Semidefinite Embedding

Semidefinite embedding (SDE) or maximum variance unfolding (MVU) is an algorithm in computer science that uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data. MVU can be viewed as a non-linear generalization of Principal component analysis.

Non-linear dimensionality reduction algorithms attempt to map high-dimensional data onto a low-dimensional Euclidean vector space. Maximum variance Unfolding is a member of the manifold learning family, which also include algorithms such as isomap and locally linear embedding. In manifold learning, the input data is assumed to be sampled from a low dimensional manifold that is embedded inside of a higher dimensional vector space. The main intuition behind MVU 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 steps:

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.

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 while preserving the nearest neighbors distances.

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.