Semidefinite embedding

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Semidefinite embedding (SDE) is an algorithm to perform non-linear dimensionality reduction of high dimensional vectorial input data. Related algorithms are Locally Linear Embedding and Isomap.

Non-linear dimensionality reduction algorithms attempt to map high dimensional data, sampled from a lower dimensional underlying manifold, onto a low dimensional Euclidean vector space. The main intuition behind Maximum Variance Unfolding (MVU) (also referred to as Semidefinite Embedding (SDE)) is to exploit the local linearity of manifolds and create a mapping that preserves local neighborhoods at every point of the underlying manifold.


MVU creates a mapping from the high dimensional input vectors to some low dimensional Euclidean vector space in the following three steps:

1. A neighborhood graph is created. Each input is connected with its k-nearest input vectors (according to Euclidean distance metric) and all k-nearest neighbors are connected with each other. If the data is sampled well enough, the resulting graph is a discrete approximation of the underlying manifold.

2. The neighborhood graph is "unfolded" with the help of semidefinite programming. Instead of learning the output vectors directly, the semidefinite programming aims to find an inner product matrix that maximizes the pairwise distances between any two inputs that are not connected in the neighborhood graph.

3. The low dimensional embedding is finally obtained by application of Multidimensional scaling on the learned inner product matrix.


The steps of applying semidefinite programming followed by a linear dimensionality reduction step to recover a low-dimensional embedding into a Euclidean space were first proposed by Linial, London, and Rabinovich in a now classical article (see below).

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