A Invertible Dimension Reduction of Curves on a Manifold

TL;DR

This paper introduces an invertible, efficient dimension reduction method for high-dimensional curves on manifolds, leveraging differential geometry tools to adapt local flat subspaces for better shape sequence representation.

Contribution

It presents a novel invertible dimension reduction technique for curves on manifolds using differential geometry, improving efficiency and local adaptability over existing methods.

Findings

The method achieves accurate shape sequence reconstruction.

Experimental results demonstrate computational efficiency.

The approach effectively captures geometric properties of curves.

Abstract

In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential geometry tools such as moving frame and parallel transportation are successfully adapted into the dimension reduction problem of high dimensional curves. Intuitively, instead of searching for a global flat subspace for curve embedding, we deployed a sequence of local flat subspaces adaptive to the geometry of both of the curve and the manifold it lies on. In practice, the experimental results of the dimension reduction and reconstruction algorithms well illustrate the advantages of the proposed theoretical innovation.