Nonlinear Dimensionality Reduction and Manifold Learning

Abstract

We have little visual guidance to help us identify any meaningful lowdimensional structure hidden in high-dimensional data. The linear projection methods of Chapter 7 can be extremely useful in discovering lowdimensional structure when the data actually lie in a linear (or approximately linear) lower-dimensional subspace (called a manifold) ℳ of input space ℜ^r. But what can we do if we know or suspect that the data actually lie on a low-dimensional nonlinear manifold, whose structure and dimensionality are both assumed unknown? Our goal of dimensionality reduction then becomes one of identifying the nonlinear manifold in question. The problem of recovering that manifold is known as nonlinear manifold learning.