Collapse Theorems in High-Dimensional Tensor Spaces under Noncommutative Geometric Frameworks

Abstract

This paper formulates a novel tensor algebraic structure based on Connes' noncommutative geometry strategy and infers fundamental collapse theorems for high-dimensional tensor spaces under specific operator actions. We demonstrate that dimensional collapse conditions provide a unified mathematical foundation for both renormalization problems in quantum field theory and vanishing gradient problems in deep neural networks. By merging functional analysis and algebraic geometry tools, we construct noncommutative differential structures with direct physical interpretations. The main outcome is a generalized collapse theorem that defines conditions for dimensional reduction in tensor networks through spectral properties of noncommutative Dirac operators. Numerical simulations validate our theoretical predictions, showing dramatic agreement between the theoretical collapse boundaries and experimentally measured critical points in both quantum field theories and deep learning models.