Wasserstein Mahalanobis Distances for Recovering Latent Geometry

• URL: http://arxiv.org/abs/2608.06560v1
• Date: Thu, 06 Aug 2026 20:16:43 GMT
• Title: Wasserstein Mahalanobis Distances for Recovering Latent Geometry
• Abstract summary: We introduce a Wasserstein Mahalanobis distance to recover latent geometry from nonlinear observations.<n>We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis.<n>Results establish a distribution-valued analog of classical Mahalanobis geometry.
• Score: 3.337790639927531
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.

Related papers

• Nested Inductive Bias Framework for SPD Manifold Learning [0.0]
  In Geometric Deep Learning, inductive biases serve two primary functions: enforcing manifold constraints and embedding relational priors.<n>We introduce a Nested Inductive Bias framework to pull back non-Euclidean target geometries onto the SPD manifold.<n>We also propose the Rational Conformal Metric to establish state-of-the-art robustness against outliers.
  arXiv  Detail & Related papers  (2026-09-03T20:50:18Z)
• A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning [52.452902154360565]
  We introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning.<n>MCBP explicitly models the intrinsic curvature of the data manifold and computes pointwise mean curvature without requiring principled manifold parametrization.<n>Experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance.
  arXiv  Detail & Related papers  (2026-05-05T20:19:09Z)
• On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry [0.0]
  In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-root parameter spaces, finite-sample behavior can deviate systematically from predictions.<n>We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a ((,g)) with Fisher-constrained-Rao metric.<n>We describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case.
  arXiv  Detail & Related papers  (2026-04-14T13:40:13Z)
• Learning Eigenstructures of Unstructured Data Manifolds [47.81117132002129]
  We introduce a novel framework that learns a spectral basis for shape and manifold analysis from unstructured data.<n>By replacing the traditional operator selection, construction, and eigendecomposition with a learning-based approach, our framework offers a principled, data-driven alternative to conventional pipelines.
  arXiv  Detail & Related papers  (2025-11-30T22:06:49Z)
• Wasserstein Regression as a Variational Approximation of Probabilistic Trajectories through the Bernstein Basis [41.99844472131922]
  Existing approaches often ignore the geometry of the probability space or are computationally expensive.<n>A new method is proposed that combines the parameterization of probability trajectories using a Bernstein basis and the minimization of the Wasserstein distance between distributions.<n>The developed approach combines geometric accuracy, computational practicality, and interpretability.
  arXiv  Detail & Related papers  (2025-10-30T15:36:39Z)
• Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation [79.27003481818413]
  We introduce FlatVI, a training framework that regularises the latent manifold of discrete-likelihood variational autoencoders towards Euclidean geometry.<n>By encouraging straight lines in the latent space to approximate geodesics on the decoded single-cell manifold, FlatVI enhances compatibility with downstream approaches.
  arXiv  Detail & Related papers  (2025-07-15T23:08:14Z)
• Procrustes Wasserstein Metric: A Modified Benamou-Brenier Approach with Applications to Latent Gaussian Distributions [0.0]
  We introduce a modified Benamou-Brenier type approach leading to a Wasserstein type distance.<n>This distance is defined by penalizing the action with a costless movement of the particle that does not change the direction and speed of its trajectory.
  arXiv  Detail & Related papers  (2025-03-20T12:34:22Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate.<n>We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Non-Parametric Manifold Learning [0.0]
  We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. A consequence is a proof of consistency for (untruncated) manifold distances. The estimator resembles, and in fact its convergence properties are derived from, a special case of the Kontorovic dual reformulation of Wasserstein distance known as Connes' Distance Formula.
  arXiv  Detail & Related papers  (2021-07-16T19:32:34Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.