Manifolds

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

Monocular dense prediction has recently seen remarkable success by repurposing pre-trained diffusion models. This opens a promising yet challenging avenue for more efficient multi-task learning paradigm. However, existing multi-task diffusion methods often introduce parameter-heavy adapters, experts, or learnable task tokens, leading to computational redundancy. In this paper, we reveal an inherent mechanism within one-step diffusion models: the native, fixed sinusoidal timestep embedding can be repurposed as an endogenous task steering signal. Based on this discovery, we propose Multi-task Unified eStimation via timestep Embedding (MUSE), a parameter-free, single-model multi-tasking approach for dense prediction. We interpret this mechanism via Manifold Decoupling, where discrete, fixed timestep values deterministically steer the generation process towards decoupled, task-specific manifolds in the latent space. Extensive experiments across 10 datasets demonstrate that MUSE achieves highly competitive performance on both monocular depth and normal estimation, and its efficacy generalizes across U-Net and DiT architectures. Our work offers a concise and efficient path toward generalist vision models by simply unlocking the latent potential of existing generation infrastructure.

Recent text-to-video (T2V) diffusion models rely heavily on auxiliary reward signals (e.g., via reward models or DPO) to align generated content with human aesthetics and improve realism. These signals, however, incur substantial computational overhead, require costly human annotations, and often yield limited improvement in fine-grained local details. In this paper, we argue that your data manifold is secretly a reward model. By explicitly modeling the manifold structure of high-quality Supervised Fine-Tuning (SFT) data and encouraging video latents to lie on this manifold, we derive dense, differentiable, and nearly cost-free reward signals that significantly improve video quality, particularly in mitigating low-level distortions. Our modeling builds upon Local Coordinate Coding (LCC), which captures the skeleton' of the manifold. However, directly applying LCC suffers from mean regression, pulling latents toward the geometric mean and losing high-frequency details. We therefore extend it to Shell Local Coordinate Coding (Shell-LCC), which models the manifold surface' as an isotropic shell to align with the true high-density region. Experiments demonstrate that our approach improves realism, enhances high-frequency details, reduces over-smoothing artifacts, and alleviates motion blur.

Recent studies suggest that diffusion models can recover geometric structure in the data manifolds they are trained on, yet the supporting evidence has so far come mostly from natural-image data, where the underlying geometry itself is unknown. We study this question in a setting where the geometry is analytically tractable: constrained inverse kinematics (IK). Each task-space constraint defines a configuration-space manifold with known intrinsic dimension, giving direct ground truth for evaluating the geometry learned by the model. For each of the 6-DoF UR5 and 7-DoF Franka, we train a single conditional diffusion model across seven constraint families, spanning solution manifolds from discrete IK branches to self-motion manifolds. Our empirical results reveal that the intrinsic dimension recovered from the model's score function matches the analytical degrees of freedom of the corresponding constraint manifold across both robots. Moreover, linear interpolation in the latent space leads to generated solutions that remain close to the appropriate constraint manifold, indicating that the learned representation further captures geometric structure of the constraint family beyond intrinsic dimension alone. Constrained IK therefore offers a controlled setting for studying the intrinsic geometry learned by diffusion models.

What is the geometry of a visual percept? The most widely used protocols for decomposing neural network representations into interpretable parts treat concepts as isolated directions, yet recent work shows that concepts are often realized as geometric structures in low dimensional regions of activation space. We turn to the literature of Structured sparsity to close this gap, and show that block sparsity, which groups directions into blocks, is the prior matched to a generative model in which a representation is a sparse sum of low-dimensional manifolds: the modern, learned form of a classical idea in visual neuroscience, where a visual feature is carried by a coordinated group of neurons rather than a single tuned one. We implement three variants of block-sparse featurizers (BSFs) and, through a minimum-description-length analysis, show that all three describe activations more compactly than direction-based featurizers, with the recovered concepts typically two- to four-dimensional. We then use BSFs to (i) recontextualize prior work, showing that curve detectors in InceptionV1 actually read from a single continuous curve manifold, (ii) discover novel manifolds including shadows and lighting in DINOv3, and (iii) support interpretable control of image generation in diffusion models (SDXL) via manifold steering.

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.

Foundation models provide highly descriptive representations for medical images, yet their reliability degrades under distribution shifts arising from changes in patients, devices, or acquisition conditions. Reliable out-of-distribution (OOD) detection is therefore essential for safe deployment. Recent post-hoc detectors efficiently exploit frozen embeddings (e.g., kNN), whereas reconstruction-based OOD detection in latent feature space has seen limited adoption due to inconsistent performance. In this work, we show that the limitation of reconstruction-based methods in latent space does not stem from poor reconstruction quality, but from how reconstruction errors are scored. Standard L2 residual norms collapse the anisotropic residual structure, thereby suppressing informative deviations. To address this limitation, we introduce MaRS (Mahalanobis Residual Scoring), a label-free OOD detector that learns an in-distribution manifold using a lightweight autoencoder and measures deviation via a Mahalanobis distance on reconstruction residuals, yielding variance-aware OOD scores. Across three imaging modalities, multiple types of distribution shift, and different model families and scales, MaRS outperforms established confidence-, distance-, and reconstruction-based baselines, while remaining fully post-hoc and lightweight. The code is available at https://github.com/francescodisalvo05/mars.

Recently, several learning paradigms have been introduced to enforce idempotency in generative models. The goal is to ensure that repeated application of a model leaves samples unchanged once they lie on the target data manifold. In practice, however, many of these approaches fail to achieve exact fixed points, leading to instability and drift under repeated application. In this work, we argue that a key reason for this failure is a geometric mismatch between the manifolds learned by the encoder and decoder. The encoder projects inputs onto one latent manifold, while the decoder implicitly learns to reconstruct data from a different manifold. This discrepancy prevents the model from learning truly idempotent mappings. To address this issue, we propose a new training framework that explicitly closes this gap by forcing the encoder and decoder to learn consistent representations of the same underlying data manifold. By aligning the geometry of these components, our method encourages stable projections. Empirically, we show that our approach achieves significantly lower idempotency error and consistently regenerates identical outputs under repeated application, compared to existing methods. We demonstrate the effectiveness of the proposed framework on both image generation and image editing tasks. Finally, we show that enforcing idempotency in this manner improves identity preservation and information stability, leading to more realistic and controllable generative editing models.

A widely held intuition in deep learning is that stochastic gradient descent (SGD) implicitly favors flat minima and that flat minima generalize better, but standard Euclidean measures of flatness such as the trace or maximum eigenvalue of the loss Hessian are not invariant under reparametrizations that preserve the network function, which undermines the theoretical foundations of this narrative. In this study we resolve this issue by grounding flatness in the Riemannian geometry of the statistical manifold induced by the Fisher Information Matrix (FIM). We define Riemannian sharpness mathematically and prove that it is invariant under smooth, function-preserving reparametrizations, which directly addresses the critique of Dinh et al. in the paper ``Sharp minima can generalize for deep nets''.We note that this invariance is a property of the true FIM; the diagonal empirical estimator used in practice (and in all experiments below) inherits invariance only approximately, and exact invariance under arbitrary reparametrizations would require structured estimators such as K-FAC. We formalize the gradient noise of mini-batch SGD as having a covariance structure proportional to the FIM, derive the stationary distribution of the resulting stochastic differential equation, and then show that the probability mass is exponentially concentrated at Riemannian-flat minima. A PAC-Bayes generalization bound controlled explicitly by SR formally links this geometric bias to test performance. Our experiments on MNIST and CIFAR-10 confirm that SR reliably tracks generalization in ways that Euclidean sharpness does not, and that its scaling with $η/B$ matches the theoretical predictions. Together these results provide a rigorous, reparametrization-invariant account of why flat minima generalize.

We present a unified framework that turns a set of 6-DoF inspection viewpoints into a time-optimal, collision-free route for a 9-DoF robotic system. Unlike modular pipelines that fix a single inverse-kinematics (IK) configuration per viewpoint, build an all-pairs travel-time map, and then route, our method jointly optimizes the visiting order and the per-viewpoint configuration in a single global search. The three-dimensional self-motion manifold of each viewpoint is parameterized in closed form so that the pose constraint holds by construction, the rest-to-rest travel time is approximated by a closed-form admissible double-integrator surrogate, and the tour is encoded by random keys. A derivative-free optimizer (CMA-ES) minimizes a cheap penalized objective over order and configuration, after which direct-collocation trajectory optimization is applied only to the edges of the selected route to certify dynamic feasibility and torque limits, and to return exact timings. This reduces the trajectory solves from quadratic to linear in the number of viewpoints and removes the decoupling that prevents modular pipelines from being globally time-optimal. Simulations and real-robot experiments on a KUKA LBR iiwa with a 2-DoF linear stage validate feasibility, smooth execution, and reduced end-to-end inspection time relative to modular and naive distance-based baselines.

In Self-Supervised Learning (SSL), preventing representation collapse by explicitly enforcing a uniform distribution on the unit hypersphere has proven to be effective. However, current frameworks typically rely on sliced statistical regularizers such as SIGReg (used in LeJEPA) and SUSReg (used in SPHERE-JEPA), which approximate this continuous objective via Monte Carlo sampling along random 1D directions. This stochasticity injects projection variance into the training gradients, destabilizing optimization, and hindering convergence. In this work, we first show that analytically integrating out these random projections natively yields a deterministic Maximum Mean Discrepancy (MMD), bypassing the variance of sliced methods. Motivated by this equivalence, we formulate full-dimensional objectives for MMD, Kernel Stein Discrepancy (KSD), and Kullback-Leibler (KL) divergence directly on the sphere to enforce a uniform distribution. To prevent spatial bias, we equip these tests with rotationally invariant kernels constructed via spectral theory, systematically evaluating two canonical families: smooth exponential decay (Heat) and strict frequency cutoff (Bandlimited) filters. Empirically, removing projection-induced noise results in more stable optimization, faster convergence, and consistent improvements over stochastic sliced regularizers on ImageNet and Galaxy10. Furthermore, we reveal that the choice of the statistical test shapes the geometry of the learned latent space: MMD and KSD favor locally clustered organization suitable for object-centric domains, whereas the continuous KDE-based KL divergence promotes fine-grained instance separation, yielding the strongest results on unclustered procedural texture retrieval.

Gaussian width is a central geometric complexity measure in high-dimensional probability, compressed sensing, convex optimization, and learning theory. It quantifies the average extent of a set along random directions, thereby capturing the effective dimension of constraint sets, hypothesis classes, and descent cones. However, this notion is intrinsically Euclidean. Statistical models instead carry a natural Riemannian geometry induced by the Fisher information metric, where directions are scaled according to statistical distinguishability rather than ambient Euclidean length. We introduce Fisher width, a Fisher-geometric analogue of Gaussian width for statistical manifolds. At a parameter point $θ$, Fisher width replaces the Euclidean identity by the local metric tensor $G(θ)^{1/2}$, measuring the Gaussian width of the Fisher-rescaled set. This makes the resulting quantity sensitive to local statistical curvature and invariant under smooth reparameterizations. We develop the basic theory of Fisher width, showing that it retains key structural features of Gaussian width, including concentration, metric perturbation stability, and spectral comparison bounds with the Euclidean baseline, while also capturing anisotropic geometric effects invisible to Euclidean measures. As an application, we prove a generalization bound for Fisher-Lipschitz hypothesis classes and propose computable estimators, which we evaluate empirically on MNIST across three model classes. Fisher width is to statistical manifolds what Gaussian width is to Euclidean convex bodies. This work lays the foundation for studying complexity and learning on curved statistical manifolds.

In this paper, we introduce SP$^3$, a novel Plug-and-Play algorithm that accelerates maximum a posteriori image restoration by replacing denoisers with Spherical Encoders (SE) as generative priors. SP$^3$ approximates the intractable proximal prior step by utilizing the SE tightly structured latent space as a robust projection onto the natural image manifold. Alternating this projection with a closed-form data-consistency step, via Half-Quadratic Splitting, achieves stable convergence without requiring gradient computation during inference. This unique formulation unlocks "anytime" restoration capabilities, producing sharp, plausible images from the first iteration. Evaluations across a variety of image restoration tasks demonstrate that SP$^3$ achieves perceptual quality comparable to state-of-the-art zero-shot diffusion and flow methods while being $3$-$630\times$ faster.

A significant gap exists between theory and practice in deep learning. Generalization and approximation error bounds are often derived for simplified models or are too loose to be informative. Many rely on the manifold hypothesis and on geometric regularity such as intrinsic dimension, curvature, and reach. Progress requires insight into data-manifold geometry and suitable benchmarks, yet existing options are polarized: analytic manifolds with known geometry but limited applicability, or real-world datasets where geometry is only coarsely estimable. We introduce a benchmarking framework for studying data geometry. We repurpose and extend dSprites and COIL-20 with additional transformation dimensions and dense, axis-aligned sampling, and pair them with finite-difference estimators that recover curvature, reach, and volume at near-ground-truth accuracy in a regime where general-purpose estimators are unreliable or difficult to deploy. The framework is intended as a controlled testbed, useful as a calibration environment for geometric estimators and a sandbox for probing theoretical assumptions. To illustrate its use, we present two application studies, namely assessing the scaling behavior of the bounds of Genovese et al. and Fefferman et al., and tracking the layer-wise geometry of a $β$-VAE, highlighting the behavior of current bounds and the value of controlled benchmarks for guiding and validating future theory. A reference implementation is available at https://github.com/koulakis/manifold-microscope.

Most existing anomaly detection methods rely on estimating a probability density or learning an enclosing decision boundary, implicitly assuming that normal data occupies a region of non-zero volume in the ambient space. In contrast, structural anomaly detection considers data that lies near a low-dimensional manifold, creating a mismatch between the inductive bias of existing methods and the structure of the data, often resulting in degraded performance. To address this mismatch, we introduce a geometric perspective. Specifically, we learn a projection operator onto the manifold of normal samples and define a sample as anomalous if it is altered by this projection. This formulation naturally integrates the inductive bias of manifold-supported data and reframes anomaly detection in terms of a projection residual, thereby resolving issues arising from modeling degenerate distributions. Notably, it provides a unifying interpretation of reconstruction-based methods by explaining their success and failure in terms of projection quality. In particular, it explains the strong generalization ability of projection-aligned models as a consequence of contraction behavior toward the manifold. Moreover, by decoupling anomaly detection from probabilistic modeling, it reduces the tendency to misclassify rare but normal samples, a widely recognized limitation of existing approaches. Empirically, we demonstrate that projection-aligned methods achieve strong performance, outperforming boundary-based methods while improving upon existing reconstruction-based approaches.

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Weight-space geometry plays a central role in neural network optimization, yet manifold constraints are often applied uniformly across all weight matrices. In this work, we ask whether different transformer modules prefer different manifold geometries. We study Manifold Muon for GPT-2 pretraining and compare layer-wise assignments of Stiefel and DGram constraints across attention and MLP blocks. Our results show a clear asymmetry: constraining attention layers with Stiefel geometry while assigning DGram geometry to MLP layers gives the best performance among the tested configurations, whereas the inverted assignment and all-DGram configuration become unstable under the shared hyperparameter setting. We trace this failure to singular value growth in DGram-constrained attention weights, which can amplify attention logits and induce softmax saturation. These findings suggest that symmetry-aware and geometry-aware optimization for transformers should be module-specific rather than uniform.

Neural speech codecs based on Vector-Quantized VAEs (VQ-VAEs) are core audio tokenizers for speech LLMs, yet their reconstruction fidelity is bottlenecked by quantization error. Modifying the quantizer or increasing model capacity are common fixes, but they complicate downstream language modeling. Our core idea is to align the decoder's internal feature manifolds when processing both the quantized tokens and their original continuous embeddings, using a lightweight feature-mapping loss. This requires minimal training overhead and no inference-time changes. Applied to XCodec2, self-guidance improves all reconstruction metrics, achieving state-of-the-art low-bitrate performance. Notably, it enables a 4x codebook reduction without fidelity loss, which downstream TTS experiments show significantly improves LLM-based synthesis by simplifying the token modeling space. Multiple statistical observations and visualizations corroborate the enhanced internal manifold alignment in the decoder. Extensive experiments confirm its generality across various inductive biases. Self-guidance thus establishes an efficient, broadly applicable method for high-fidelity neural audio coding.

Low-Rank Adaptation (LoRA) significantly reduces compute and memory costs for finetuning Deep Learning models but is often harder to tune than dense training: when using factor-wise optimizers such as AdamW, it is sensitive to initialization choices, its optimal learning rates transfer poorly across ranks, and it often fails to beat dense baselines. We derive LoRA-Muon by applying the Muon optimizer's spectral steepest-descent rule to the low-rank setting. Along with our split weight-decay rule, our main claim is that LoRA-Muon is a good low-rank proxy for full-rank Muon and Shampoo-family optimizers. Its optimal learning rates transfer across rank, width, depth, and factor-rescaling. In our compute-matched TinyShakespeare study, a rank-$2$ proxy recovers the dense best tested learning rate, and a rank-$32$ LoRA-Muon run attains lower mean validation loss than the dense baseline in the seed-averaged sweep. We further show that the Spectron optimizer depends on arbitrary factor scaling, so it would likely be a poor fit when finetuning starts from badly imbalanced factors, and that LoRA-RITE's simplified QR-coordinate core implements the same spectral update. LoRA-Muon computes that update without QR-decomposition and avoids storing second moments, making it more accelerator-friendly and memory-efficient.

Designing dynamically feasible trajectories for rigid bodies is a fundamental problem in robotics. While direct methods are widely used, the existing constrained optimizers typically operate in Euclidean space and ignore the manifold structure of rigid body motions. This mismatch may introduce singularities or lead to poorly conditioned optimization problems. To bridge this gap, we develop a structure-aware framework for constrained trajectory optimization directly on matrix Lie groups. Our approach is based on the second-order rigid body models utilizing Lie group structures, which enables efficient Newton-type updates while preserving the underlying geometry. Building on this model, we propose a line-search Lie Group Interior Point Method (LieIPM) to handle constraints on the manifolds. We instantiate the framework for rigid body motion planning using Lie group variational integrators and derive closed-form intrinsic derivatives that exploit group symmetries. The LieIPM preserves the topology of rotation motions by construction and avoids singularities. Numerical results demonstrate superior robustness and faster convergence compared to general-purpose solvers and structure-exploiting optimal control methods.

Multivariate time series forecasting requires capturing the continuously evolving correlation structure among interacting variables. Existing state-space models process time series by scanning tokenized temporal or spatial sequences, discarding the evolutionary geometric structure. We address this limitation by introducing manifold constraints into state-space modeling: treating the cross-variable correlation structure as a continuous trajectory on the symmetric positive definite manifold, whose Riemannian geometric features, tangent space linearity, and Frechet mean centrality act as a principled geometric regularizer that guides and stabilizes the selective scanning dynamics of SSMs. We propose SPDM, a geometry-aware SSM architecture that realizes this principle through two cooperating mechanisms: a manifold trajectory path that projects dynamically evolving covariance matrices from the SPD manifold to a Euclidean tangent space, and a geometric gating scheme that directly modulates SSM's internal selective parameters based on geometric signals derived from the manifold trajectory. The parameterization preserves the linear-time complexity of the Mamba parallel scan while embedding rich structural constraints, making the architecture preserve prediction accuracy and computational efficiency simultaneously. Extensive experiments on eleven real-world benchmark datasets establish state-of-the-art forecasting performance, and further studies confirm that geometrically constrained state-space dynamics are the dominant architectural factor behind its performance gains.

Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data. When the data lies on a manifold with non-Euclidean topology, the standard Gaussian prior introduces a topological mismatch that degrades reconstruction quality and prevents faithful representation. We present a constructive mathematical framework that resolves this mismatch for all manifolds that admit a product covering space. These are manifolds expressible as products of elementary factors (circles, intervals, or lines) or as quotients of such products by a finite symmetry group. The class includes cylinders, tori, Möbius strips, Klein bottles, and real projective spaces. Factorized distributions over the elementary factors yield product topologies with closed-form, decoupled KL divergences, so that each latent factor can be shaped independently while keeping training tractable. We catalogue reparametrizable encoder-prior pairs for periodic, bounded, and unbounded supports, and provide coordinate transformations that allow standard neural networks to output non-Euclidean parameters with smooth gradients. For quotient manifolds, the decoder receives group-invariant features of the covering-space coordinates, so that identified points produce identical outputs. Anchor constraints fix the coordinate system relative to the data or create soft topological holes. Experiments on synthetic manifolds and real-image datasets (rotated and cyclically shifted MNIST) confirm that a topology-matched prior aligns KL regularization with the data manifold. The resulting topology-aware models outperform the Gaussian baseline at all practically relevant regularization strengths. The code is available at https://github.com/JvHulst/VAE-Topology.

The rapid evolution of generative image models challenges existing AI-generated image detectors, particularly in open-world settings with unseen generators. Recent training-free approaches measure robustness gaps in frozen vision foundation models (VFMs), detecting fakes via perturbation-induced embedding drift. However, these methods rely on fixed invariance geometry inherited from pretraining and lack principled adaptation to the detection task. We instead formulate AI-generated image detection as learning a structured invariance manifold of real images under one-class supervision. Building upon a frozen VFM, we introduce lightweight projection heads that decompose representation space into complementary robust and fragile subspaces. The robust subspace is explicitly trained to suppress variations induced by physically plausible imaging transformations, approximating tangent directions of a real-image manifold, while the fragile subspace retains sensitivity to edit-like perturbations. A structured ordering margin enforces hierarchical separation between physical invariance and edit-induced variability, enabling detection as a margin-violation test relative to the learned manifold. At inference, multi-scale patch-wise drift under both transformation families yields a dual-channel invariance signature and interpretable localization. Extensive experiments demonstrate strong open-world generalization across unseen generators and resolutions, consistently outperforming training-free robustness-based baselines while providing interpretable invariance-violation maps.

Blind image restoration requires recovering clean images from observations corrupted by unknown and potentially mixed degradations. While recent deterministic flow-based methods model restoration as transport processes that map degraded images to clean ones, they typically rely on Euclidean interpolation, implicitly assuming linear degradation geometry. In this paper, we explicitly model degradations as points on a low-dimensional Riemannian manifold and formulate restoration as geodesic transport on the joint image-manifold space. Using a geodesic flow matching objective, we learn intrinsic transport dynamics that respect the curvature of degradation space. This framework generalizes linear flow matching, provides a principled treatment of mixed degradations as geodesic compositions, and yields a clean theoretical interpretation for generalization beyond observed degradations.

Across contemplative, philosophical, and psychological accounts, human consciousness is often described along a similar spectrum, ranging from reactive and self-focused patterns to more integrative and coherent ones. Understanding whether language models encode such a structured, human-interpretable consciousness spectrum in representation space is important for model guidance, evaluation and alignment. In this work, we study the geometric structure and dynamics of patterns along this spectrum in transformer embedding spaces. We show that embeddings exhibit a globally organized geometry aligned with this spectrum: sentences associated with similar states cluster into locally coherent regions, forming a structured manifold. In particular, higher-level and lower-level regions exhibit convexity-like stability, while intermediate regions form a transition corridor. Dynamically, both utility-guided and geometry-only greedy trajectories consistently traverse from lower- to higher-level regions, passing through intermediate tiers, indicating that navigability is an intrinsic property of the representation space, guided but not dictated by a global directional signal. These results suggest that embedding spaces encode structured and navigable geometry aligned with a hypothesized consciousness-spectrum taxonomy, broadly inspired by recurring structural descriptions of human consciousness across contemplative traditions, philosophy, and modern psychology, providing a representation-level perspective for analyzing and guiding model behavior.

Neighbor embedding algorithms reveal correlations in high-dimensional data by constructing an equivalent graph representation in a lower-dimensional space. An increasingly popular algorithm is Uniform Manifold Learning and Projection (UMAP), which uses algebraic topology to map distances between the two spaces. While it works well on many types of data sets, UMAP has trouble adding out-of-sample points to a pre-existing mapping. In particular, UMAP often places new points on the periphery of the found clusters, rather than in their interiors with their correlated neighbors. Here, we overcome this ``repulsion effect'' by optimizing pairwise interactions within the original k-nearest-neighbor graph. Moreover, we show that parameterizing UMAP obtains better embeddings than non-parametric algorithms, particularly as the data gets more complex (e.g., medical images). We also show that the repulsion effect is naturally mitigated when a parameterized UMAP is employed to embed the data. We characterize different UMAP approaches using trustworthiness, nearest neighbor classifiers, and by analyzing attractive and repulsive forces in the embeddings.

Symmetric nonnegative matrix factorization (Symmetric NMF) approximates a matrix as $WW^T$ with nonnegative rectangular factor $W$. It has broad applications in graph clustering and machine learning. In contrast to the NMF, projected gradient methods for the symmetric problem had been associated with slow convergence. To address this, we introduce SNMPBB, the first adaptation of nonmonotone projected Barzilai-Borwein methods to Symmetric NMF, demonstrating that gradient algorithms are significantly more effective than previously understood. We further extend SNMPBB to graph clustering using the graph Laplacian regularization (Graph-SNMPBB) and to large problems with low-rank approximations (LAI-SNMPBB). For all variants we prove global convergence to first-order stationary points and also that Barzilai-Borwein curvature information is preserved with randomized approximations. On synthetic data, SNMPBB achieves 6 times speedup over the alternative SymANLS for similar residuals, with advantages growing at higher ranks. Across six real-world clustering benchmarks, Graph-SNMPBB matches or exceeds SymANLS accuracy. Lastly, LAI-SNMPBB outperforms state-of-the-art LAI-SymPGNCG on 34 SuiteSparse matrices in both runtime and residual quality.

Generating physically consistent 3D tabletop scenes is a fundamental yet underexplored problem for interactive and generalist robotic learning. The challenge stems from dense object hierarchies and irregular affordances. Here, an interactive scene denotes a physically valid, collision-free environment directly loadable into physics simulators. Existing methods, ranging from decoupled symbolic solvers to end-to-end regression models, often suffer from error propagation or overfitting to noisy supervision containing widespread physical violations. To address these limitations, we introduce PhyScene3D, a framework that reformulates generation as a Human-Mimetic Constructive Process. The proposed Cognitive Topological Reasoning Chain (CTRC) factorizes scene synthesis into a sequential, anchor-conditioned process. It employs a 3D AABB-based placement scheme that imposes a strong structural inductive bias. To address imperfect supervision and physical infeasibility, we introduce Physics-Aware Denoising Alignment (PADA). It integrates a differentiable Signed Distance Field (SDF) with Test-Time Optimization (TTO) to project generated scenes onto a physics-feasible manifold while preserving semantic intent. Experiments demonstrate that PhyScene3D outperforms state-of-the-art approaches in both semantic accuracy and physical validity, achieving a 40% reduction in scene-wise collision rate relative to the human-annotated training data.

Sparse high-dimensional representations are conducive to uncovering nontrivial structures in unsupervised exploration of data. Such a representation can deal with the dense connectivity in graphs relevant to community detection problems. However, sparse high-dimensional representations are capable of doing more, including manifold tiling and feature learning. Conventional algorithms optimize in the space of computationally intractable completely positive matrices or relax the problem to the space of doubly nonnegative matrices that scale with sample size in a way rendering them impractical for large data sets. Some of these methods also impose a row sum constraint, such as double stochasticity. Row sum constraints have the added advantage of being shift-invariant, in the context of manifold tiling. Constraints on the row sum of output similarity matrices require nontrivial online learning rules. Addressing these needs, we propose a versatile online biologically plausible learning algorithm capable of learning sparse shift-invariant representations, useful for clustering, manifold tiling, or sparse coding, depending on the data structure.

MLLMs frequently hallucinate objects inconsistent with visual inputs. This issue is typically attributed to the over-reliance on language priors, which can override the visual context. Recent training-free decoding strategies address this by penalizing language priors. However, these methods overlook the dual nature of language priors, where they can be both helpful and harmful depending on the alignment with visual evidence. In particular, blindly suppressing language priors often disrupts the model's semantic manifold, leading to performance degradation, a phenomenon we term Manifold Departure. To address this, we propose Manifold-Guided Adaptive Projection (MGAP), a geometry-aware, training-free decoding method that mitigates hallucinations while preserving representation structure. MGAP first constructs a language-prior subspace from blind hidden states via SVD. During decoding, MGAP projects each multimodal hidden state onto this subspace and applies a consistency-aware gate to adaptively attenuate only the projected prior component, yielding a subspace-selective update that largely preserves the orthogonal semantic components. Extensive experiments on POPE and CHAIR show that MGAP outperforms prior decoding baselines, achieving stronger hallucination suppression without sacrificing coherence.

A simple way to improve the performance of almost any machine learning model is not to train a single but several models with diverse algorithms which will make slightly distinct kinds of predictions and errors on the same data, and thus improve the average predictions and robustness. However, making predictions using a whole ensemble of models is cumbersome and computationally too expensive to allow deployment to a large number of users, especially if the models are large neural nets. In response to this, we introduce a layer and point wise projection mapping, which maps student and teacher representations into an aligned high-dimensional embedding space during training process. The proposed approach combined with LoRA injection reduces the student model trainable parameters to less than 1% of the teacher model, while significantly improving word error rate (WER) compared to other distillation methods, as demonstrated in ablation studies. Unlike a mixture of experts, our method can be trained rapidly and in parallel.

Riemannian diffusion models generalize score-based generative modeling to manifold-supported data via stochastic diffusion equations on the manifold. However, training requires sampling from and differentiating the manifold heat kernel, which is rarely available in closed form beyond a few highly symmetric manifolds. We propose a general approach that approximates the heat kernel by directly solving the manifold heat equation with a physics-informed neural network (PINN). Given an explicit manifold specification, we choose a coordinate system, derive the corresponding heat (Fokker--Planck) equation and a short-time asymptotic approximation, and then train a PINN to learn the log heat kernel. The resulting surrogate enables both forward noising (heat-kernel sampling) and conditional-score evaluation for denoising score matching. We demonstrate the method on diverse manifolds including $S^2$, $SO(3)$, $\mathrm{SPD}(n)$, and permutation-quotiented point clouds.

Cross-domain EEG decoding remains challenging despite advances in Riemannian deep learning: covariance matrices from different subjects occupy systematically distinct regions of the SPD manifold, yet existing domain adaptation methods either require target-domain calibration data or learn subject-specific components that cannot generalise across domains. We propose dynamic Stiefel routing: a pool of $K$ expert projection filters on the Stiefel manifold, each specialised for a different region of the SPD manifold, with each input covariance routed to the most appropriate filter via cross-attention, adapting the subspace projection per sample. A central finding is that this approach, implemented naively, provably collapses to ensemble averaging: when routing weights are uniform, the adaptive filter reduces exactly to an equal-contribution combination of experts, indistinguishable from a single fixed filter. Three structural properties break this degeneracy: a symmetric anchor $W_{\mathrm{base}} \in \mathrm{St}(n,k)$ that removes proximity bias among experts; a frozen domain-discriminative query encoder that decouples routing from task optimisation; and a decoupled key alignment loss that trains expert keys toward stable domain attractors. Together they produce the first genuinely committed and domain-structured routing on SPD manifolds, with consistent gains across three datasets: balanced accuracy improves from $0.773\to 0.823$, $0.757\to 0.809$, and $0.801\to 0.839$, with the alignment strategy determined automatically by a single data-driven rule and no dataset-specific hyperparameter search.

Detecting subtle visual anomalies in images remains challenging, particularly when only normal samples are available a priori. Such unsupervised anomaly detection is typically solved by measuring feature similarity of a query patch to a memory of normal patches. However, similarity alone does not reveal how strongly a query patch violates the structure of the normal feature manifold. We propose a training-free Laplacian graph energy optimization formulation, named ANoCo that scores Anomaly by the cost of Non-Conformity of a query patch to align with a fixed normal manifold. For each query patch, we construct a bipartite query to normal graph weighted by cosine affinity, explicitly removing query-query and normal-normal edges to prevent evidence dilution. We formulate anomaly scoring as a convex Laplacian energy with anchored normal nodes, and solve in closed form. In particular, we do not use the optimized features themselves-the anomaly score is the magnitude of the update required to satisfy normality constraints, reframing the graph Laplacian as a non-conformity operator rather than a smoothing prior. The proposed method introduces no learnable parameters, message passing, or sampling, and has complexity comparable to a single linear solve. Across standard benchmarks, it delivers strong image-level AUROC, stable localization maps, and improved robustness over prior methods, demonstrating the effectiveness of using optimization-induced feature drift as anomaly measure.

We take a Gromov-Wasserstein perspective on Vision-Language-Action (VLA) learning, where the goal is to make the relational geometry of action representations compatible with the semantic geometry of VL embeddings. However, this alignment is non-trivial due to the mathematical heterogeneity between the domains: the semantic space of vision-language is topologically linear and isotropic, whereas the physical manifold of robotic action is non-Euclidean and anisotropic. Their disjoint metric structures render direct regression ill-posed. To resolve this incompatibility, we introduce LAST (Lie-algebraic Action Space Tokenizer), which reconstructs the action space to establish local metric compatibility with the VL modality via a two-stage transformation: (1) Global Topological Linearization: linearizing the action manifold via Lie-algebraic mapping, converting trajectories into a fixed-length, physically additive representation. (2) Local Metric Discretization: hierarchically discretizing the representation into schemas and whitened residuals, yielding approximately isotropic local charts that are statistically aligned with the semantic metric. By resolving the structural mismatch at both global and local levels, LAST enables VLA models with superior convergence and generalizability.

A fundamental open question in self-supervised learning (SSL) is the explicit characterization of the optimal geometry of the learned representations. Recently, LeJEPA identified isotropic Gaussian embeddings as optimal for minimizing downstream prediction risk in Euclidean spaces. However, the corresponding problem for distributions supported on lower-dimensional manifolds, such as the hypersphere, remains unexplored. In this work, we demonstrate that extending this minimax analysis to smooth distributions on Riemannian manifolds fundamentally changes the optimal solution. We show that, under a worst-case formulation, both k-nearest neighbors and kernel ridge regression induce hyperspherical uniformity. More precisely, we show that uniform distributions on manifolds are optimal for k-nearest neighbors, and that the uniform distribution on the sphere is optimal for kernel ridge regression with both the exponential dot-product kernel and the linear kernel. This theoretical insight reveals a fundamental limitation of Gaussian embeddings: their non-uniform density induces anisotropic k-NN neighborhoods, severely biasing the estimator. To correct this, we introduce SPHERE-JEPA, a theoretically grounded SSL framework. We adapt LeJEPA's Cram{é}r-Wold projection mechanism to enforce hyperspherical uniformity rather than a Gaussian prior. Empirically, SPHERE-JEPA yields significant improvements, boosting texture retrieval mAP by over 6%, while consistently matching or outperforming LeJEPA on standard benchmarks-including a +1.8% linear probing gain on ImageNet-1K (ViT-B/14).

Robotic systems with redundant degrees of freedom can achieve the same task outcome using multiple configurations, resulting in solution sets that form manifolds in the configuration space. Existing approaches typically exploit such redundancy locally through Jacobian-based techniques to compute individual solutions or trajectories. While effective for solution computation, these methods do not retain a representation of the geometry of the solution set itself. In this work, we adopt a representation-centric approach to estimate the geometric structure of the solution space. We consider solution manifolds induced by general task-defining maps and construct an implicit scalar field over the configuration space, whose zero-level set corresponds to the solution manifold. To this end, we generate samples in the neighborhood of the solution manifold using a Jacobian-guided exploration strategy, which efficiently captures its local and global structure. The resulting implicit representation is defined over the configuration space and naturally induces a continuous, distance field that encodes proximity to the solution manifold. Experiments on a planar three-link robot and a seven-degree-of-freedom Franka manipulator demonstrate the effectiveness of the proposed representation. Furthermore, the framework enables consistent modeling of solution spaces across families of tasks with continuous variation.

Image generative models aim to sample data points from the underlying data manifold, a task that requires learning and decoding a dense, low-dimensional, and compact parameterization space. To achieve this, we propose the Data Manifold-aware Image diffusioN moDel (MIND), a novel framework that explicitly models manifold geometry by integrating discrete patch tokenization into the score function of a continuous diffusion model. This approach successfully leverages both the structural quantification capabilities of discrete tokens and the parallel generation flexibility of continuous diffusion. Moreover, we enable end-to-end differentiable training via a novel soft top-$k$ aggregation mechanism and introduce dual-branch high-frequency feature embedding layers to alleviate the spectral bias of transformer backbones on low-dimensional inputs. Furthermore, for inference, we design a multi-stage transition sampling scheme that dynamically adjusts the sampling scheme based on timestep. Extensive experiments on ImageNet 256$\times$256 demonstrate the effectiveness of MIND. After 80-epoch training, our base model achieves an FID of 22.73 without guidance, nearly halving the 43.47 FID of the vanilla DiT-B/2 baseline. The proposed method reduces FID by 15.95 and 9.06 on average compared with the baselines DiT and SiT, respectively. For image generation on ImageNet-256$\times$256 with guidance, the proposed MIND-B with only 130M parameters achieves an FID of 2.06, superpassing the LlamaGen-3B with 3.1B parameters. The proposed MIND-XL with 715M parameters further reduces the FID to 1.95. Our MIND introduces a fresh perspective on diffusion-based image generation, paving the way for future research and innovation in this community. The code will be publicly available.

The generation of accurate 3D molecular conformations is a pivotal challenge in computational chemistry and drug discovery. Recently, diffusion and flow matching models have achieved remarkable success. However, there is a critical misalignment between their mathematical formulation and the physical reality of molecules. Existing approaches predominantly treat molecules as unstructured point clouds in Cartesian space, overlooking the intrinsic hierarchical mechanics where bond lengths and bond angles are relatively stiff, whereas torsion angles constitute the dominant flexible degrees of freedom. This lack of manifold awareness forces models to relearn fundamental geometric constraints from scratch, often leading to physically implausible intermediate structures. To address this, we propose GO-Flow that aligns generative modeling with molecular geometry via manifold decomposition. Instead of forcing motion through Euclidean space, GO-Flow decomposes the generation process into three physically motivated subspaces: translation space with linear optimal transport, rotation space with geodesic flows on $SO(3)$, and conformation space with entropic optimal transport. This decomposition injects geometric inductive biases and makes the generative paths better aligned with molecular degrees of freedom. When combined with equivariant neural architectures, it encourages rotation-consistent generation and improves geometric validity. Extensive experiments on GEOM-Drugs and GEOM-QM9 demonstrate that GO-Flow achieves state-of-the-art generation quality. Notably, by learning straighter probability paths on the correct manifolds naturally, our method enables high-fidelity sampling with as few as 50 steps, effectively bridging the gap between structural precision and computational efficiency.

We study denoising score matching (DSM) when the latent distribution is supported on a smooth embedded manifold $M \subset \mathbb{R}^D$. Under ambient Gaussian corruption, the tangent denoising target contains a singular normal-fiber noise channel whose variance diverges as $d/σ^2$ as $σ\to 0^+$. We show that conditioning on the nearest-point projection $π(X)$ canonically removes this singularity: the resulting conditional expectation is the unique $L^2$-optimal Rao-Blackwellized predictor of the tangent DSM target among all estimators depending only on the projected observation $π(X)$. We then compute the small-noise expansion of this canonical target and show that it equals the intrinsic Riemannian score up to an explicit order-$σ^2$ correction that decomposes into an intrinsic Tweedie term and an extrinsic curvature term involving the Weingarten and Ricci operators. In the flat case, the construction reduces exactly to ordinary lower-dimensional Gaussian DSM, while on $S^d$ the extrinsic correction simplifies to the scalar factor $(1-d/2)\nabla_M \log q$; this extrinsic $σ^2$ correction cancels identically on $S^2$, though the intrinsic Tweedie term remains.

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{tΔ}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly defined, and we recover the classical graph-Laplacian pointwise rate $O(N^{-2/(d+6)})$ up to logarithmic factors, for diffusion times $t$ up to $O(1)$ and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of $Q_t f$ at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density $p$ is $C^3$ and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.

Recent advances in generative models highlight the power of geometry-aware modeling in manifold-constrained settings. Yet, for natural images, the field remains confined to Euclidean assumptions, failing to exploit the potential of intrinsic geometric structures within the data. In this work, we investigate the geometry of natural images and observe that semantic information is predominantly encoded in directional components, while norm components can be approximated by the global average. This property holds across both RGB and latent spaces, suggesting that natural images can be effectively modeled on a hypersphere. Building on this finding, we introduce Spherical Optimal Transport Flow Matching (SOT-CFM), which utilizes angular distance, and Spherical Flow Matching (SFM), which constrains dynamics directly on the manifold. Our experiments demonstrate that these geometry-aware methods achieve superior performance against Euclidean baselines. Ultimately, this work provides a novel perspective that bridges the gap between Riemannian manifold-based modeling and natural image generation.

We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for continuous symmetry of manifold geometry. Thereby we achieve learnable and stable dynamics on the underlying manifold for general Lie group, and we are able to utilize the powerful representation capability of Lie group such as SO(3) and SE(3) to solve real world engineering problems in areas such as robotics, graphics, and control. Two core challenges are: (i) General Lie groups are incompatible with addition arithmetic, which is necessary for neural network interactions. (ii) The dynamics evolve in the nonlinear representation space of special algebra rather than the normal Euclidean space, which violates the paradigm of common neural ODEs. To address these two challenges, we firstly introduce adjoint Lie group action on the Lie algebra, which induces a linear mapping and transfer to the block-wise structure of weight matrices, such that addition could operate on the Lie algebra as a vector space. Then we parameterize the Lie algebra and the adjoint action as linear transformation so that the architecture is aligned with neural network perceptrons. Explicitly, this embedding appears as block-wise manifold constraints on weights, and we develop algorithms to learn the equilibrium with stability guarantees of the temporal neural network dynamics. Experiments are implemented on a specific Lie group SE(3), with the application scenario of telescopic manipulators.

The rapid evolution of generative models has enabled the creation of hyper-realistic facial deepfakes, exposing a critical vulnerability in modern digital forensics: the inability of detectors to generalize to unseen manipulation techniques. Traditional networks suffer from representation collapse, overfitting to localized artifact fingerprints of specific training generators. This work investigates whether modern Vision Foundation Models can serve as generalizable, out-of-the-box feature extractors capable of tracking forensic anomalies across entirely unseen generative manifolds. We conduct a systematic cross-domain evaluation comparing three foundational learning paradigms: fully supervised macro-semantic features (RoPE-ViT), pure self-supervised geometric features (DINOv3), and multi-teacher agglomerative representations (NVIDIA C-RADIOv4-H). By deploying frozen backbones subjected to downstream linear probing, we map the performance limitations of these architectures on the challenging DF40 benchmark. Our empirical findings expose the intrinsic trade-offs between pre-training paradigms and parameter scale, proving that while foundation models retain high discriminative capabilities for entire face synthesis, localized face editing techniques expose fundamental boundaries in linear probe evaluation structures. Source code and model weights are available in http://github.com/mribrahim/deepfake

Low-dimensional embeddings are widely used as visual summaries of high-dimensional data and to enable downstream scientific discoveries. Yet, popular nonlinear dimension reduction methods, such as t-SNE and UMAP, are often selected based on visual appeal alone and without rigorous quantitative validation. A major reason is that manifold embeddings typically do not provide an out-of-sample map nor an inverse back to the original feature space; this makes held-out validation, the gold standard in supervised learning, all but impossible. To address these challenges, we develop a novel framework, MEDAL (Manifold Embedding Distillation via Autoencoder Learning), which distills a fitted manifold embedding into a reusable encoder--decoder model. MEDAL trains a constrained autoencoder whose bottleneck exactly matches any teacher embedding while the decoder reconstructs the original input; this yields an explicit map for new samples, an approximate inverse, and a pointwise reconstruction-based measure of distortion in the manifold space. This converts static manifold embeddings into models that can be evaluated on held-out data, enabling quantitative validation including comparing different dimension reduction methods as well as hyperparameter tuning. Across multiple benchmark and scientific case studies, we show that MEDAL enables held-out validation to determine optimal manifold embeddings and hyperparameters, reveals biologically coherent regions that are difficult to preserve in two dimensional embeddings, and detects distribution shift when new samples are mapped into a fixed reference manifold. MEDAL provides a general validation wrapper to any existing dimension reduction technique that will improve the rigor and reliability of dimension reduction in scientific workflows.

Generative priors in Image Super-Resolution (SR) often compromise faithful restoration, we attribute this limitation to a fundamental spectral misalignment between isotropic objectives and the intrinsic natural image manifold. While Direct Preference Optimization offers a path to alignment, its reliance on spectrally flat Gaussian noise fails to distinguish authentic high-frequency details from hallucinations. To bridge this geometric gap, we propose ASASR, a theoretically grounded framework that recasts the generative flow into a Sobolev-induced Riemannian geometry by explicitly coloring the noise transition kernel to mirror natural spectral decay. Driving this geometric alignment, we integrate a parametric adversary grounded in the Riesz Representation Theorem, which synthesizes targeted negative samples equivalent to worst-case Sobolev gradients to direct optimization along the tangent space of plausible structural failures. Extensive evaluations demonstrate that ASASR outperforms leading generative baselines, particularly in preserving spectral consistency and structural fidelity, offering a robust solution that effectively mitigates artifacts.

Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities -- clean intersection and intrinsic transversality -- are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.

Spectral clustering largely depends on the affinity graph, yet constructing a graph that preserves reliable local connectivity while adapting to heterogeneous data structures remains challenging. Existing granular-ball-based spectral clustering methods usually reduce graph complexity by using coarse-grained representatives. However, the learned local regions are often treated as graph nodes or anchors, and their structural information is not sufficiently used to regularize the original sample-level graph. To address this issue, this paper proposes a Minimum Description Length based Granular-Ball Tree-Regularized Spectral Clustering method, termed MDL-GBTRSC. The proposed method constructs a granular-ball tree through local MDL model selection, with reciprocal neighborhood continuity used to discourage splits that break reliable local connections. The stable leaf balls obtained from the tree provide coding-scale information for regularizing the sample-level affinity graph. In addition, a shared-neighbor bridge code is introduced to adjust weak local bridge relations without requiring an additional user-specified threshold. In this way, MDL-GBTRSC connects interpretable local representation learning with affinity graph construction in a unified spectral clustering framework. Experiments on real and synthetic datasets show that MDL-GBTRSC achieves the best average ARI and NMI under the adopted fixed-configuration protocol compared with classical spectral clustering baselines and representative granular-ball, micro-cluster, and anchor-based methods.

Correlation matrices are fundamental summaries of functional brain networks, yet standard analyses often treat entries independently, ignoring the curved geometry of correlation space. Existing geometric methods frequently lack closed-form operations or depend on arbitrary region ordering, limiting scalability. We introduce a scalable geometric framework with two components: (i) the Off-log metric, a smooth transformation mapping correlation matrices to symmetric zero-diagonal matrices. This enables closed-form expressions for distances, Frechet means, and linear models, allowing standard statistical modeling without complex manifold optimization. (ii) Grassmannian subspace discrimination, which compares subjects via principal-angle distances between eigenvector subspaces, resolving inherent sign and basis ambiguities. Both components integrate into standard machine-learning workflows for inference, regression, and classification. Validated across two clinical cohorts (Parkinson's and psychosis) and three ageing fMRI datasets, the Off-log metric increased sensitivity in permutation tests and matched or exceeded Riemannian and Euclidean baselines in classification. Brain-age prediction performance was comparable, with Riemannian metrics excelling in two of three cohorts. The Grassmannian method consistently outperformed Euclidean baselines, highlighting disease-relevant networks. Overall, geometry-aware representations improve sensitivity and predictive performance while remaining straightforward to deploy at scale.

Large language models frequently produce errors in reasoning tasks despite possessing the underlying knowledge required for correct reasoning. One possible approach to improve reasoning consistency is through activation steering. However, existing activation steering approaches apply fixed, pre-computed correction vectors, ignoring where the model currently sits along its generation trajectory; the result is indiscriminate perturbation that disrupts already-correct steps as freely as erroneous ones. We propose Manifold-Guided Attention Steering (MAGS), a trajectory-aware inference-time intervention grounded in a geometric observation: the output activations of specific attention heads diverge from a low-dimensional correctness manifold at the point of error, and this deviation compounds through subsequent steps. For each identified attention head, we learn a low-dimensional subspace from contrastive pairs of correct and incorrect traces that capture the directions along which error behavior deviates from correct behavior. During inference, we monitor each head's proximity to this manifold and apply a targeted projection correction when deviation exceeds a learned threshold, steering the attention output back toward the correct subspace before the error propagates. MAGS consistently outperforms both unsteered baselines and static steering approaches across benchmarks spanning mathematical reasoning (MATH-500, GSM8K), code generation (HumanEval, MBPP), and molecular generation (SMILES), suggesting that correctness manifolds are a general feature of LLM attention geometry.

This paper presents a low-complexity, model-free, output-feedback controller for a class of unknown time-varying nonlinear systems with unknown input constraints. The controller achieves the preset control accuracy when the actuator is not saturated and maintains flexible control accuracy after actuator saturation. This result extends existing constraint control methods for linear manifolds to a more general form, including the construction of nonlinear manifolds and various types of constraints, thereby achieving preset control accuracy within finite or fixed time. Additionally, flexible control under unknown saturation is achieved through the construction of an error-driven flexible constraint. Finally, second-order and higher-order control examples and simulations are provided.

Converting hand-drawn sketches into structured 3D geometries remains challenging due to the difficulty of representing non-Euclidean surfaces and maintaining topological consistency. Existing generative models such as GANs, NeRFs, and diffusion architectures often fail to produce editable manifolds directly usable in downstream design workflows. We present Sketch2MinSurf, a hybrid vision-language and geometric optimization framework that integrates vision-language guidance with minimal-surface theory to generate smooth and editable 3D surfaces from hand-drawn sketches. The core of our approach is a spatial-topological encoding that represents geometry as tuples of node coordinates and real/virtual edge skeletons, enabling stable topological control during generation. We further introduce the Sketch2MinSurf Structural Loss (S2MS-Loss), a reward-modulated objective that jointly constrains geometric reconstruction and topological coherence. On a test set of 100 sketches, Sketch2MinSurf achieves a topological similarity score of 0.844, outperforming existing sketch-to-shape baselines. The generated manifolds are directly editable and free from non-manifold artifacts. A public art installation at a university showcases the method's potential for human-intent-driven 3D form generation. The dataset and code are available at https://anonymous.4open.science/r/Sketch2MinSurf/.

Deep neural networks (DNNs) experience significant performance degradation when processing noisy labels, primarily due to overfitting on mislabeled data. Current mainstream approaches attempt to mitigate this issue by passively filtering clean samples during training. However, simple sample filtering within feature spaces degraded by noise struggles to distinguish between challenging samples and noisy samples, creating a bottleneck for model performance. We highlight for the first time the fundamental importance of actively reshaping feature space geometry for learning from noisy data. We propose a novel Geometry-aware Manifold Regularization Paradigm whose core idea is to explicitly construct energy barriers between data manifolds by actively synthesizing virtual outlier samples. By imposing geometric constraints that promote intra-class compactness and inter-class separation, this approach enhances the discriminability between hard and noisy samples, leading to the learning of more robust representations. Our regularization mechanism exhibits high universality, with effectiveness independent of any prior assumptions about noise patterns. It can be integrated as a standalone mechanism into existing sample selection frameworks, providing stronger robustness against diverse noisy environments. Experiments demonstrate that our paradigm achieves performance surpassing current state-of-the-art (SOTA) methods on multiple benchmarks, including CIFAR-10, with particularly pronounced advantages under more challenging asymmetric noise conditions. Furthermore, this paradigm significantly enhances the model's capability in Out-of-Distribution (OOD) detection, ensuring superior reliability and safety for deployment in open-world scenarios.

Machine unlearning is a fundamental mechanism that enforces the right to be forgotten. Existing unlearning studies that rely on label manipulation or task-gradient reversal often deliver limited unlearning effectiveness. Moreover, they can undermine the original learning objective and typically do not guarantee equivalence to standard unlearning by retraining. In this paper, we propose \textbf{ManiF-SMC} (\textbf{Mani}fold \textbf{F}orgetting with \textbf{S}elf \textbf{M}ode \textbf{C}onnectivity), motivated by the observation that a model retrained on the remaining data tends to classify erased samples by their semantic similarity to the retained data. We begin with systematically recasting the approximate unlearning as pushing each erased sample away from its original learned manifold representation centroid toward its nearest semantic neighbors in the retained data. This reformulation aligns unlearning with retraining behavior and operates purely in representation space, reducing reliance on labels and task-specific gradients. To tackle the manifold representation-based unlearning problem, ManiF-SMC encapsulates the unlearning and representation preservation goals in a margin-based triplet loss. Because finding a suitable margin for unlearning is challenging, we propose a self-mode-connectivity module that rapidly reconstructs the local manifold to guide the adaptive margins generation for each unlearning case. Extensive experiments on four representative datasets show that ManiF-SMC achieves unlearning effectiveness comparable to state-of-the-art approximate methods while operating solely within the model's representation space.

Diffusion and flow-based generative models dominate visual synthesis, with guidance aligning samples to user input and improving perceptual quality. However, Classifier-Free Guidance (CFG) and extrapolation-based methods are heuristic linear combinations of velocities/scores that ignore the generative manifold geometry, breaking probability conservation and driving samples off the learned manifold under strong guidance. We analyse guidance through the continuity equation and show its effect decomposes into a divergence term and a score-parallel term defined invariantly across parameterisations. We prove the divergence term blows up structurally as sampling approaches the data manifold, motivating a time-dependent schedule alongside score-parallel attenuation. The resulting plug-and-play rule, Adaptive Manifold Guidance (AdaMaG), bounds both terms at no additional inference cost. Finally, we show that most empirical heuristics for reducing saturation or improving generation quality correspond directly to the two terms in our decomposition. Across image generation benchmarks, AdaMaG improves realism, reduces hallucinations, and induces controlled desaturation in high-guidance regimes.

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph weighted by pairwise distances, without requiring any prior vector representation of the data. The embedding is obtained by solving a variational problem that matches local, on-graph distances to the Euclidean metric, induced by the differentials of the embedding functions. The resulting Euler-Lagrange equations are derived in a coordinate-free form, enabling direct evaluation of all operators from the distance graph alone. Though non-linear and missing an explicit expression for their non-linearity, these equations are shown to be resolved as an iteratively updated sparse linear problem. The main contributions of the proposed approach are (a) the derivation of the functional equations governing the optimal Euclidean embedding in the continuum, (b) a representation-free formulation that requires only a neighborhood distance graph and no feature vectors and (c) an estimation procedure based exclusively on local graph operations. We experimentally evaluate the resulting non-parametric algorithm on synthetic manifolds and real datasets, demonstrating consistent preservation of local metric structure and neighboring relations, while approximating the global isometric embedding.

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

This paper introduces the Manifold Probe, a supervised method for discovering representation manifolds in superposition. The method generalizes linear regression probes by learning the space of features of a concept that can be linearly predicted from the representations, and then learning the directions used to encode them. We demonstrate the probe on representations of time and space in Llama 2-7b, finding manifolds which linearly represent an interpretable set of features in each case. In the case of time, we show that by steering along the manifold, we can influence the model's completions about the years in which famous songs, movies and books were released, providing evidence that the Manifold Probe can discover manifolds which are causally involved in model behaviour.

Recent Singing Voice Synthesis (SVS) advances enable highly realistic but potentially malicious AI covers, making singing voice deepfake detection (SVDD) crucial. Self-Supervised Learning (SSL)-based detectors achieve state-of-the-art performance by fine-tuning speech SSL backbones to capture singing-specific spoof artifacts. Existing adversarial attacks often fail against SSL-SVDD, creating a false impression of inherent robustness. We reveal this stems from two challenges. First, at the objective level, attacks optimize cross-entropy on local surrogates, crossing surrogate-specific boundaries rather than suppressing shared spoof evidence. Second, at the method level, attacks follow the surrogate's dominant gradient direction. In SSL-SVDD, this aligns with fine-tuned artifact-sensitive directions, limiting transferability to unseen detectors - a geometric failure we term the Linearity Trap. To properly evaluate robustness, we propose MARS (Meta-Adversarial Regression of Semantics), a transfer-based black-box framework tailored to SSL-SVDD. Structurally, MARS shifts to hypothesis-evidence manipulation by constructing a natural semantic anchor from the pre-trained SSL space and an artifact anchor from the fine-tuned space. Algorithmically, MARS escapes the Linearity Trap via bi-level optimization: the inner stage induces tangential exploration, while the outer stage guides the audio toward the natural semantic manifold. Experiments on the CtrSVDD benchmark show MARS improves Attack Success Rate (ASR) in in-distribution transfer (13%), out-of-distribution transfer (10%), and cross-task evaluation (36%), highlighting the urgent need for robust SVDD systems.

Score-based generative models are trained in high-dimensional ambient spaces, yet many data distributions are supported on low-dimensional nonlinear structures. We prove that, for compact $d$-dimensional smooth manifolds $\mathcal{M} \subset [0,1]^D$ with $d > 2$ and $β$-Hölder densities strictly positive on $\mathcal{M}$, a variance-preserving SGM estimator attains the intrinsic Wasserstein--1 sample exponent $\tilde{\mathcal{O}}(D^{\mathcal{O}_β(d)}n^{-(β+1)/(d+2β)})$, up to logarithmic factors and explicit geometry and density factors. The full nonasymptotic bound explicitly isolates the finite-order geometry envelope, Hölder radius, density lower bound, ambient dependence, and finite-order correction terms. The analysis separates score approximation into a large-noise tangent-cell regime and a small-noise projection-centered, de-Gaussianized Laplace regime. The key technical ingredient is a ReLU implementation of nearest-projection coordinates via finite intrinsic anchors and Gauss--Newton iterations, rather than approximating the manifold projection as a black-box high-dimensional smooth map. Consequently, for families with polynomially controlled geometry and density lower bounds, the constructed score-network parameters have polynomial ambient dependence.