Pattern recognition algorithms based on space-filling curves and orthogonal expansions

The classical problem of constructing a multidimensional pattern classifier in the Bayesian framework is considered. Preprocessing of the learning sequence by a quasi-inverse of a space-filling curve is proposed and properties of space-filling curves which are necessary to obtain Bayes risk consistency are indicated. The learning sequence transformed into the unit interval is used to estimate the coefficients in an orthogonal expansion of the Bayes decision rule. To transform a new observation into the unit interval requires O(d) elementary operations, where d is the dimension of the observation space. Strong Bayes risk consistency of the classifiers is proved when distributions of the random pair of the observation vector and its class are absolutely continuous with respect to the Lebesgue measure. Attainable convergence rate of the Bayes risk is discussed. Details of the classification algorithm based on the Haar series and its properties are presented. Distribution-free consistency of the classifier is established. The performance of such a classifier is tested both on simulated data and on the standard benchmarks providing misclassification errors comparable to, or even better than the k nearest neighbors (k-NN) method.