Manifold learning: what, how, and why

TL;DR

This survey explains manifold learning as a set of non-linear dimension reduction techniques that uncover the geometric structure of high-dimensional data, aiding visualization, de-noising, and interpretation.

Contribution

It provides a comprehensive overview of the principles, methods, and statistical foundations of manifold learning from a practitioner's perspective.

Findings

Manifold learning reveals the geometric shape of high-dimensional data.

Trade-offs and statistical considerations are crucial for reliable results.

The survey discusses parameter and algorithm choices for effective ML.

Abstract

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.