Manifold learning: what, how, and why
Marina Meilă Hanyu Zhang
Manifold learning (ML), known also as non-linear dimension reduction, is a
set of methods to find the low dimensional structure of data. Dimension
reduction for large, high dimensional data is not merely a way to reduce the
data; the new representations and descriptors obtained by ML reveal the
geometric shape of high dimensional point clouds, and allow one to visualize,
de-noise and interpret them. This survey presents the principles underlying ML,
the representative methods, as well as their statistical foundations from a
practicing statistician's perspective. It describes the trade-offs, and what
theory tells us about the parameter and algorithmic choices we make in order to
obtain reliable conclusions.