Fixed Boundary Flows

Abstract

We consider the fixed boundary flow with canonical interpretability as principal components extended on the non-linear Riemannian manifolds. We aim to find a flow with fixed starting and ending point for multivariate datasets lying on an embedded non-linear Riemannian manifold, differing from the principal flow that starts from the center of the data cloud. Both points are given in advance, using the intrinsic metric on the manifolds. From the perspective of geometry, the fixed boundary flow is defined as an optimal curve that moves in the data cloud. At any point on the flow, it maximizes the inner product of the vector field, which is calculated locally, and the tangent vector of the flow. We call the new flow the fixed boundary flow. The rigorous definition is given by means of an Euler-Lagrange problem, and its solution is reduced to that of a Differential Algebraic Equation (DAE). A high level algorithm is created to numerically compute the fixed boundary. 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 illustrate how the fixed boundary flow can be used and interpreted, and its application in real data