This paper concerns the use of threshold decision lists for classifying data into two classes. The use of such methods has a natural geometrical interpretation and can be appropriate for an iterative approach to data classification, in which some points of the data set are given a particular classification, according to a linear threshold function (or hyperplane), are then removed from consideration, and the procedure iterated until all points are classified. We analyse theoretically the generalization properties of data classification techniques that are based on the use of threshold decision lists and the subclass of multilevel threshold functions. We obtain bounds on the generalization error that depend on the levels of separation — or margins — achieved by the successive linear classifiers.

We apply techniques from probabilistic learning theory to analyse theoretically the accuracy of data classification techniques that are based on the use of threshold decision lists. 1

A new upper bound is given on the number of ways in which a set of N points in R n can be partitioned by k parallel hyperplanes. This bound improves upon a result of Olafsson and Abu-Mostafa [IEEE Trans. Pattern Analysis and Machine Intelligence 10(2), 1988: 277-281]; it agrees with the known (tight) result for the case k = 1; and it is, for fixed k and n, tight to within a constant. A previously published claimed improvement to the bound of Olafsson and Abu-Mostafa is shown to be incorrect. 1