A General Setting for Spectral Dimensionality Reduction

To effectively analyse spectral dimensionality reduction and the associated problems it is useful to frame the methodology within a general setting. As the name suggests, at the heart of spectral dimensionality reduction is the spectral decomposition of a square symmetric feature matrix. Different techniques can be distinguished based on the construction of this feature matrix and the eigenvectors that are subsequently used (i.e. smallest or largest). This feature matrix aims to capture certain properties of the 

Strange and R. Zwiggelaar, Open Problems in Spectral Dimensionality Reduction, SpringerBriefs in Computer Science, DOI 10.1007/978-3-319-03943-5 2, 

Definition 1 Given a set of D-dimensional data X = e that is sampled 

Due to the vagueness of this definition, many questions naturally present themselves. For example, what constitutes a faithful representation of the original data Is the high-dimensional data expected to lie on or near a low-dimensional subspace or a low-dimensional submanifold How is the data sampled from this subspace or submanifold These types of questions underlie the general assumptions that different approaches make about the setting within which dimensionality reduction takes place and will be returned to in due course. 

A more formal definition of spectral dimensionality reduction can be obtained by filling in some of the gaps found in Definition 1. As previously mentioned, a feature matrix is built from X that aims to capture certain properties of the data and will often represent subspace or submanifold properties. Given the original data X, the feature matrix F is built such that 

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