19 research outputs found
Efficient Classification for Metric Data
Recent advances in large-margin classification of data residing in general
metric spaces (rather than Hilbert spaces) enable classification under various
natural metrics, such as string edit and earthmover distance. A general
framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004]
left open the questions of computational efficiency and of providing direct
bounds on generalization error.
We design a new algorithm for classification in general metric spaces, whose
runtime and accuracy depend on the doubling dimension of the data points, and
can thus achieve superior classification performance in many common scenarios.
The algorithmic core of our approach is an approximate (rather than exact)
solution to the classical problems of Lipschitz extension and of Nearest
Neighbor Search. The algorithm's generalization performance is guaranteed via
the fat-shattering dimension of Lipschitz classifiers, and we present
experimental evidence of its superiority to some common kernel methods. As a
by-product, we offer a new perspective on the nearest neighbor classifier,
which yields significantly sharper risk asymptotics than the classic analysis
of Cover and Hart [IEEE Trans. Info. Theory, 1967].Comment: This is the full version of an extended abstract that appeared in
Proceedings of the 23rd COLT, 201
Faster Clustering via Preprocessing
We examine the efficiency of clustering a set of points, when the
encompassing metric space may be preprocessed in advance. In computational
problems of this genre, there is a first stage of preprocessing, whose input is
a collection of points ; the next stage receives as input a query set
, and should report a clustering of according to some
objective, such as 1-median, in which case the answer is a point
minimizing .
We design fast algorithms that approximately solve such problems under
standard clustering objectives like -center and -median, when the metric
has low doubling dimension. By leveraging the preprocessing stage, our
algorithms achieve query time that is near-linear in the query size ,
and is (almost) independent of the total number of points .Comment: 24 page