Algebraic Statistics and Singular Models for AI: Polynomial Models, Identifiability, Singular Learning, and Latent Geometry

Why does parameter count so often fail to measure the true complexity of modern AI models? Algebraic Statistics and Singular Models for AI develops a unified mathematical answer. It connects polynomial and semialgebraic model representations, parameter fibres, symmetry, identifiability, stratified geometry, and singular asymptotics to show how latent structure and over-parameterisation change the foundations of statistical learning.

Beginning with algebraic representations, Gröbner bases, elimination, toric and determinantal models, the book builds a geometric language for understanding what data can actually identify. It then turns to factor models, finite mixtures, tensor decompositions, low-rank structures, and latent-variable models, before developing the local geometry of singular truths, nonstandard likelihood-ratio limits, resolution of singularities, zeta functions, and the real log canonical threshold (RLCT). These tools lead naturally to singular Bayesian asymptotics, marginal likelihood, generalisation error, WAIC, WBIC, and singular BIC.

The final part connects this theory to modern AI systems. Examples include LoRA and low-rank adaptation, redundant and functionally equivalent neural-network parameterisations, deep linear and specified ReLU networks, mixture-of-experts degeneracies, variational latent models, posterior collapse, model merging, and local learning-coefficient diagnostics. Throughout, the book carefully distinguishes exact theorems, controlled model transfers, and empirical large-model evidence.

Designed for advanced graduate students and researchers in statistics, machine learning, applied mathematics, and theoretical AI, this volume provides a rigorous bridge between algebraic statistics and singular learning theory—and a practical conceptual framework for understanding why, in modern AI, statistical complexity is often governed by geometry rather than by the raw number of parameters.