Quantization of Ricci Curvature in Information Geometry

Ricci curvature is a fundamental concept in differential geometry for encoding local geometric structure, and its graph-based analogues have recently gained prominence as practical tools for reweighting, pruning, and reshaping network geometry. We propose Sobolev-Ricci Curvature (SRC), a graph Ricci curvature canonically induced by Sobolev transport geometry, which admits efficient evaluation via a tree-metric Sobolev structure on neighborhood measures. We establish two consistency behaviors that anchor SRC to classical transport curvature: (i) on trees endowed with the length measure, SRC recovers Ollivier-Ricci curvature (ORC) in the canonical W1 setting, and (ii) SRC vanishes in the Dirac limit, matching the flat case of measure-theoretic Ricci curvature. We demonstrate SRC as a reusable curvature primitive in two representative pipelines. We define Sobolev-Ricci Flow by replacing ORC with SRC in a Ricci-flow-style reweighting rule, and we use SRC for curvature-guided edge pruning aimed at preserving manifold structure. Overall, SRC provides a transport-based foundation for scalable curvature-driven graph transformation and manifold-oriented pruning.

This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.

Classical Fisher-information asymptotics describe the covariance of regular efficient estimators through the local quadratic approximation of the log-likelihood, and thus capture first-order geometry only. In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-constrained parameter spaces, finite-sample behavior can deviate systematically from these predictions. We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a Riemannian manifold $(Θ,g)$ with Fisher--Rao metric, immersed in $L^2(μ)$ through the square-root density map. Under suitable regularity and moment assumptions, we derive an $n^{-2}$ correction to the leading $n^{-1}I(θ)^{-1}$ covariance term for score-root, first-order efficient estimators. The correction is governed by a tensor $P_{ij}$ that decomposes canonically into three parts, an intrinsic Ricci-type contraction of the Fisher--Rao curvature tensor, an extrinsic Gram-type contraction of the second fundamental form, and a Hellinger discrepancy tensor encoding higher-order probabilistic information not determined by immersion geometry alone. The extrinsic term is positive semidefinite, the full correction is invariant under smooth reparameterization, and it vanishes identically for full exponential families. We then extend the picture to singular models, where Fisher information degenerates. Using resolution of singularities under an additive normal crossing assumption, 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. This framework also suggests geometric diagnostics of weak identifiability and curvature-aware principles for regularization and optimization.