Mathematical Foundations of Symmetrized Hyperbolic Tangent Activation in Neural Networks

This paper presents a comprehensive theoretical analysis of neural network operators activated by the symmetrized hyperbolic tangent function, with a focus on robustness and convergence in multivariate function approximation. The symmetrized hyperbolic tangent activation function has emerged as a promising tool in neural network research due to its ability to enhance approximation accuracy. This study extends the current understanding by introducing novel theorems that demonstrate the stability and accuracy of these operators in more general function spaces and under adversarial conditions. The research delves into the mathematical formulation of the symmetrized hyperbolic tangent activation function and its associated density functions, highlighting their symmetry properties crucial for neural network analysis. The paper introduces several key theorems that explore the convergence properties of these operators in general function spaces, their robustness under adversarial perturbations, stability in high-dimensional spaces, uniform convergence in Sobolev spaces, and adaptive robustness under noise. The findings reveal that the symmetrized hyperbolic tangent activation function exhibits enhanced convergence rates in general function spaces, maintaining stability even in high-dimensional settings. Furthermore, the study shows that these operators are robust to adversarial conditions and noise, making them suitable for real-world applications where data integrity cannot be guaranteed. By providing a solid theoretical foundation, this work contributes to the development of more reliable and efficient neural network models for complex approximation tasks. The insights gained from this analysis have the potential to inform the design of neural network architectures and training algorithms, ultimately advancing the field of machine learning and its applications.