Bochner integrals and neural networks
This work provides a theoretical foundation for neural networks in functional analysis, which is incremental as it builds on existing mathematical frameworks.
The paper develops a functional analytic theory of neural networks by deriving a Bochner integral formula to represent functions using weights and parametrized families, establishing norm inequalities and studying variation spaces and tensor products, showing that variation spaces are Banach spaces.
A Bochner integral formula is derived that represents a function in terms of weights and a parametrized family of functions. Comparison is made to pointwise formulations, norm inequalities relating pointwise and Bochner integrals are established, variation-spaces and tensor products are studied, and examples are presented. The paper develops a functional analytic theory of neural networks and shows that variation spaces are Banach spaces.