Geometric Measure and Non-Archimedean Geometric Neural Network: A Novel Approach

Introduction

In the quest for advanced machine learning models, we continuously seek to integrate sophisticated mathematical concepts into neural network design. This morning I began leveraging ideas from Geometric Measure Theory and Non-Archimedean Geometry and found that we can create novel architectures that enhance the network’s ability to capture complex patterns in data. This article explores a neural network architecture inspired by these theories, applied to the CIFAR-10 image classification task.

Mathematical Theories

Geometric Measure Theory

Geometric Measure Theory (GMT) combines techniques from differential geometry and measure theory to study sets that are too irregular to be analysed using traditional methods. It deals with generalized surfaces and their properties, making it a powerful tool for understanding fractal structures and irregular shapes.

Key Concepts

1. Hausdorff Measure: This concept extends the idea of Lebesgue measure to account for the “size” of fractal sets. It’s defined using limits and covers of sets with small diameters, allowing the quantification of dimensions that are not necessarily integers (e.g…