We describe a procedure which finds a hierarchical clustering by hillclimbing. Thecost function we use is a hierarchical extension of the k-means cost; our local moves are tree restructurings and node reorderings. Weshow these can be accomplished efficiently, by exploiting special properties of squared Euclidean distances and by using techniques from scheduling algorithms.

In this paper, we study bootstrapping algorithms for learning from unlabeled data. The general idea in bootstrapping is to use some initial labeled data to build a (possibly partial) predictive labeling procedure; then use the labeling procedure to label more data; then use the newly labeled data to build a new predictive procedure and so on. This process can be iterated until a fixed point is reached or some other stopping criterion is met. Here we give P AC style bounds on generalization error which can be used to formally justify certain boostrapping algorithms. One well-known form of bootstrapping is the EM algorithm (Dempster, Laird and Rubin, 1977).

The rule-based bootstrapping introduced by Y arowsky, and its co-training variant by Blum and Mitchell, have met with considerable empirical success. Earlier work on the theory of co-training has been only loosely related to empirically useful co-training algorithms. Here we give a new P ACstyle bound on generalization error which justifies both the use of confidences -- partial rules and partial labeling of the unlabeled data -- and the use of an agreement-based objective function as suggested by Collins and Singer. Our bounds apply to the multiclass case, i.e., where instances are to be assigned one of

The rule-based bootstrapping introduced by Yarowsky, and its cotraining variantby Blum and Mitchell, have met with considerable empirical success. Earlier work on the theory of co-training has been only loosely related to empirically useful co-training algorithms. Here we give a new PACstyle bound on generalization error which justifies both the use of confidences -- partial rules and partial labeling of the unlabeled data -- and the use of an agreement-based objective function as suggested byCollins and Singer. Our bounds apply to the multiclass case, i.e., where instances are to be assigned one of labels for