Riemannian Optimization

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely eliminating inner iterative loops. In both regimes, per-iteration complexity scales linearly with dataset size, and we provide a rank-sufficiency certificate for global optimality verification. Extensive experiments across a range of problem sizes demonstrate that our regularization-free first- and second-order solvers achieve faster convergence and superior performance over existing state-of-the-art low-rank OT solvers.

We propose a fast and correspondence-free local point cloud registration method that leverages geometric surface structure and reproducing kernel Hilbert space (RKHS) embeddings. The method represents point clouds as continuous functions with point-wise anisotropic kernels that encode local geometry. This formulation improves alignment along surface normals while relaxing alignment along tangential directions. To solve the resulting registration problem, we propose a second-order on-manifold optimization scheme with approximate Riemannian Hessians, achieving a speedup of up to 10x over the first-order solvers used in prior correspondence-free RKHS-based methods. We demonstrate improved frame-to-frame LiDAR and RGB-D tracking accuracy across diverse indoor and outdoor datasets. On a LiDAR tracking registration task in the driving domain, we achieve a reduction of $>55\%$ in both translational and rotational drift in challenging feature-sparse environments. On object registration benchmarks, we show improved robustness over ICP-based methods and further gains when refining global initialization, particularly under moderate misalignment.

We explore Riemannian optimization techniques for rank-factored matrix parameters, targeting contemporary deep learning applications. We examine ten points in the algorithm design space: two geometries for rank-$r$ matrices, three geometries for rank-$r$ partial isometries, and block-matrix variants of these five, where factors are shared across block-rows and block-columns. We apply our methods to the multihead attention parameters in small language models. After tuning learning rates, our methods do not conclusively outperform an AdamW baseline. Our implementations are available online.

The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response. Existing gradient-based estimators either operate in the ambient space and suffer from the curse of dimensionality, or localize in the reduced space at a per-outer-iteration cost at least quadratic in the sample size. We show that minimizers of the population Minimum Average Variance Estimation (MAVE) risk approximate the same Grassmannian target as the Outer Product of Gradients (OPG), and recast the empirical criterion as a smooth maximization on the Stiefel manifold with closed-form Riemannian gradient. The resulting algorithm, SMAVE, combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent. A simplified version comes with almost-sure convergence and a non-asymptotic rate matching the standard non-convex stochastic first-order scaling. Empirically, SMAVE matches or improves on RMAVE's synthetic subspace recovery at moderate-to-high ambient dimension, and on four real datasets it uniformly improves over OPG and is competitive with or outperforms RMAVE at orders of magnitude lower runtime.

While finetuning effectively adapts foundation models to specialized downstream tasks, it can degrade nontarget capabilities acquired during pretraining. Existing forgetting aware methods typically seek safer updates through specialized initialization or fixed constraints, but do not regulate the adaptation preservation trade-off during training. We propose Foundation Preserving LoRA (FoLoRA), a forgetting aware optimization framework. Guided by a first order preservation condition, FoLoRA defines a forgetting penalty over pretraining-proxy activations and a task utility over downstream task activations. It then scores update directions by task utility per unit forgetting penalty via a generalized Rayleigh quotient. The resulting spectral coordinate system enables direction wise gated Adam updates, attenuating low utility to penalty directions during training. To estimate the forgetting penalty, FoLoRA constructs pretraining proxy calibration data by sampling from the pretrained model rather than relying on a single proxy dataset. Experiments on math, code, and instruction following adaptation show that FoLoRA achieves the strongest preservation adaptation balance over baselines, improving target task performance with best aggregate preservation of non target capabilities.

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.

We present a framework for distributed Pose Graph Optimization (PGO) by formulating the problem as a second-order continuous-time dynamical system evolving on Lie groups. By modeling pose variables as massive particles subject to damping, the equilibrium points of the resulting Riemannian dynamics coincide with first-order critical points of the original PGO problem. Using the governing damped Euler--Poincaré equations and a semi-implicit geometric integrator, we design an optimization algorithm that generalizes existing algorithms such as Riemannian gradient descent and Gauss--Newton. In multi-robot settings, we present a fully distributed and parallel method based on block-diagonal mass and damping matrices, where each robot solves an ordinary differential equation for its own poses with minimal communication overhead. Moreover, modeling both state and velocity enables principled neighbor prediction that significantly improves convergence under delayed communication. Theoretically, we present an analysis and establish sufficient condition that ensures energy dissipation under the employed geometric discretization scheme. Experiments on benchmark PGO datasets demonstrate that the proposed solver achieves superior performance compared to state-of-the-art distributed baselines in both synchronous and asynchronous regimes.

Muon and related norm-constrained matrix optimizers have become central to large-scale learning problems. They are formulated as a linear maximization oracle (LMO) over an ambient matrix-norm ball in unconstrained Euclidean space. However, these do not generalize cleanly to manifold-valued parameters such as low-rank factorizations, orthogonality constraints, or symmetric positive definite (SPD) matrices. Naively restricting the Muon LMO to the tangent space (i) breaks quotient symmetries and (ii) couples the tangent-space constraint with an ambient norm bound, thereby obstructing closed-form solutions on various manifolds of interest. We resolve both issues with a single observation: every Riemannian metric canonically lifts a unitarily invariant Euclidean norm to an intrinsic norm on each tangent space, and the resulting intrinsic norm constrained LMO is symmetry preserving. Building on this, we introduce intrinsic Muon (iMuon), a unified framework that yields closed-form updates on the fixed-rank, SPD, Stiefel, and Grassmann manifolds for any unitarily invariant norm, including the spectral, Frobenius, and nuclear norms. We establish convergence guarantees for both deterministic and stochastic iMuon with rate constants that depend only on the manifold dimension. Notably, on the fixed-rank manifold this constant depends only on the rank, making the rate independent of factor conditioning and removing the runtime factor-rescaling required by prior work. Experiments on LoRA finetuning of LLMs, image classification, and subspace learning illustrate the efficacy of the proposed approach.

Sign-based optimization algorithms, such as SignSGD and Muon, have garnered significant attention for their remarkable performance in training large foundation models. Despite this empirical success, we still lack a theoretical understanding of when and why these sign-based methods outperform vanilla SGD. The core obstacle is that under standard smoothness and finite variance conditions, SGD is known to be minimax optimal for finding stationary points measured by $\ell_2$-norms, thereby fundamentally precluding any complexity gains for sign-based methods in standard settings. To overcome this barrier, we analyze sign-based optimizers leveraging $\ell_1$-norm stationarity, $\ell_\infty$-smoothness, and a separable noise model, which can better capture the coordinate-wise nature of signed updates. Under this distinct problem geometry, we derive matched upper and lower bounds for SignSGD and explicitly characterize the problem class in which SignSGD provably dominates SGD. Specifically, we compare the \emph{upper bound of SignSGD} with the \emph{lower bound of SGD}, illustrating that SignSGD effectively reduces the complexity by a factor of $d$ under \emph{sparse noise}, where $d$ is the problem dimension. Furthermore, we elevate this framework to the matrix domain, providing an equivalent optimal lower bound for the Muon optimizer, proving that extending the sign operator to matrices preserves this optimal scaling with dimensionality. Finally, we bridge our theoretical bounds to practice, demonstrating that the theoretical superiority of SignSGD accurately predicts its faster convergence during the pretraining of a 124M parameter GPT-2 model.

The empirical success of large language model (LLM) pre-training relies heavily on heuristic stabilization techniques, such as explicit normalization layers and weight decay. While recent constrained optimization approaches that explicitly restrict weights may improve numerical stability and performance, the mechanism and motivation for adding constraints still remain elusive. This paper systematically demystifies the role of explicit manifold constraints in LLM pre-training. By introducing the Msign-Aligned Constrained Riemannian Optimizer (MACRO)-a provably convergent, single-loop optimization framework-our study disentangles weight regularization heuristics from interacting mechanisms like RMS normalization and decoupled weight decay. Theoretical analyses and comprehensive empirical evaluations reveal that manifold constraints independently bound forward activation scales and enforce stable rotational equilibrium, thereby subsuming the roles of these heuristic mechanisms. Evaluations on large-scale LLM architectures demonstrate that MACRO achieves highly competitive performance while rigorously preserving the theoretical guarantees of exact Riemannian optimization.

Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing literature often presents results at a high level of abstraction, omitting the detailed coordinate-level derivations required for implementation and algorithm development. This work provides a self-contained and rigorous treatment of the foundations of Riemannian geometry, with a focus on explicit derivations tailored to Riemannian optimization. We systematically develop the key geometric structures -- including tangent and cotangent spaces, tensor calculus, metric tensors, Levi-Civita connections, curvature, and geodesics -- emphasizing step-by-step derivations in coordinates and matrix form. Building on these foundations, we derive the Riemannian gradient, Hessian, exponential map, and retraction in a form suitable for numerical computation. We further specialize these constructions to important matrix manifolds, including the Stiefel, Grassmann, and SPD (Symmetric Positive Definite) manifolds, providing explicit formulas widely used in optimization and geometric machine learning. This monograph develops a unified and implementation-oriented treatment of Riemannian geometry for optimization on manifolds. Its main contribution is the systematic organization and detailed derivation of classical geometric constructions in forms directly usable for algorithm design and numerical implementation. By connecting coordinate-level differential geometry with matrix-manifold formulas, the monograph bridges the gap between abstract theory and practical computation, and provides a reference for researchers and practitioners working in Riemannian optimization and related fields.

Assigning one of K options to each of N groups under a total cost budget is a recurring problem in efficient AI, including mixed-precision quantization, non-uniform pruning, and expert selection. The objective, typically model loss, depends jointly on all assignments and does not decompose across groups, preventing combinatorial solvers from directly optimizing the true objective and forcing reliance on proxy formulations. Methods such as evolutionary search evaluate the actual loss but lack gradient information, while penalty-based approaches enforce the budget only approximately and often require extensive hyperparameter tuning. We present a new approach by showing that, under softmax relaxation, the budget constraint defines a smooth Riemannian manifold in logit space with unusually simple geometry. The normal vector admits a closed-form expression, shifting logits along the cost vector changes expected cost monotonically, and vector transport reduces to a single inner product. Building on these properties, we propose Riemannian Constrained Optimization (RCO), which augments a standard Adam step with tangent projection, binary-search retraction, and momentum transport. Combined with Gumbel straight-through estimation and budget-constrained dynamic programming for discrete feasibility, RCO enables first-order optimization of the actual loss under exact budget enforcement without introducing constraint-specific hyperparameters. Across both synthetic benchmarks and realistic LLM compression settings, RCO matches or exceeds state-of-the-art methods while often requiring substantially less wall-clock time. Source code is available at https://github.com/IST-DASLab/RCO.

Optimization over the Stiefel manifold $\mathrm{St}(p,d)$, the set of $p \times d$ column-orthonormal matrices, is fundamental in statistics, machine learning, and scientific computing, yet remains challenging in the presence of non-convex, non-smooth, or black-box objectives. Existing methods largely rely on either convex relaxations or gradient-based Riemannian optimization, limiting applicability in derivative-free and highly multimodal settings. We propose \textsc{BOOOM} (Black-box Optimization Over Orthonormal Manifolds), a general-purpose framework for loss-function-agnostic optimization on $\mathrm{St}(p,d)$. The key idea is a global Givens rotation-based parametrization that maps the manifold to an unconstrained Euclidean angle space while preserving feasibility exactly. Building on this representation, BOOOM employs a structured, parallelizable, derivative-free search based on Recursive Modified Pattern Search, enabling systematic exploration through plane-wise rotations without requiring gradient information and facilitating escape from poor local optima. We establish a unified theoretical framework showing equivalence between angle-space and manifold optimization, transfer of stationarity, and global convergence in probability under mild conditions. Empirical results across diverse problems, including heterogeneous quadratic optimization, low-rank and sparse matrix decomposition, independent component analysis, and orthogonal joint diagonalization, among other widely studied settings, demonstrate strong performance relative to state-of-the-art methods, particularly in non-smooth and highly multimodal regimes. We further illustrate its practical utility through a novel supervised PCA formulation applied to metabolomics data in colorectal cancer.

Modern world models are becoming too complex to admit explicit dynamical descriptions. We study safety-critical contextual control, where a Planner must optimize a task objective using only feasibility samples from a black-box Simulator, conditioned on a context signal $ξ_t$. We develop a sample-based Penalized Predictive Control (PPC) framework grounded in online Riemannian optimization, in which the Simulator compresses the feasibility manifold into a score-based density $\hat{p}(u \mid ξ_t)$ that endows the action space with a Riemannian geometry guiding the Planner's gradient descent. The barrier curvature $κ(ξ_t)$, the minimum curvature of the conditional log-density $-\ln\hat{p}(\cdot\midξ_t)$, governs both convergence rate and safety margin, replacing the Lipschitz constant of the unknown dynamics. Our main result is a contextual safety bound showing that the distance from the true feasibility manifold is controlled by the score estimation error and a ratio that depends on $κ(ξ_t)$, both of which improve with richer context. Simulations on a dynamic navigation task confirm that contextual PPC substantially outperforms marginal and frozen density models, with the advantage growing after environment shifts.

We present a geometric framework for Reinforcement Learning (RL) that views policies as maps into the Wasserstein space of action probabilities. First, we define a Riemannian structure induced by stationary distributions, proving its existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapped from the state space to the tangent space of probability measures over the action space. Next, we formulate a general RL optimization problem and construct a gradient flow using Otto's calculus. We compute the gradient and the Hessian of the energy, providing a formal second-order analysis. Finally, we illustrate the method with numerical examples for low-dimensional problems, computing the gradient directly from our theoretical formalism. For high-dimensional problems, we parameterize the policy using a neural network and optimize it based on an ergodic approximation of the cost.

Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others. Current nearly optimal algorithms work only in affine spaces due to the loss of one degree of freedom when working with general convex constraints. Obtaining an accelerated algorithm that makes nearly optimal $\widetilde{O}(1/(γ\sqrt{\varepsilon}))$ first-order queries to a $γ$-quasar convex smooth function \emph{with constraints} was independently asked as an open problem in Martínez-Rubio (2022); Lezane, Langer, and Koolen (2024). In this work, we solve this question by designing an inexact accelerated proximal point algorithm that we implement using a first-order method achieving the aforementioned rate and, as a consequence, we improve the complexity of the accelerated geodesically Riemannian optimization solution in Martínez-Rubio (2022). We also analyze projected gradient descent and Frank-Wolfe algorithms in this constrained quasar-convex setting. To the best of our knowledge, our work provides the first analyses of first-order methods for quasar-convex smooth functions with general convex constraints.