Fixed Boundary Flows

Overview

This project focuses on fixed boundary flows with canonical interpretability as principal components extended on non-linear Riemannian manifolds. Our primary objective is to identify a flow with fixed starting and ending points for noisy multivariate data sets lying near an embedded non-linear Riemannian manifold. In geometric terms, the fixed boundary flow represents an optimal curve that traverses the data cloud, maintaining two unchanging endpoints. At any given point along this flow, we seek to maximize the inner product between the locally calculated vector field and the tangent vector of the flow. The formal definition arises from an optimization problem utilizing the intrinsic metric of the manifolds.

An implementation in R is available on Github:
Detailed description and discussion can be found in paper:
To cite:

@article{yao2023fixed,
  title={Random Fixed Boundary Flows},
  author={Yao, Zhigang and Xia, Yuqing and Fan, Zengyan},
  journal={arXiv preprint arXiv:1904.11332},
  year={2023}
}

Selected Talks

First Symposium of Geometry and Statistics in China

Yanqi Lake Beijing Institute of Mathematical Sciences and Applications,
\(\quad\) Yau Center at Tsinghua University
July 29, 2023