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Manifold Learning

Last update: 08 Dec 2024 12:11
First version: 25 September 2014

Suppose that we observe random vectors $X$ in some $p$-dimensional space, most of ordinary Euclidean space $\mathbb{R}^p$. Sometimes, the probability distribution of $X$ is supported on a big volume in $\mathbb{R}^p$, maybe even the whole space; this is the classical situation considered by multivariate statistics. Sometimes, however, the support is concentrated on, or at least near, some geometric structure with a dimension $q < p$. If the low-dimensional structure is a finite set of 0-dimensional points, we have clutsering. If the low-dimensional structure is a linear subspace, we have the situation dealt with by factor analysis. More interesting to me is the more general situation where the low-dimensional structure is a smooth but curved manifold. Then the goal of manifold learning is to try to reconstruct the underlying manifold from observations of $X$. We might also be content with knowing some geometric or topological properties of the manifold, like its dimension.

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