**Manifold learning**, aka **manifold estimation**,
are methods that extract the geometric structure of data,
typically by embedding the data in a low-dimension space, i.e.
manifold learning as nonlinear dimensionality reduction

Alternatively, manifold learning is also viewed as a subset of topological data analysis
[@Wasserman2018].

Relation with graph embedding:
graph $G = \{X, W\}$ has sample $X$ and affinity matrix $W$.

## Preservation of (Global) Mutual Distances

Isometric embedding:
Multi-dimensional scaling (MDS) [@Schoenberg1937];
ISOMAP [@Tenenbaum2000].

Low Lipschitz distortion:
randomizing the search for an embedding with small distortion [@Johnson1984].

Kernel principal component analysis (kernel PCA) [@Scholkopf1998].
All kernel-based manifold learning methods are special cases of kernel PCA [@Ham2004].

## Local Geometry

Local linear embedding (LLE) [@Roweis2000];

Local tangent space alignment (LTSA) [@ZhangZY2004];

### Spectral methods

Laplacian eigenmaps (LE) [@Belkin2003];
Hessian eigenmaps [@Donoho2003];

Geometric diffusion:
diffusion maps [@Coifman2005a; @Coifman2006a];
geometric harmonics [@Coifman2005b; @Coifman2006b];

## Misc

Graph embedding? [@Goyal2018]:
Stochastic Neighbor Embedding (SNE) [@Hinton2002];
t-Distributed Stochastic Neighbor Embedding (t-SNE) [@Maaten2008], for visualization;
Uniform Manifold Approximation and Projection (UMAP) [@McInnes2018], for visualization;

Random walk methods;

Deep networks:
Automated Transform by Manifold Approximation (AUTOMAP) [@ZhuB2018], for image reconstruction;

🏷 Category=Manifold