Likelihood-based regularization and differential log-likelihood in kernel-based topographic map formation

Two new principles for kernel-based density estimation and kernel-based topographic map formation are introduced: likelihood-based regularization and differential log-likelihood. The former makes that every kernel has an equal probability of generating data points. The differential log-likelihood is an unbiased metric with which to judge the quality of the density estimate. We apply these principles to kernel-based topographic map formation based on log-likelihood maximization. We restrict ourselves to Gaussian kernels and homogeneous, homoscedastic mixings. We show that the negative log-likelihood equals the quantization error of the map up to a scale factor