A0538
Title: Random fixed boundary flows: A twin sister of principal flow
Authors: Zhigang Yao - National University of Singapore (Singapore) [presenting]
Abstract: The focus is on fixed boundary flows with canonical interpretability as principal components extended on non-linear Riemannian manifolds. The aim is to find 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 is defined as an optimal curve that moves in the data cloud with two fixed endpoints. At any point in the flow, the inner product of the vector field is maximized, 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 data sets, the fixed boundary flow is named the random fixed boundary flow, and its limiting behavior is analyzed under noisy observed samples. It is shown 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. It is further proven that the random fixed boundary flow converges largely to the population fixed boundary flow with high probability. Finally, it illustrates how the random fixed boundary flow can be used and interpreted and demonstrates its application in real data sets.
