Simplicial Nonlinear Principal Component Analysis

We present a new manifold learning algorithm that takes a set of data points lying on or near a lower dimensional manifold as input, possibly with noise, and outputs a simplicial complex that fits the data and the manifold. We have implemented the algorithm in the case where the input data can be triangulated. We provide triangulations of data sets that fall on the surface of a torus, sphere, swiss roll, and creased sheet embedded in a fifty dimensional space. We also discuss the theoretical justification of our algorithm.