call this procedure Riemannian Mean Pooling (RMP). Across three datasets with non-trivial linguistic structure (CoLA, CREAK, RTE), RMP outperforms Euclidean mean pooling, while on FEVER-Symmetric, a benchmark constructed to remove annotation-driven lexical artifacts, the method correctly stays at chance. Ablations show that a randomly initialised encoder combined with Fréchet aggregation already beats Euclidean pooling on two of the three signal-bearing datasets, localising the source of the gain to the geometric aggregation rather than to learned manifold structure; the trained encoder contributes additional signal specifically on CREAK, the most knowledge-heavy of the three signal-bearing datasets.

Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

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.

Cross-subject EEG stress detection remains challenging because discriminative stress-related patterns are both subject-dependent and frequency-specific. Conventional Riemannian methods model spatial covariance mainly in the time domain, overlooking neural oscillations that are critical for high-level cognitive state decoding, while standard temporal tokenization often fragments inter-slice temporal coherence. To address these limitations, we propose \method{}, an Intra-Inter Riemannian Manifold Attention Network for EEG-based stress detection. \method{} constructs spatial covariance matrices independently at each frequency point and maps them to the SPD tangent space, preserving channel-wise geometry together with frequency-specific discriminative cues. It further introduces frequency cluster aggregation to select informative spectral components and reduce redundancy by forming compact, data-driven frequency clusters aligned with EEG rhythms. Finally, an intra-inter slice attention module adaptively integrates local slice-level spectral dynamics and global temporal context across EEG sequences. Experiments on three datasets show that \method{} consistently outperforms five state-of-the-art baselines, achieving up to 82.78% balanced accuracy while remaining efficient with only 1.60M parameters and 31.95M FLOPs.

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.

Multi-site functional MRI (fMRI) studies are essential for robust neuropsychiatric diagnosis yet suffer severe domain shifts from scanner heterogeneity, demographics, and site-specific acquisition protocols. Traditional domain adaptation requires concurrent source and target data access, violating clinical privacy regulations. Moreover, functional connectivity matrices lie on the Symmetric Positive Definite (SPD) manifold, where Euclidean operations cause geometric distortions corrupting diagnostic patterns. We propose BrainRiem, a source-free domain adaptation framework learning compact Riemannian brain prototypes via manifold-aware bi-level optimization. It employs the Log-Euclidean Metric to ensure prototypes remain valid SPD matrices, while Dirichlet Energy spectral calibration aligns their frequency characteristics with real brain networks. Only anonymized prototypes are transmitted to target sites, serving as stable anchors for training local models without source data access and reducing leakage under the evaluated attacks. Comprehensive experiments on ABIDE and REST-meta-MDD show BrainRiem consistently outperforms state-of-the-art source-free, traditional, and graph domain adaptation methods across diverse scanners and demographics. Notably, learned prototypes exhibit biologically interpretable connectivity patterns aligning with established neuroscience findings, validating the necessity of Riemannian geometry for brain network analysis.

Brain-Computer Interface (BCI) based on electroencephalography (EEG) enables direct interaction between the brain and external environments and has significant applications in assistive technologies, medical rehabilitation, and entertainment. Recently, EEG decoding methods based on Symmetric Positive Definite (SPD) learning have demonstrated superior performance. However, these methods typically employ basic network architectures and do not explicitly capture local relationships between EEG signals. This limitation is problematic for EEG signals due to their inherently low Signal-to-Noise Ratio (SNR). Moreover, most existing Riemannian manifold-based methods are restricted to specific metrics. The most widely used is the Affine-Invariant Metric (AIM). However, it has a quadratic dependency on the SPD matrices and cannot handle ill-conditioned SPD matrices, which hinders the effectiveness of networks. In contrast, the Bures-Wasserstein Metric (BWM) exhibits linear dependence on SPD matrices and demonstrates superior performance for ill conditioning. To overcome these challenges, we propose a Riemannian self-attention network based on the BWM. Additionally, the recently introduced power-deformed generalized Bures-Wasserstein metric reveals a nonlinear relationship between SPD matrices and matrix power deformation. This metric provides a more nuanced representation of the geometric structure of the SPD manifold. Consequently, we extend our model to a learnable version. For simplicity, we refer to it as GBWAtt. Experimental results on three EEG benchmarking datasets validate the robustness and effectiveness of our proposed method. The code is available at https://github.com/jissc/GBWAtt.

One-step diffusion editors are fast because they avoid inversion and iterative optimization, but a single transport update must be aggressive enough to realize the target prompt and conservative enough to preserve the source image--and no fixed update strength satisfies both demands across edit types. We treat this tension as a post-hoc candidate-selection problem on top of energy-field transport rather than as a new editing model. Our proposed method, Riemannian Residual Line Search, first builds a stronger edit by estimating the local time curvature of the prompt-delta field and projecting the corrected direction back onto the update norm of the original first-order energy-field transport estimation. It then forms a small residual path from the source image to this strong edit, retains the original first-order output as one candidate, and picks the final image by maximizing target-prompt CLIP alignment. On a 700-sample PIE-Bench++ evaluation across 10 edit type IDs, our method achieves state-of-the-art (SOTA) performance among current one-step update algorithms.

Pose graph optimization (PGO) is a key back-end component for state estimation in networked multi-robot simultaneous localization and mapping (SLAM). In object-based multi-robot SLAM, the problem becomes more tightly coupled because robots must jointly estimate both their trajectories and the poses of persistent objects observed by multiple agents. Existing decentralized solutions often assume that the communication graph closely matches the physical interaction topology, which is restrictive in realistic deployments where communication is sparse, intermittent, or time-varying. This paper presents a fully decentralized Riemannian optimization framework for object-based multi-robot PGO that decouples the coupled estimation problem via a consensus mechanism, enabling flexible communication topologies. To improve convergence under limited communication budgets, we further develop a distributed approximate-Newton scheme that exploits local second-order information while operating directly on the SE(d) manifold to preserve geometric consistency, and we establish the convergence to Riemannian first-order stationary points and provide a local condition-number analysis explaining the benefit of approximate second-order information over first-order Riemannian descent. The resulting method reduces iteration count and communication overhead without sacrificing estimation accuracy. Extensive evaluations on public benchmarks, large-scale simulations, and real-world multi-robot experiments demonstrate improved accuracy, runtime efficiency, scalability across network topologies, and robustness to communication failures.

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.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Imitation learning for robotic manipulation relies on large sets of human demonstration trajectories, which are often noisy and temporally irregular due to variable operator speed, intermittent pauses, and inconsistent action density. A common preprocessing strategy is time-uniform downsampling to shorten sequences, but it cannot effectively remove speed-induced non-uniformity or redundant pauses. This mismatch degrades data quality and hinders policy learning. To address this issue, we propose Information-Standardized Trajectory Resampling (ISR), an offline preprocessing method for effective imitation learning. ISR resamples each trajectory by enforcing approximately equal information distance between adjacent points. Specifically, we map trajectories onto an information-modulated Riemannian manifold and perform geodesic-equidistant parameterization. We construct an information-intensity field from velocity and acceleration norms: the velocity term removes small-motion redundancy, while the acceleration term preserves high-curvature and fine-manipulation phases. We evaluate ISR on three real-world manipulation tasks with mainstream imitation learning policies. Compared with the baseline time-uniform 3x downsampling, ISR improves task success rates by about 25%, remains robust across datasets collected from different operators, and reduces both dataset size and training cost. The code and videos are publicly available at https://d-robotics-ai-lab.github.io/isr.page.

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.

Traditional approaches place intelligence in the agent, whether as a learned policy or a search procedure. We instead place intelligence in the space itself: a scene induces a Riemannian metric on the configuration manifold, and action reduces to following the geodesics of that metric rather than invoking a separate planner or collision checker. A single Encoder-Router network realizes this idea through three complementary parameter groups -- frame parameters that orient the generators, modulation parameters that govern their spatial propagation, and basic coefficients that determine their strength. These groups combine through a shared semigroup-superposition mechanism to produce a single Riemannian metric field, yielding a compact architecture whose geometry scales naturally with scene complexity. Trained on a single two-obstacle scene, the model demonstrates robust zero-shot generalization across unseen obstacle configurations, with orders-of-magnitude separation between collision-free and obstacle-penetrating path costs.

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.

High-dimensional datasets often concentrate near low-dimensional structures, but estimating their geometry from samples typically relies on graphs and kernels that scale poorly with dataset size and dimension. We propose Riemannian metric matching: a denoising probabilistic framework for learning the Riemannian geometry of data using neural networks. Specifically, we learn the carré du champ operator, which, using diffusion geometry, gives us access to the Riemannian geometry toolkit for downstream machine learning and statistical tasks. Our key observation is that the carré du champ operator can be formulated as a conditional expectation over random perturbations of the data, which can be exploited for sample-wise training and constant cost, amortized inference without explicit kernel construction. Empirically, metric matching rivals or improves the accuracy of $k$-NN-based diffusion geometry estimators, while enabling amortized inference that is up to $400\times$ faster, and supports graph-free geometric analysis on high-dimensional images where nearest neighbors break down.

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, $$(κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}.$$ of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

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.