Multi-Composite Activated Neural Networks Treated as Positive Linear Operators

Multi-composite activated neural network operators can be understood as positive linear operators, allowing them to be analyzed using standard, established theory. Formed by composing multiple general activation functions, these operators act upon continuous real-valued functions defined on a compact interval. This work presents a quantitative analysis of how quickly these operators converge to the unit operator. Utilizing general inequalities based on the modulus of continuity—applicable to either the function itself or its derivative—this study establishes both uniform and Lp approximation results. Furthermore, the analysis incorporates the convexity of functions to produce related, specific results.