Unified Framework for Spectral Dimensionality Reduction, Maximum Variance Unfolding, and Kernel Learning By Semidefinite Programming: Tutorial and Survey

TL;DR

This paper provides a comprehensive tutorial and survey on unifying spectral dimensionality reduction methods, kernel learning via SDP, and Maximum Variance Unfolding, highlighting their connections and various extensions for data manifold learning.

Contribution

It unifies spectral methods as kernel PCA, explains multiple MVU variants, and discusses kernel learning by SDP for manifold unfolding and out-of-sample extensions.

Findings

Unified spectral methods as kernel PCA.

Detailed explanation of MVU and its variants.

Discussion on kernel learning via SDP for manifold unfolding.

Abstract

This is a tutorial and survey paper on unification of spectral dimensionality reduction methods, kernel learning by Semidefinite Programming (SDP), Maximum Variance Unfolding (MVU) or Semidefinite Embedding (SDE), and its variants. We first explain how the spectral dimensionality reduction methods can be unified as kernel Principal Component Analysis (PCA) with different kernels. This unification can be interpreted as eigenfunction learning or representation of kernel in terms of distance matrix. Then, since the spectral methods are unified as kernel PCA, we say let us learn the best kernel for unfolding the manifold of data to its maximum variance. We first briefly introduce kernel learning by SDP for the transduction task. Then, we explain MVU in detail. Various versions of supervised MVU using nearest neighbors graph, by class-wise unfolding, by Fisher criterion, and by colored MVU are explained. We also explain out-of-sample extension of MVU using eigenfunctions and kernel mapping. Finally, we introduce other variants of MVU including action respecting embedding, relaxed MVU, and landmark MVU for big data.