Simplicial Nonlinear Principal Component Analysis

Abstract

We present simplicial principal component analysis (SNPCA), a new manifold reconstruction algorithm that takes a set of data points lying near a lower dimensional manifold as input, possibly with noise, and extracts a simplicial complex that fits the data and the manifold. We have implemented the algorithm in the case where the input point cloud can be triangulated by a complex of 2-simplices, but is embedded in a space of arbitrary dimension. The algorithm is easily parallelizable. We are working to extend the algorithm to data sets lying on higher dimensional manifolds and more complex shapes, possibly of varying dimension. We provide the output of our algorithm for data that fall on the surface of a torus with and without noise, a swiss roll, and creased sheet, all embedded in R. We chose these manifolds to demonstrate that the algorithm does not require a smooth underlying manifold, or a manifold without boundary. We also discuss the theoretical justification of our algorithm.