Geometric Formulation for Discrete Points and its Applications
This work offers a foundational theory that could unify discrete frameworks in multiple disciplines, though it appears incremental in building on existing mathematical concepts.
The authors introduced a novel geometric formulation for discrete points using universal differential calculus, which provides a unified mathematical framework consistent with differential geometry and applicable across fields like spectral graph theory, random walks, probability theory, physics, and machine learning.
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.