What is the difference between manifold learning and non-linear dimensionality reduction?

Question

What is the difference between manifold learning and non-linear dimensionality reduction?

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I have seen these two terms being used interchangeably. For example:

http://www.cs.cornell.edu/~kilian/research/manifold/manifold.html :

Manifold Learning (often also referred to as non-linear dimensionality reduction) pursuits the goal to embed data that originally lies in a high dimensional space in a lower dimensional space, while preserving characteristic properties.

http://www.stat.washington.edu/courses/stat539/spring14/Resources/tutorial_nonlin-dim-red.pdf :

In this tutorial ‘manifold learning’ and ‘dimensionality reduction’ are used interchangeably.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3337666/ :

Dimensionality reduction methods are a class of algorithms that use mathematically defined manifolds for statistical sampling of multidimensional classes to generate a discrimination rule of guaranteed statistical accuracy.

However, http://scikit-learn.org/stable/modules/manifold.html is more nuanced:

Manifold learning is an approach to non-linear dimensionality reduction.

One first difference I can see is that a manifold can be linear, therefore one should compare non-linear manifold learning and non-linear dimensionality reduction.

Answer 1

Non-linear dimensionality reduction occurs when method used for reduction assumes that manifold on which latent variables are lying is, well... non-linear.

So for linear methods manifold is a n-dimensional plane, i.e. affine surface, for non-linear methods it's not.

"Manifold learning" term usually means geometrical/topological methods that learn non-linear manifold.

So we can think about manifold learning as a subset of non-linear dimensionality reduction methods.