Random Fixed Boundary Flows

We consider fixed boundary flows 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 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 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 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 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 data sets.

我们将具备规范可解释性的固定边界流（fixed boundary flow）视作在非线性黎曼流形（Riemannian Manifold）上拓展的主成分（Principal Components）。我们的目标是为分布于嵌入型非线性黎曼流形附近的带噪多变量数据集，找到具备固定起止点的流形路径。从几何视角来看，固定边界流被定义为一条以两个固定端点为起止、在数据点云中穿行的最优曲线。在该流形路径上的任意一点，我们最大化局部计算得到的向量场与该路径切向量的内积。其严格定义源于基于流形内蕴度量的优化问题。针对随机数据集，我们将该固定边界流命名为随机固定边界流，并分析其在带噪观测样本下的极限行为。我们构建了用于计算随机固定边界流的高层算法，并给出了该算法的收敛性证明。我们证明，固定边界流可由三段曲线串联而成；当流形退化为欧几里得空间时，其中一段将与常规主成分流（principal flow）完全重合。我们进一步证明，随机固定边界流以高概率在大范围内收敛于总体固定边界流（population fixed boundary flow）。最后，我们阐释了随机固定边界流的应用与解释方法，并展示了其在真实数据集上的应用效果。