Deep Neural Networks: A Formulation Via Non-Archimedean Analysis
This work addresses the challenge of robust function approximation in neural networks for theoretical machine learning, though it appears incremental as it builds on existing approximation theory with a novel mathematical formulation.
The paper tackles the problem of designing deep neural networks with robust universal approximation capabilities by introducing a new class based on non-Archimedean analysis, resulting in networks that approximate real-valued functions on specific rings and square-integrable functions on the unit interval.
We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.