5 research outputs found
Clustering is difficult only when it does not matter
Numerous papers ask how difficult it is to cluster data. We suggest that the
more relevant and interesting question is how difficult it is to cluster data
sets {\em that can be clustered well}. More generally, despite the ubiquity and
the great importance of clustering, we still do not have a satisfactory
mathematical theory of clustering. In order to properly understand clustering,
it is clearly necessary to develop a solid theoretical basis for the area. For
example, from the perspective of computational complexity theory the clustering
problem seems very hard. Numerous papers introduce various criteria and
numerical measures to quantify the quality of a given clustering. The resulting
conclusions are pessimistic, since it is computationally difficult to find an
optimal clustering of a given data set, if we go by any of these popular
criteria. In contrast, the practitioners' perspective is much more optimistic.
Our explanation for this disparity of opinions is that complexity theory
concentrates on the worst case, whereas in reality we only care for data sets
that can be clustered well.
We introduce a theoretical framework of clustering in metric spaces that
revolves around a notion of "good clustering". We show that if a good
clustering exists, then in many cases it can be efficiently found. Our
conclusion is that contrary to popular belief, clustering should not be
considered a hard task
On the practically interesting instances of MAXCUT
The complexity of a computational problem is traditionally quantified based
on the hardness of its worst case. This approach has many advantages and has
led to a deep and beautiful theory. However, from the practical perspective,
this leaves much to be desired. In application areas, practically interesting
instances very often occupy just a tiny part of an algorithm's space of
instances, and the vast majority of instances are simply irrelevant. Addressing
these issues is a major challenge for theoretical computer science which may
make theory more relevant to the practice of computer science.
Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a
clustering problem. Using a variety of techniques, we investigate practically
interesting instances of this problem. Specifically, we show how to solve in
polynomial time distinguished, metric, expanding and dense instances of MAXCUT
under mild stability assumptions. In particular, -stability
(which is optimal) suffices for metric and dense MAXCUT. We also show how to
solve in polynomial time -stable instances of MAXCUT,
substantially improving the best previously known result
A correspondence between type checking via reduction and type checking via evaluation. Accompanying code overview
This is an accompanying technical report for the paper with the corresponding title, published in Information Processing Letters, volume 112, issues 1--2, pages 13--20. This document contains detailed listings of different semantic artifacts for type checking with explanations on the performed transformations.nrpages: 18status: publishe