# Random Fixed Boundary Flows

## TL;DR

This paper introduces the concept of fixed boundary flows on non-linear Riemannian manifolds, providing a method to analyze noisy multivariate data with fixed start and end points, and demonstrates its convergence and practical applications.

## Contribution

It defines and analyzes random fixed boundary flows on manifolds, develops an algorithm for computation, and proves its convergence to the population flow.

## Key findings

- The fixed boundary flow decomposes into three segments, including a principal flow in Euclidean space.
- The random fixed boundary flow converges to the population flow with high probability.
- The method is applicable to real data sets, demonstrating interpretability and utility.

## Abstract

We consider fixed boundary flow with canonical interpretability as principal components extended on non-linear Riemannian manifolds. We aim to find a flow with fixed starting and ending points for noisy multivariate data sets lying on an embedded non-linear Riemannian manifold. In geometric term, 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 derives from an optimization problem using the intrinsic metric on the manifolds. For random data sets, 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 the convergence of the algorithm is provided. We show that the fixed boundary flow yields a concatenate of three segments, of which one 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. We illustrate how the random fixed boundary flow can be used and interpreted, and showcase its application in real data sets.