A Source Coding Approach to Classification by Vector Quantization and the Principle of Minimum Description Length

An algorithm for supervised classi cation using vector quantization and entropy coding is presented. The classi cation rule is formed from a set of training data f(Xi; Yi)g n i=1, which are independent samples from a joint distribution PXY . Based on the principle of Minimum Description Length (MDL), a statistical model that approximates the distribution PXY ought to enable e cient coding of X and Y . On the other hand, we expect a system that encodes (X;Y ) e ciently to provide ample information on the distribution PXY . This information can then be used to classify X, i.e., to predict the corresponding Y based on X. To encode both X and Y , a two-stage vector quantizer is applied to X and a Hu man code is formed for Y conditioned on each quantized value of X. The optimization of the encoder is equivalent to the design of a vector quantizer with an objective function re ecting the joint penalty of quantization error and misclassi cation rate. This vector quantizer provides an estimation of the conditional distribution of Y given X, which in turn yields an approximation to the Bayes classi cation rule. This algorithm, namely Discriminant Vector Quantization (DVQ), is compared with Learning Vector Quantization (LVQ) and CART on a number of data sets. DVQ outperforms the other two on several data sets. The relation between DVQ, density estimation, and regression is also discussed.