  This study broadens the scope of theoretical frameworks in deep learning by
delving into the dynamics of neural networks with inputs that demonstrate the
structural characteristics to Gaussian Mixture (GM). We analyzed how the
dynamics of neural networks under GM-structured inputs diverge from the
predictions of conventional theories based on simple Gaussian structures. A
revelation of our work is the observed convergence of neural network dynamics
towards conventional theory even with standardized GM inputs, highlighting an
unexpected universality. We found that standardization, especially in
conjunction with certain nonlinear functions, plays a critical role in this
phenomena. Consequently, despite the complex and varied nature of GM
distributions, we demonstrate that neural networks exhibit asymptotic behaviors
in line with predictions under simple Gaussian frameworks.
