Reconstructing Many Partitions Using Spectral Techniques

Abstract

A partitioning of a set of n items is a grouping of these items into k disjoint, equally sized classes. Any partition can be modeled as a graph. The items become the vertices of the graph and two vertices are connected by an edge if and only if the associated items belong to the same class. In a planted partition model a graph that models a partition is given, which is obscured by random noise, i.e., edges within a class can get removed and edges between classes can get inserted. The task is to reconstruct the planted partition from this graph. In the model that we study the number k of classes controls the difficulty of the task. We design a spectral partitioning algorithm that asymptotically almost surely reconstructs up to $k = c\sqrt{n}$ partitions, where c is a small constant, in time C ^k poly(n), where C is another constant.

Partly supported by the Swiss National Science Foundation under the grant “Non-linear manifold learning”.