geodesic distance computation

This paper addresses the challenges of evaluating dimensionality reduction (DR) effectiveness and estimating intrinsic data dimensionality. We propose a geometric profiling method based on sectional curvature in discrete metric spaces, which characterizes large-scale data geometry via metric relationships among point triplets. For the first time, this approach systematically introduces differential-geometric curvature into quantitative DR quality assessment and intrinsic dimension estimation—without requiring embedded coordinates or manifold assumptions, thus ensuring both theoretical rigor and computational feasibility. Experiments across diverse synthetic and real-world datasets demonstrate that our method robustly discriminates DR algorithm performance, achieves significantly lower intrinsic dimension estimation error than state-of-the-art methods (e.g., MDS- and PCA-based estimators), and successfully uncovers latent negative curvature in empirical networks—including social and biological networks.

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.

DBD methods face three key bottlenecks in high-dimensional metric learning: (1) poor convergence of Fermat distances due to biased density estimation, (2) limited scalability of graph-based approaches, and (3) non-smooth geodesic paths. To address these, we propose the first differentiable joint density–path modeling framework. Our method innovatively integrates neural spline flows with denoising score matching for accurate, scalable density estimation; introduces variational path relaxation coupled with Riemannian geometric constraints to optimize smooth geodesics; and designs a dimension-adaptive Fermat distance to mitigate the curse of dimensionality. Experiments demonstrate significant improvements in DBD estimation accuracy and geodesic smoothness in high-dimensional spaces. The framework consistently enhances clustering, visualization, and generation-guided learning performance on both image and synthetic benchmarks.

To address robust clustering of unstructured point clouds, this paper proposes a novel spectral clustering method based on the global topological contribution of point pairs. The core innovation lies in constructing a simplicial complex from the point cloud and incorporating the Hodge–Laplacian operator to analyze its higher-order spectral properties, thereby capturing multi-order (beyond pairwise) topological relationships among points. This work is the first to systematically integrate Hodge–Laplacian spectral analysis into a point cloud clustering framework, synergizing sparse eigenvector computation with topological data analysis (TDA) principles to yield interpretable and noise-resilient cluster partitions. Extensive experiments on synthetic and real-world datasets demonstrate that the proposed method achieves significantly higher clustering accuracy and stability than classical spectral clustering—particularly under challenging conditions including noise corruption, non-convex cluster shapes, and multi-scale structures.

To address the problem of spurious inter-manifold edges in nearest-neighbor graphs—induced by noisy data—that distort low-dimensional manifold structure, this paper proposes ORC-ManL, a novel manifold learning algorithm. Methodologically, it is the first to incorporate Ollivier–Ricci curvature (ORC) into manifold learning, establishing a theoretical link between negative ORC and spurious edges; it further integrates local metric distortion estimation to enable geometry-driven, provably convergent graph pruning. The framework unifies ORC computation, topological optimization, and persistent homology evaluation. Empirically, ORC-ManL significantly outperforms existing pruning methods across manifold learning, intrinsic dimension estimation, and single-cell RNA-seq clustering tasks, yielding 15–32% improvements in downstream accuracy. Crucially, its theoretical convergence guarantee is empirically validated.

This work addresses the challenge that existing graph embedding methods struggle to accurately recover the geodesic geometry of manifolds under non-uniform sampling or in the presence of spurious short-circuit edges. The authors propose a path ensemble based on maximum-entropy random walks, constructing a free-energy distance by aggregating over all k-step paths. In the short-time limit, this distance approximates the squared geodesic distance, effectively balancing local neighborhood structure with global geometry. By integrating Varadhan’s heat kernel formula with symmetric kernel Gram decomposition, the method establishes a precise connection between free-energy distances and kernel methods. Scalability is enhanced through landmark-based projection and diffusion-potential pseudo-time. Experiments demonstrate that the approach significantly outperforms current diffusion- or shortest-path-based techniques on both synthetic manifolds and single-cell datasets, exhibiting superior robustness in preserving geodesic structure under non-uniform sampling and branching trajectory scenarios.

This paper addresses the modeling and comparison of parametric network families—such as time-varying metric spaces, evolving social networks, and random graph models. Methodologically, it introduces a unified analytical framework grounded in Gromov–Wasserstein (GW) optimal transport. The core contribution is a family of computable parametric GW distances that unify and generalize existing network metrics, provide theoretical approximation guarantees, and—crucially—establish the first explicit connection between GW distances and statistical functionals of random graphs. To ensure computational tractability, the approach integrates empirical estimation with generative modeling, yielding an efficient lower-bound approximation algorithm. Extensive experiments across diverse generative models and real-world datasets demonstrate that the proposed distance consistently approximates both random graphs and random metric spaces, achieving a favorable balance among theoretical rigor, computational feasibility, and empirical stability.

This study addresses the problem of recovering a globally consistent Euclidean embedding from local distance graphs alone. To this end, the authors propose a purely graph-based distance modeling framework that operates without access to original features, relying solely on neighborhood distance information. By employing a variational approach, the method aligns local graph distances with the Euclidean metric induced by the embedding. Key contributions include the formulation of a functional equation characterizing Euclidean embeddings in the continuous domain, the derivation of a coordinate-invariant Euler–Lagrange equation, and the design of a sparse linear iterative scheme for localized computation. Experiments demonstrate that the proposed approach effectively preserves local metric structure and neighborhood relationships on both synthetic manifolds and real-world datasets, achieving embeddings that closely approximate global isometries.

Estimating the intrinsic dimensionality of high-dimensional data is crucial for machine learning and computer vision, yet existing methods often fail due to reliance on specific geometric or distributional assumptions. This work proposes a nonparametric estimator based on nearest-neighbor distance ratios that requires no prior assumptions about the underlying data manifold or distribution. For the first time, it is theoretically proven that this estimator consistently converges to the true intrinsic dimension under arbitrary data distributions. Extensive experiments demonstrate that the method achieves state-of-the-art performance on both synthetic manifolds and real-world datasets, exhibiting high accuracy, strong robustness, and broad applicability.

This paper addresses the problem of recovering the intrinsic geometry—specifically, the true geodesic distances—of a one-dimensional unit-diameter manifold from noisy pairwise distance observations under dense sampling. We propose a robust distance recovery framework that abandons classical assumptions of i.i.d. additive noise with known moments, instead accommodating broader noise models and missing observations. Our method employs a geometrically aware clustering scheme built upon nonparametric estimation and the L² norm, integrating constraints from manifold curvature and injectivity radius for principled geometric inference. Theoretically, we establish that when the sampling density satisfies Ω(ε⁻²ᵈ⁻² log(1/ε)), the estimated distances achieve an additive error bound of O(ε log ε⁻¹), with computational complexity sub-cubic in the number of points (o(N³)). This significantly improves upon state-of-the-art approaches in both statistical accuracy and scalability.