Simplicial Nonlinear Principal Component Analysis

TL;DR

This paper introduces a novel manifold learning algorithm that constructs a simplicial complex fitting data on or near a lower-dimensional manifold, demonstrated on various complex surfaces in high-dimensional space.

Contribution

The paper proposes a new simplicial nonlinear PCA algorithm with theoretical justification, capable of handling noisy data and producing triangulations of complex manifolds.

Findings

Triangulations of data on torus, sphere, swiss roll, and creased sheet.

Algorithm effectively captures manifold structures in high-dimensional data.

Theoretical analysis supports the validity of the approach.

Abstract

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.