An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry

• URL: http://arxiv.org/abs/2512.21451v1
• Date: Thu, 25 Dec 2025 00:18:41 GMT
• Title: An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry
• Authors: Bing Cheng, Howell Tong,
• Abstract summary: Being infinite dimensional, non-parametric information geometry has long faced an "intractability barrier"<n>This paper introduces a novel framework to resolve the intractability with an Orthogonal Decomposition of the Tangent Space.<n>By defining the Information Capture Ratio, we provide a rigorous method for estimating intrinsic dimensionality in high-dimensional data.
• Score: 0.6138671548064355
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Being infinite dimensional, non-parametric information geometry has long faced an "intractability barrier" due to the fact that the Fisher-Rao metric is now a functional incurring difficulties in defining its inverse. This paper introduces a novel framework to resolve the intractability with an Orthogonal Decomposition of the Tangent Space ($T_fM=S \oplus S^{\perp}$), where S represents an observable covariate subspace. Through the decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as $G_f$, which is a finite-dimensional and computable representative of information extractable from the manifold's geometry. Indeed, by proving the Trace Theorem: $H_G(f)=\text{Tr}(G_f)$, we establish a rigorous foundation for the G-entropy previously introduced by us, thereby identifying it not merely as a gradient-based regularizer, but also as a fundamental geometric invariant representing the total explainable statistical information captured by the probability distribution associated with the model. Furthermore, we establish a link between $G_f$ and the second-order derivative (i.e. the curvature) of the KL-divergence, leading to the notion of Covariate Cramér-Rao Lower Bound(CRLB). We demonstrate that $G_f$ is congruent to the Efficient Fisher Information Matrix, thereby providing fundamental limits of variance for semi-parametric estimators. Finally, we apply our geometric framework to the Manifold Hypothesis, lifting the latter from a heuristic assumption into a testable condition of rank-deficiency within the cFIM. By defining the Information Capture Ratio, we provide a rigorous method for estimating intrinsic dimensionality in high-dimensional data. In short, our work bridges the gap between abstract information geometry and the demand of explainable AI, by providing a tractable path for revealing the statistical coverage and the efficiency of non-parametric models.

Related papers

• Manifold Percolation: from generative model to Reinforce learning [0.26905021039717986]
  Generative modeling is typically framed as learning mapping rules, but from an observer's perspective without access to these rules, the task becomes disentangling the geometric support from the probability distribution.<n>We propose that continuum percolation is uniquely suited to this support analysis, as the sampling process effectively projects high-dimensional density estimation onto a geometric counting problem on the support.
  arXiv  Detail & Related papers  (2025-11-25T17:12:42Z)
• Provable Non-Convex Euclidean Distance Matrix Completion: Geometry, Reconstruction, and Robustness [8.113729514518495]
  The Euclidean Distance Matrix Completion problem arises in a broad range of applications, including sensor network localization, molecular robustness, and manifold learning.<n>In this paper, we propose a low-rank matrix completion task over the space of positive semi-definite Gram matrices.<n>The available distance measurements are encoded as expansion coefficients in a non-orthogonal basis, and optimization over the Gram matrix implicitly enforces geometric consistency through nonnegativity and the triangle inequality.
  arXiv  Detail & Related papers  (2025-07-31T18:40:42Z)
• CP$^2$: Leveraging Geometry for Conformal Prediction via Canonicalization [51.716834831684004]
  We study the problem of conformal prediction (CP) under geometric data shifts.<n>We propose integrating geometric information--such as geometric pose--into the conformal procedure to reinstate its guarantees.
  arXiv  Detail & Related papers  (2025-06-19T10:12:02Z)
• IIKL: Isometric Immersion Kernel Learning with Riemannian Manifold for Geometric Preservation [15.82760919569542]
  Previous research generally mapped non-Euclidean data into Euclidean space during representation learning.<n>In this paper, we propose a novel Isometric Immersion Kernel Learning (IIKL) method.<n>We show that our method could reduce the inner product invariant loss by more than 90% compared to state-of-the-art methods.
  arXiv  Detail & Related papers  (2025-05-07T12:08:33Z)
• Finsler Multi-Dimensional Scaling: Manifold Learning for Asymmetric Dimensionality Reduction and Embedding [41.601022263772535]
  Dimensionality reduction aims to simplify complex data by reducing its feature dimensionality while preserving essential patterns, with core applications in data analysis and visualisation.<n>To preserve the underlying data structure, multi-dimensional scaling (MDS) methods focus on preserving pairwise dissimilarities, such as distances.
  arXiv  Detail & Related papers  (2025-03-23T10:03:22Z)
• Proper Latent Decomposition [4.266376725904727]
  We compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization.<n>With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold.<n>This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.
  arXiv  Detail & Related papers  (2024-12-01T12:19:08Z)
• The Fisher-Rao geometry of CES distributions [50.50897590847961]
  The Fisher-Rao information geometry allows for leveraging tools from differential geometry. We will present some practical uses of these geometric tools in the framework of elliptical distributions.
  arXiv  Detail & Related papers  (2023-10-02T09:23:32Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Inverting brain grey matter models with likelihood-free inference: a tool for trustable cytoarchitecture measurements [62.997667081978825]
  characterisation of the brain grey matter cytoarchitecture with quantitative sensitivity to soma density and volume remains an unsolved challenge in dMRI. We propose a new forward model, specifically a new system of equations, requiring a few relatively sparse b-shells. We then apply modern tools from Bayesian analysis known as likelihood-free inference (LFI) to invert our proposed model.
  arXiv  Detail & Related papers  (2021-11-15T09:08:27Z)
• Improving Metric Dimensionality Reduction with Distributed Topology [68.8204255655161]
  DIPOLE is a dimensionality-reduction post-processing step that corrects an initial embedding by minimizing a loss functional with both a local, metric term and a global, topological term. We observe that DIPOLE outperforms popular methods like UMAP, t-SNE, and Isomap on a number of popular datasets.
  arXiv  Detail & Related papers  (2021-06-14T17:19:44Z)
• Geometric variational inference [0.0]
  Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques are used to go beyond point estimates. This work proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation.
  arXiv  Detail & Related papers  (2021-05-21T17:18:50Z)
• 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)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)

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.