Quantization Functionals and Regularized Principal Manifolds

Many settings of unsupervised learning can be viewed as quantization problems, namely of minimizing the expected quantization error subject to some restrictions. This has the advantage that tools known from the theory of (supervised) risk minimization like regularization can be readily applied to unsupervised settings. Moreover, one may show that this setting is very closely related to both, principal curves with a length constraint and the generative topographic map. Experimental results demonstrate the feasibility of the proposed method. In a companion paper we show that uniform convergence bounds can be given for algorithms such as a modiied variant of the principal curves problem.