Generative Topographic Map - Details of The Algorithm

Details of The Algorithm

The approach is strongly related to density networks which use importance sampling and a multi-layer perceptron to form a non-linear latent variable model. In the GTM the latent space is a discrete grid of points which is assumed to be non-linearly projected into data space. A Gaussian noise assumption is then made in data space so that the model becomes a constrained mixture of Gaussians. Then the model's likelihood can be maximized by EM.

In theory, an arbitrary nonlinear parametric deformation could be used. The optimal parameters could be found by gradient descent etc.

The suggested approach to the nonlinear mapping is to use a radial basis function network (RBF) to create a nonlinear mapping between the latent space and the data space. The nodes of the RBF network then form a feature space and the nonlinear mapping can then be taken as a linear transform of this feature space. This approach has the advantage over the suggested density network approach that it can be optimised analytically.