Marius Willner, Maximilian Scharf, André Uschmajew +2math.OC cs.CV physics.comp-ph
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.
Traditional Text-based Person Search (TPS) is typically limited to matching static appearance attributes, severely neglecting dynamic action information. The Text-based Person Anomaly Search (TPAS) task bridges this gap, requiring models to locate micro-level specific abnormal behaviors while matching macro-level appearance of pedestrians. However, current TPAS methods face fundamental limitations: external explicit pose estimators are fragile in unconstrained surveillance scenarios, and implicit learning encounters visual decoupling failure under pixel-level entanglement, causing dominant appearance information to easily swallow and contaminate subtle action features. Furthermore, performing contrastive optimization on hard negative samples (``same appearance, different actions'') in conventional Euclidean spaces induces severe shortcut learning. To address these, we propose the Lightweight Action Inversion and Riemannian rectification network (LightAIR). First, it introduces textual semantic priors as anchors via a lightweight action inversion operator to extract pure action features, thereby overcoming visual-inherent coupling. Subsequently, it employs orthogonal null-space projection to constrain appearance features within the orthogonal complement space of action features, guaranteeing strict forward decoupling. Finally, we designed a gradient rectification module that computes the Riemannian gradient to constrain the backpropagation trajectory, forcing the gradient flow to update strictly along the tangent space that preserves decoupling properties, thereby cutting off harmful shortcuts. Extensive experiments on the widely used TPAS and TIPR datasets demonstrate that LightAIR significantly outperforms existing state-of-the-art methods. Codes are available at https://github.com/rainy-london/LightAIR
Jie Zhang, Cheng-Fang Su, Yi-Jui Huang +1cs.LG cs.AI cs.CL
Residual connections are the de facto mechanism for training deep neural networks stably. Geodesic Normalization (GeoNorm) recasts them on a Riemannian manifold, orthogonalizing each layer output against the current hidden state and applying the resulting update through the Riemannian exponential map. Every hidden state thus keeps a constant $\ell_{2}$-norm, confining the residual stream to a hypersphere. The exponential map, however, is only one member of a broad family of retraction maps. We show that on the hypersphere this entire family collapses to a single scalar design choice. What distinguishes one retraction from another is only how the magnitude of an update is converted into a rotation angle within the plane spanned by the hidden state and the update. This view places Euclidean residual connections and GeoNorm in one framework. Instantiating it with the metric projection retraction and the Cayley retraction yields Proj-SpheretNorm and Cay-SpheretNorm, which are exactly norm-preserving yet require only algebraic operations. Both prove to be members of a one-parameter family of angular retractions, $p$-SpheretNorm, whose rotation angle saturates rather than growing without bound. The two methods above are recovered exactly at $p = 1$ and $p = 2$, while the identity map and GeoNorm arise only as limits at either end. On nanoGPT, all three methods outperform existing lightweight deep connection schemes, and the best validation loss is attained at finite $p$, indicating that the exponential map is not the preferred retraction for spherical residual streams but merely one end of a spectrum.
The Discrete Fourier Transform (DFT), the Discrete Cosine Transform (DCT), and their block-wise variants underpin most deployed image and video codecs. Their effectiveness rests on three properties: their runtime is near-linear (up to a polylogarithmic factor) in the image size, they are exactly invertible, and they carry few to no parameters. In this work, we generalize these bases to isometric multilinear bases, allowing a small number of extra parameters (polylogarithmic in the image size), while preserving all three properties. We develop a scheme to train a better transformation for a given image dataset: we use isometric tensor networks, inspired by quantum many-body theory, to parameterize the basis, and train it with Riemannian optimization. We show that training consistently improves performance, as our parameterized bases can represent the traditional DFT and DCT-IV (a variant of the DCT). Evidence is shown across natural photographs and line drawings. On Quick Draw line-drawing compression, for example, the best trained basis outperforms the block cosine transform used in the JPEG format by $20\%$ in terms of compressed data size.
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.
Kernel regression with tensor trains and Hadamard overparameterization (KReTTaH) is introduced as a training-data-free, interpretable, and nonparametric framework for multi-way data imputation. The imputation problem is reformulated as regression in reproducing kernel Hilbert spaces (RKHS), where the tensor regression coefficients are explicitly constrained to lie on fixed-rank tensor-train (TT) manifolds and structured via Hadamard overparameterization to promote sparsity and high representational efficiency. Rather than relying on costly cross-validation, KReTTaH jointly optimizes the TT coefficient tensors and the kernel covariance matrices within a Riemannian product-manifold framework -- the former on fixed-rank TT manifolds, the latter on the manifold of positive-definite matrices -- thereby enabling automated kernel-hyperparameter selection. Numerical tests on two challenging applications -- imputation of high-dimensional functional magnetic resonance imaging (fMRI) data and recovery of missing edge flows in dynamic graphs -- demonstrate that KReTTaH consistently outperforms state-of-the-art tensor-, Bayesian-, and neural-network-based baselines in terms of modeling accuracy.
Ludwig Winkler, Andrew Leaver-Fay, Joseph Kleinhenz +1cs.LG
Generative models learn data distributions that reside on a low-dimensional manifold within a higher-dimensional ambient space. Optimizing differentiable objectives on this manifold is challenging: the ambient loss landscape is high-dimensional, rugged, and non-convex. Direct gradient descent, blind to the manifold's geometry, quickly drifts off it. Diffeomorphic optimization starts from the observation that diffusion and flow models provide a map from the data manifold to a much simpler base space in which we perform gradient descent. Using differential geometry, we show this is equivalent to Riemannian gradient descent on the data manifold up to $\mathcal{O}(λ^2)$ corrections, keeping trajectories on-manifold by construction and yielding a smoother optimization surface. For protein design, we extend diffeomorphic optimization to the matrix Lie groups $\mathrm{SO}(3)$ and $\mathrm{SE}(3)$, deriving an autograd-compatible $\mathrm{SO}(3)$ gradient and a generalized adjoint-state method for backpropagation through Lie-group ODE solvers. Diffeomorphic optimization improves over tuned guidance on secondary-structure targeting with FrameFlow ($91.3\%$ vs. $63.3\%$ of residues in the Ramachandran target), outperforms OC-Flow on peptide binding affinity at $2\times$ the speed, and reduces Rosetta energies by thousands of units across the PDB test set for structures with hundreds of residues.
Recently, diffusion models have been widely adopted in generative modeling and have served as foundational models for many image generation tasks. To control the generation without costly re-training or fine-tuning, many works seek inference-time guidance methods to steer the latent via a differentiable objective at inference time. However, these methods cannot effectively preserve the original Gaussian distribution because they introduce distributional drift, thereby degrading the sample quality. To address this gap, we propose DiffRGD, a distribution-aware guidance framework that explicitly preserves the latent Gaussian structure. DiffRGD formulates each sampling step as a constrained optimization problem on a spherical manifold induced by the latent Gaussian distribution, and solves it efficiently via Riemannian Gradient Descent (RGD). DiffRGD is a plug-and-play method that can be seamlessly integrated into any pre-trained diffusion model. Extensive experiments demonstrate that DiffRGD outperforms previous methods in most image restoration and conditional generation tasks. Our project page is available at https://diffrgd.github.io/.
Ray Zhang, Marcus Greiff, Thomas Lew +1cs.CV cs.AI cs.RO
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.
Kang An, Jiaxiang Li, Donald Goldfarb +1cs.LG cs.AI math.OC
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.