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

arXiv: https://arxiv.org/abs/2106.15379 · HTML: https://mlsanity.com/paper/2106.15379/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 63 references that are in the map.
- First posted on arXiv 29 Jun 2021.

## Builds on (most cited first, 9 of 9)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Eigenvalue and Generalized Eigenvalue Problems: Tutorial](https://mlsanity.com/paper/1903.11240/) (2019)
- [Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Survey](https://mlsanity.com/paper/2106.08443/) (2021)
- [Feature Selection and Feature Extraction in Pattern Analysis: A Literature Review](https://mlsanity.com/paper/1905.02845/) (2019)
- [Locally Linear Embedding and its Variants: Tutorial and Survey](https://mlsanity.com/paper/2011.10925/) (2020)
- [Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey](https://mlsanity.com/paper/2009.08136/) (2020)
- [Unsupervised and Supervised Principal Component Analysis: Tutorial](https://mlsanity.com/paper/1906.03148/) (2019)
- [Laplacian-Based Dimensionality Reduction Including Spectral Clustering, Laplacian Eigenmap, Locality Preserving Projection, Graph Embedding, and Diffusion Map: Tutorial and Survey](https://mlsanity.com/paper/2106.02154/) (2021)
- [A Summary Of The Kernel Matrix, And How To Learn It Effectively Using Semidefinite Programming](https://mlsanity.com/paper/1709.06557/) (2017)

## Built on by (most cited first, 1 of 1)

- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)

## Abstract (arXiv)

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.