manifold-anchored em

This study addresses the challenging problem of fitting an unknown number of hyperplanes to data, which involves non-convexity, non-differentiability, and uncertainty in model order. To tackle these difficulties, the authors propose a two-stage unsupervised learning approach grounded in a unit-sphere manifold framework. In the first stage, they integrate Riemannian expectation-maximization with heavy-tailed kernel density estimation to robustly infer posterior probabilities. The second stage employs hard assignment annealing to obtain geometrically consistent local optima. Key innovations include a manifold optimization framework for handling non-convex constraints, a projection-based density estimation scheme for initialization, and the two-stage optimization strategy itself. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art baselines in both geometric accuracy and robustness.

This work addresses the challenges of modeling complex dependencies and mitigating redundancy in high-dimensional parameter spaces for discrete data generation. It introduces, for the first time, a Riemannian geometric structure with isometric properties into the exponential parameter space of product manifolds over categorical distributions, thereby constructing a low-dimensional latent subspace. By leveraging the Riemannian metric, geodesics within this subspace become straight lines, enabling consistent and efficient flow-matching training. The proposed approach substantially reduces the dimensionality of latent variables while preserving strong representational capacity for discrete data distributions. Experimental results demonstrate that the model achieves accurate and efficient discrete data generation using a significantly lower-dimensional latent space, effectively balancing computational efficiency with modeling performance.

Existing generative AI approaches typically treat image geometry—e.g., manifold structure—and probabilistic modeling—e.g., latent-space distributions—in isolation, often assuming uniform priors or neglecting the intrinsic Riemannian metric of the data manifold, thereby limiting generation quality. This paper proposes the Manifold-Probabilistic Projection Model (MPPM), the first framework unifying differential-geometric and probabilistic perspectives for image generation. MPPM formalizes generation as a deterministic projection onto the “high-quality image” manifold, jointly characterizing geometric structure and distributional properties in both pixel and latent spaces. Eschewing stochastic sampling, MPPM constructs a differentiable projection operator via kernel density estimation and manifold learning, revealing that diffusion models fundamentally implement manifold projection. Empirical evaluation shows that its latent-space variant, LMPPM, outperforms Latent Diffusion Models (LDMs) across multiple benchmarks, achieving significant gains in generation fidelity and inpainting accuracy.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

This work systematically uncovers the decisive role of problem geometry—specifically, the curvature of the constraint set and the structure of gradients—in governing the statistical-computational trade-offs of stochastic and online optimization algorithms. We introduce the first geometric measure quantifying the deviation of a constraint set from quadratic convexity, rigorously identifying the geometric origins of suboptimality in subgradient methods. We prove that diagonal-preconditioned SGD achieves minimax-optimal convergence rates under quadratic convex constraints. For non-Euclidean, non-quadratically-convex domains—such as ℓₚ-balls with p < 2—we establish tight convergence bounds for mirror descent and adaptive gradient methods, and uncover, for the first time, a precise correspondence between their convergence rates and the accuracy-computation trade-off in Gaussian sequence estimation. Our results provide geometric criteria for algorithm selection and unify the understanding of when nonlinear updates—e.g., via mirror descent—are necessary to attain statistical optimality.

This work addresses a critical limitation in existing deep latent variable models, which employ Euclidean averaging to estimate Gaussian mixture priors, causing subpopulation prototypes to deviate from the data manifold and leading to performance degradation as the number of subpopulations increases. To overcome this, the authors propose a manifold-anchored variational learning framework that, for the first time, integrates manifold constraints into the M-step of the EM algorithm. Specifically, they construct a heat kernel–weighted latent graph and select graph medoids with the highest diffusion centrality as prototypes, guaranteeing their strict adherence to the manifold. Additionally, Dirichlet energy regularization is introduced to enhance geometric smoothness in the latent space. The method is both general-purpose and capable of providing unsupervised uncertainty scores, achieving state-of-the-art accuracy in cardiac scar and brain MRI tasks while generating the clearest and most stable prototypes to date, significantly outperforming all baselines.

Traditional linear dimensionality reduction methods often fail to effectively uncover the intrinsic low-dimensional manifold structure embedded in high-dimensional data. This work systematically traces the historical development of manifold fitting and, for the first time, categorizes it into three distinct phases: nonparametric statistics, mathematically inspired analysis, and modern practical statistics. It clarifies manifold fitting’s role as an independent geometric data analysis tool and delineates its conceptual boundaries from related techniques such as manifold embedding and denoising. By integrating nonparametric methods, differential geometry, and contemporary statistical learning approaches, the paper explores cutting-edge applications of manifold fitting in neural networks and bioinformatics, offering a comprehensive reference framework that elucidates both its theoretical limits and practical utility.

This study investigates how optimizer geometry influences spectral bias and the emergence of low-rank solutions in matrix factorization. To this end, we construct a unified dynamical framework that precisely quantifies how normalization, curvature correction, and damping regulate singular value dynamics. By theoretically analyzing algorithms including Euclidean gradient descent, SignGD, Adam approximations, Muon, Shampoo, and K-FAC, we systematically reveal how optimizer geometry and network depth jointly govern training dynamics. Our analysis elucidates the intrinsic mechanisms through which different optimizers either eliminate or preserve low-rank bias within finite time. These findings provide rigorous theoretical foundations for designing optimizers tailored to specific inductive biases.

This work addresses the limitation of existing state space models in multivariate time series forecasting, which often overlook the dynamically evolving geometric relationships among variables. To remedy this, the paper introduces, for the first time, a symmetric positive definite (SPD) manifold constraint into state space modeling, leveraging Riemannian geometric features as structural regularizers to enable geometry-aware temporal modeling. The proposed method integrates projection from the SPD manifold to its tangent space, a geometry-guided gating mechanism based on manifold-valued signals, and Mamba’s linear-complexity parallel scan architecture. Extensive experiments on eleven real-world benchmark datasets demonstrate state-of-the-art performance, underscoring the critical role of geometric constraints in enhancing forecasting accuracy.

本文针对聚类中的不确定性问题，提出基于流形假设的聚簇方法（MBC），通过几何与样本量度结合定义不确定性区间，量化数据固有的模糊性。