Geometric Formulation for Discrete Points and its Applications

TL;DR

This paper presents a new geometric framework for discrete points using universal differential calculus, unifying various discrete theories across multiple fields like physics, probability, and machine learning.

Contribution

It introduces a consistent geometric formulation for discrete sets that integrates differential geometry with spectral graph theory and random walks, offering a unified theoretical perspective.

Findings

Provides a comprehensive geometric description of discrete points

Unifies multiple discrete frameworks in probability, physics, and machine learning

Suggests an intrinsic, unified theory of discrete structures

Abstract

We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so that it is consistent with differential geometry, and works on spectral graph theory and random walks. Consequently, our formulation comprehensively demonstrates many discrete frameworks in probability theory, physics, applied harmonic analysis, and machine learning. Our approach would suggest the existence of an intrinsic theory and a unified picture of those discrete frameworks.