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];


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