{"ID":2825103,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.21451","arxiv_id":"2512.21451","title":"An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry","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 ${\\bf G}_f$, which is a finite-dimensional and computable representative of information extractable from the manifold's geometry. Significantly, by proving the Trace Theorem: $H_G(f) = \\text{Tr}({\\bf G}_f)$, we establish a rigorous foundation for the G-entropy previously introduced by us, thereby identifying it as a fundamental geometric invariant representing the total explainable statistical information captured by the probability distribution associated with a model. Furthermore, we establish a link between ${\\bf G}_f$ and the second derivative (i.e. the curvature) of the KL-divergence, leading to the notion of Covariate Cramér-Rao Lower Bound(CRLB). We demonstrate that ${\\bf G}_f$ 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 assessing the statistical coverage and the efficiency of non-parametric models.","short_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...","url_abs":"https://arxiv.org/abs/2512.21451","url_pdf":"https://arxiv.org/pdf/2512.21451v2","authors":"[\"Bing Cheng\",\"Howell Tong\"]","published":"2025-12-25T00:18:41Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\"]","methods":"[]","has_code":false}