{"id":9393,"date":"2025-12-29T07:02:33","date_gmt":"2025-12-29T07:02:33","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/12\/29\/2512-21451\/"},"modified":"2025-12-29T07:02:33","modified_gmt":"2025-12-29T07:02:33","slug":"2512-21451","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/12\/29\/2512-21451\/","title":{"rendered":"An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry"},"content":{"rendered":"

An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry
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arXiv:2512.21451v1 Announce Type: new
\nAbstract: 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 Cram’er-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.<\/div>\n

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\n Bing Cheng, Howell Tong
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An approach to Fisher-Rao metric for infinite dimensional non-parametric information geometry arXiv:2512.21451v1 Announce Type: new 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 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,112],"tags":[487,1271,458],"class_list":["post-9393","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-dimensional","tag-fisher","tag-information"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/9393"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=9393"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/9393\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=9393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=9393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=9393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}