  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.
