A sparse grid based generative topographic mapping for the dimensionality reduction of high-dimensional data

Abstract

Most high-dimensional data exhibit some correlation such that data points are not distributed uniformly in the data space but lie approximately on a lower-dimensional manifold. A major problem in many data-mining applications is the detection of such a manifold from given data, if present at all. The generative topographic mapping (GTM) finds a lower-dimensional parameterization for the data and thus allows for nonlinear dimensionality reduction. We will show how a discretization based on sparse grids can be employed for the mapping between latent space and data space. This leads to efficient computations and avoids the ‘curse of dimensionality’ of the embedding dimension. We will use our modified, sparse grid based GTM for problems from dimensionality reduction and data classification.