Information Geometry

Runtime monitoring of stochastic systems must distinguish nominal distributional relaxation from regime departure while controlling repeated-test false alarms under explicit validity assumptions. This paper links relative-entropy dissipation, information geometry, and sequential inference in a bounded first-passage monitoring architecture. For reversible Fokker--Planck dynamics, relative entropy to an invariant density is non-increasing; under exogenous forcing, its derivative decomposes into nominal dissipation and an information-space forcing term. The runtime layer uses Gaussian window surrogates, nominal-relative covariance shrinkage, a coordinate-consistent relative precision diagnostic, and randomized conformal ranks aggregated by a mixture power-martingale process. Analytical Ornstein--Uhlenbeck validation gives zero positive nominal Kullback--Leibler increments, forcing-identity residuals below 3.31 x 10^-6, and coordinate-invariance errors at numerical roundoff. On NSL-KDD, the monitor yields 0/100 alarms on internal nominal streams but 63/100 on official test-normal streams; post-change detection is 99.0% for seen and 98.53% for test-only attack types with median one-window delay. On UNSW-NB15, internal-null alarms are 0/100, whereas official test-normal alarms rise to 90/100; post-change detection is 81.33%, with 18.67% pre-change alarms. In these evaluations, calibration transport emerges as a major deployment constraint. No universal benchmark superiority, causal inference, or physical-work interpretation is claimed.

We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly Kähler-Ricci. The complex normalizing flow relates the initial and target realified densities under the complex change of variables, necessitating the log determinant of the ensemble of Wirtinger Jacobians. The Ricci curvature of a Kähler manifold is the second order mixed Wirtinger partial derivative of the log of the local density of the volume form. Therefore, we reconcile these two facts by drawing forth the connection that the log determinant used in the complex normalizing flow matches a Ricci curvature term under differentiation and conditions. The log density under the normalizing flow is kindred to a spatial information metric under an augmented Jacobian and a Bayesian perspective to the parameter, thus under the continuum limit the log likelihood matches a Fisher metric, or more closely a Kähler cross-entropy Hessian. This recovers a Kähler-Ricci flow variation up to a time derivative and expectation, or an average-valued Kähler-Einstein flow. Using this framework, we establish other relevant results, attempting to bridge the statistical and ordinary behaviors of the complex normalizing flow to the geometric features of our derived Kähler flow.

Multiplicative gating is widely used in neural architectures and has recently been applied to attention layers to improve performance and training stability in large language models. Despite the success of gated attention, the mathematical implications of gated attention mechanisms remain poorly understood. We study attention through the geometry of its representations by modeling outputs as mean parameters of Gaussian distributions and analyzing the induced Fisher--Rao geometry. We show that ungated attention operator is restricted to intrinsically flat statistical manifolds due to its affine structure, while multiplicative gating enables non-flat geometries, including positively curved manifolds that are unattainable in the ungated setting. These results establish a geometric expressivity gap between ungated and gated attention. Empirically, we show that gated models exhibit higher representation curvature and improved performance on tasks requiring nonlinear decision boundaries whereas they provide no consistent advantage on tasks with linear decision boundaries. Furthermore, we identify a structured regime in which curvature accumulates under composition, yielding a systematic depth amplification effect.

Expert specialization is fundamental to Mixture-of-Experts (MoE) model success, yet existing metrics (cosine similarity, routing entropy) lack theoretical grounding and yield inconsistent conclusions under reparameterization. We present an information-geometric framework providing the first rigorous characterization of MoE specialization dynamics. Our key insight is that expert routing distributions evolve on the probability simplex equipped with the Fisher information metric, enabling formal analysis via Riemannian geometry. We prove that standard heuristic metrics violate parameterization invariance (Theorem 1), establish that specialization corresponds to geodesic flow with quantified approximation bounds (Theorem 2), and derive a failure predictor with theoretical threshold justification (Theorem 3). The framework introduces two principled metrics: Fisher Specialization Index (FSI) achieving r=0.91+/-0.02 correlation with downstream performance, and Fisher Heterogeneity Score (FHS) predicting training failure at 10% completion with AUC=0.89+/-0.03 -- outperforming validation-loss-based early stopping by 23% while requiring 40x fewer compute cycles. We validate intervention protocols achieving 87% recovery rate when FHS>1 is detected. Comprehensive experiments across language modeling (WikiText-103, C4), vision MoE (ImageNet), and scaling studies (8-64 experts, 125M-2.7B parameters) validate our theoretical predictions.

Diffusion models often degrade in latent spaces, yet the formal causes remain poorly understood. We quantify latent-space diffusability via the rate of change of the Minimum Mean Squared Error (MMSE) along the diffusion trajectory. Our framework decomposes this MMSE rate into contributions from Fisher Information (FI) and Fisher Information Rate (FIR). We show that isometric embeddings preserve intrinsic FI and establish quantitative intrinsic-FI bounds for a broader class of bi-Lipschitz encoders with controlled weak volume distortion, whereas FIR is governed by the interplay between encoder and data geometries. Our analysis separates four geometric contributions in local stability bounds for Gaussian-smoothed FIR: dimensional compression, tangential distortion, high-frequency encoder curvature, and curvature of data manifold. Experiments across diverse autoencoding architectures provide qualitative support for the geometric mechanisms identified by the theory and show that empirical FI and FIR track several measures of generation quality and latent-space geometry in the settings tested. We establish FI and FIR as a comprehensive analytical framework for understanding latent diffusability.

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Maximum likelihood prediction (MLP) is a core task at the heart of modern large language models. Here, we study a quantum version of this task for a simplified data model consisting of independent and identically distributed samples, as a first step. The quantum maximum likelihood predictor (QMLP) is obtained by embedding of empirical probability distributions into quantum states and performing a minimization of quantum relative entropy over a given class of states. We derive non-asymptotic performance guarantees for QMLP in terms of convergence rates and concentration inequalities, both in trace norm and quantum relative entropy. Our approach provides a unified framework to handle MLP within both classical and quantum LLMs. We also consider the related problem of quantum information projection and generalize the quantum Pythagorean theorem to mixture families specified by possibly non-self-adjoint linear constraints. We further show that the Pythagorean inequality continues to hold in the infinite-dimensional setting whenever the convex information-projection problem attains a finite minimum.

Supervised dimensionality reduction maps labeled data into a low-dimensional feature space while preserving class separation. A common strategy is to learn features that maximize a measure of statistical dissimilarity between the class-conditional probability distributions. Information geometry, which is rooted in Riemannian geometry, provides an alternative framework for measuring class dissimilarity. It treats probability distributions as points in a statistical manifold and uses the Fisher information metric to define a geodesic distance--the Fisher-Rao distance--between distributions The Fisher-Rao distance is an appealing candidate for measuring class separation because the Fisher information metric is a local measure of discriminability, and because it allows a geometric interpretation. Here, we present Supervised Quadratic Feature Analysis (SQFA), a supervised dimensionality reduction method which learns linear features that maximize Fisher-Rao distances between class-conditional distributions, under Gaussian assumptions. In multiple real world datasets, we find that SQFA features support classification accuracy that is competitive with features that maximize more popular measures of dissimilarity, or that are learned by other state-of-the-art dimensionality reduction methods. Notably, the best classification accuracy is achieved by SQFA-H features, a variant of SQFA that maximizes the Hellinger distance, a rarely used objective for dimensionality reduction. These results demonstrate the potential of information geometry as a tool for supervised dimensionality reduction. We provide a Python implementation of SQFA at https://github.com/dherrera1911/sqfa.