A source coding approach to classification by vector quantization and the principle of minimum description length

An algorithm for supervised classification using vector quantization and entropy coding is presented. The classification rule is formed from a set of training data {(X_i, Y_i)}_i=1ⁿ, which are independent samples from a joint distribution P_XY. Based on the principle of minimum description length (MDL), a statistical model that approximates the distribution P_XY ought to enable efficient coding of X and Y. On the other hand, we expect a system that encodes (X, Y) efficiently to provide ample information on the distribution P_XY. 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 Huffman 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 reflecting the joint penalty of quantization error and misclassification rate. This vector quantizer provides an estimation of the conditional distribution of Y given X, which in turn yields an approximation to the Bayes classification rule. This algorithm, namely discriminant vector quantization (DVQ), is compared with learning vector quantization (LVQ) and CART^R 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.