Manifold learning: what, how, and why
It provides a comprehensive overview for statisticians and practitioners, but is incremental as a survey rather than introducing new methods.
This survey tackles the problem of understanding and applying manifold learning for high-dimensional data analysis, presenting principles, methods, and statistical foundations to guide reliable implementation.
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.