Random Fixed Boundary Flows

• 成果类型: Article
• 署名作者: Yao, Zhigang; Xia, Yuqing; Fan, Zengyan
• 署名单位: National University of Singapore; Zhejiang University of Finance & Economics; Singapore University of Social Sciences (SUSS); National University of Singapore
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2023.2257892
• 发表日期: 2024
• 页码: 2356-2368
• 关键词: nonlinear dimensionality reduction; principal; MANIFOLDS; CURVES

摘要

We consider fixed boundary flows with canonical interpretability as principal components extended on nonlinear Riemannian manifolds. We aim to find a flow with fixed start and end points for noisy multivariate datasets lying near an embedded nonlinear Riemannian manifold. In geometric terms, the fixed boundary flow is defined as an optimal curve that moves in the data cloud with two fixed end points. At any point on the flow, we maximize the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. The rigorous definition is derived from an optimization problem using the intrinsic metric on the manifolds. For random datasets, we name the fixed boundary flow the random fixed boundary flow and analyze its limiting behavior under noisy observed samples. We construct a high-level algorithm to compute the random fixed boundary flow, and provide the convergence of the algorithm. We show that the fixed boundary flow yields a concatenate of three segments, one of which coincides with the usual principal flow when the manifold is reduced to the Euclidean space. We further prove that the random fixed boundary flow converges largely to the population fixed boundary flow with high probability. Finally, we illustrate how the random fixed boundary flow can be used and interpreted, and demonstrate its application in real datasets. Supplementary materials for this article are available online.