The 10^33 Reduction for Curvature in Neural Networks: A Hierarchical Mean-Field, Kronecker-Factored, and Nonextensive q-Statistical Route Around Dense Hessian Inversion
Year: 2026
Type: preprint
Abstract: Dense second-order training of modern neural networks is obstructed by the storage, communication, factorization, inversion, and determinant costs of Hessian-, Fisher-, or generalized Gauss-Newton-scale matrices. For a model with $n$ trainable parameters, naive dense inversion scales as $O(n^3)$; for $n\sim 10^{11}$ this corresponds to a formal $10^{33}$-operation barrier. Existing practical methods such as K-FAC, Shampoo, Hessian-free optimization, diagonal adaptive methods, and quasi-Newton variants avoid this wall by imposing structure, sketching, or implicit Hessian-vector operations. This note develops a cautious but concrete research program for a hierarchical curvature proxy that combines: (i) layer-local Kronecker-factored curvature blocks, (ii) nearest-neighbor or block-tridiagonal layer-layer corrections, (iii) a low-rank global mean-field correction, and (iv) nonextensive $q$-statistical scalar closures for arithmetic/geometric curvature anisotropy. The central AM/GM index is \begin{equation} q_i^{\mathrm{AG}} =1-\frac12\ln\left(\frac{\GM_i}{\AM_i}\right), \end{equation} which is a local feature-wise arithmetic-geometric contrast, not automatically identical to a Tsallis PDF $q$, a random-matrix $q$, or a copula dependence $q$. When $q_i^{\mathrm{AG}}$ can be independently estimated or calibrated from finite-variance $q$-Gaussian fluctuation statistics, $q$-RMT spectral statistics, or $q$-copula dependence statistics, the geometric curvature scale may be recovered by \begin{equation} \GM_i=\AM_i\exp\{2(1-q_i^{\mathrm{AG}})\}, \end{equation} without direct determinant or eigenvalue-product computation. The proposed program is therefore not a claim that all $q$'s are equal, but a testable closure framework linking stochastic training diffusion, statistical geometry, and hardware-friendly structured curvature.
Author: michael, fredrick
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164034935