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Fixed Boundary Flows
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
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