geometric information decomposition

Existing approaches, such as those based on the von Mises–Fisher (vMF) distribution, model only the mean direction and thus fail to capture complex geometric structures—such as multimodality, axial symmetry, or zonal patterns—in spherical weighted empirical measures. This work proposes a Geometric Information Decomposition (GID) framework that leverages spherical harmonics to construct a nested sequence of maximum-entropy projections, hierarchically quantifying the incremental information-theoretic gaps at each level. For the first time, this enables a layered decomposition of higher-order geometric structures inherent in spherical measures, transcending the limitations of single-parameter models by fully characterizing features ranging from the mean direction to high-order anisotropy and fine angular patterns. Theoretical guarantees include invariance, consistency, and asymptotic normality, along with a quadratic-form zero-calibration test. Experiments on circular and spherical data successfully reveal latent structures invisible to vMF-based methods, demonstrating the approach’s efficacy and practical utility.

In infinite-dimensional nonparametric information geometry, the Fisher–Rao metric’s functional nature renders its inversion intractable—long posing a “computational intractability barrier.” Method: We propose an orthogonal decomposition framework on the tangent space, yielding a finite-dimensional, computable covariate Fisher information matrix (cFIM) under observable covariates. This integrates covariate projection with second-order curvature analysis of the KL divergence within a semiparametric modeling paradigm. Contributions: (i) We prove that the G-entropy equals the trace of cFIM, establishing it as a geometric invariant; (ii) we strengthen the manifold assumption into a testable condition—rank deficiency of cFIM; (iii) we define the information capture ratio, enabling rigorous intrinsic dimension estimation. Our framework provides geometrically grounded, statistically valid tools for quantifying both statistical coverage and model efficiency in explainable AI, and enables verifiable intrinsic dimension inference in high-dimensional settings.

This paper establishes an axiomatic unification framework bridging information theory and statistical thermodynamics. Method: It introduces a novel synthesis of large deviations theory, Kolmogorov conditional expectation, and information projection, using empirical frequencies as the foundational driver to construct an extended information geometry; entropy functions—including Shannon entropy, mutual information, and relative entropy—are systematically derived and endowed with thermodynamic energy interpretations via Legendre–Fenchel parametrization of the empirical mean manifold. Contribution/Results: (1) It reveals thermodynamic-style additivity and intrinsic Riemannian geometric structure of entropies in the infinite-sample limit; (2) it unifies the additive properties of the empirical mean manifold with those of statistical thermodynamics; (3) it fundamentally extends information geometry from the space of probability distributions to the space of empirical frequencies, thereby providing a new paradigm for the geometric and physical interpretation of information.

This work addresses the limitations of classical Fisher–Rao information geometry in handling non-regular statistical models—such as those lacking densities, featuring parameter-dependent support, or exhibiting biased score functions—by developing a unified weak information geometry framework. Within the setting of tempered distributions, the authors introduce weakly regular inference functions and Stein representations as tools for information extraction, constructing a Riemannian metric induced by Godambe information that transcends Fisher–Rao constraints. This approach yields a coherent geometric theory for five classes of non-regular settings, including likelihood-free models, varying-support families, transformation-based inference, and mixture models, while revealing a geometric correspondence between inferential non-identifiability and block-diagonal metric structure. Leveraging tempered distribution theory, Schwartz kernels, quadratic Stein discrepancies, and reproducing kernel Hilbert spaces, the study derives closed-form weak Godambe metrics for models such as Cantor location families, uniform scale families, shifted exponentials, hierarchical mixtures, and α-stable noise-driven lattice heat equations, demonstrating their geometric validity and stability.

This paper addresses the lack of a unified algebraic framework for geometrically characterizing statistical models in information geometry. It establishes an elementary, integrated framework bridging differential geometry (affine connections, curvature), the geometry of probability measure spaces, and (pre-)Frobenius manifold theory. The work systematically constructs, for the first time, a natural correspondence between exponential families and (pre-)Frobenius manifolds. Its contributions are threefold: (1) it uncovers the intrinsic algebraic–geometric structure of exponential families; (2) it imports foundational ideas from topological field theory into elementary information geometry, thereby opening a new pathway for geometric statistics; and (3) it designs a three-stage, teaching–research integrated pedagogical framework—accessible to undergraduate students and math enthusiasts—that provides a concrete, operational perspective on geometric modeling while stimulating research into open problems at this interdisciplinary interface.

This work addresses the limitation of classical Fisher information asymptotics, which captures only the first-order geometry of parameter estimation covariance and fails to accurately characterize bias in finite samples. By viewing regular parametric families as Riemannian manifolds equipped with the Fisher–Rao metric and embedding square-root densities into an L² space, the authors derive a second-order (n⁻²) correction term for the covariance. They innovatively unify intrinsic Ricci curvature, extrinsic second fundamental form, and the Hellinger divergence tensor to construct a coordinate-invariant, curvature-aware framework for higher-order covariance expansion, extending it to singular statistical models. This geometric approach elucidates the mechanisms linking learning rates and posterior mean squared error, offering new principles for diagnosing and optimizing weak identifiability.

This work addresses the lack of a unified theoretical foundation in existing regularization methods and their inability to guarantee thermodynamic energy efficiency during learning. By integrating information geometry, the principle of maximum entropy, and thermodynamic optimality, the study proposes that regularization should minimize the Fisher–Rao distance between belief states and a reference state. It is rigorously shown that this metric is the unique geometric structure satisfying both parameter invariance and the maximum entropy assumption. Optimal regularization forms are derived on hyperbolic and von Mises manifolds for Gaussian and circular distribution models, respectively. This establishes, for the first time, a theoretical framework linking regularization to thermodynamic efficiency and yields experimentally testable predictions.

This study addresses the decomposition of generalization error in unsupervised learning by leveraging information geometry. For the first time, it rigorously decomposes the KL generalization error into three non-negative components—model error, data bias, and variance—using the generalized Pythagorean theorem and a dual e-mixture variance identity. Under an ε-PCA model with isotropic Gaussian assumptions, closed-form expressions for each component are derived, revealing that the optimal feature retention threshold coincides precisely with the noise level ε. The analysis further constructs a phase diagram delineating three distinct regimes—retain-all, interior, and collapse—that characterize phase transitions in the error structure. Comprehensive numerical experiments validate the theoretical predictions.

This study addresses the challenge of effectively quantifying the directional distribution and morphological characteristics of three-dimensional tree-like geometric structures. To this end, it proposes a novel framework that integrates quadratic form theory with the Fisher metric from information geometry. By modeling tree-like structures as 3D geometric graphs, the method leverages quadratic forms to capture their directional spread and introduces, for the first time, a hexplot model to enable visualization and statistical analysis of directional distributions. This approach delivers the first analytical tool for 3D arboreal structures that simultaneously ensures geometric rigor and statistical interpretability, significantly enhancing the quantification of morphological features and the efficiency of cross-structure comparisons in fields such as neuroscience and botany.