The Dynamics of Deforming Manifold: A Mathematical Model

TL;DR

This paper introduces a new mathematical model for deforming manifolds in embedded space, incorporating physical meanings and constraints, with applications in data dimension reduction demonstrated through simulations.

Contribution

It presents a modified differential dynamic model with a novel autonomous deforming field for geometric data analysis and dimension reduction.

Findings

Model effectively describes manifold deformation with physical interpretability.

Autonomous deforming field demonstrates flattening effect in simulations.

Potential applications in practical data dimension reduction tasks.

Abstract

In order to meet the requirements of practical applications, a model of deforming manifold in the embedded space is proposed. The deforming vector and deforming field are presented to precisely describe the deforming process, which have clear physical meanings. The proposed model is a modification of the general differential dynamic model, with constraints of spatial and temporal continuity on the deforming field. The deformation integral and derivative are presented as compact expressions of manifold deforming process. Moreover, a specific autonomous deforming field with flattening effect is defined, which provides a novel geometric viewpoint on data dimension reduction. The effectiveness of this autonomous deforming field is proved by numerical computation simulations, which indicate the promising potential of the proposed model in practical dimension reduction tasks.