Exact Recovery of Mangled Clusters with Same-Cluster Queries

TL;DR

This paper presents an algorithm for exactly recovering arbitrarily ellipsoidal clusters in a semi-supervised setting with minimal queries, extending previous spherical cluster assumptions and demonstrating efficiency and accuracy through experiments.

Contribution

It introduces a new algorithm that relaxes spherical cluster assumptions to arbitrary ellipsoids with margin, achieving exact recovery with logarithmic query complexity.

Findings

Queries scale logarithmically with input size

Algorithm recovers clusters with zero misclassification

Experimental validation confirms effectiveness

Abstract

We study the cluster recovery problem in the semi-supervised active clustering framework. Given a finite set of input points, and an oracle revealing whether any two points lie in the same cluster, our goal is to recover all clusters exactly using as few queries as possible. To this end, we relax the spherical $k$-means cluster assumption of Ashtiani et al.\ to allow for arbitrary ellipsoidal clusters with margin. This removes the assumption that the clustering is center-based (i.e., defined through an optimization problem), and includes all those cases where spherical clusters are individually transformed by any combination of rotations, axis scalings, and point deletions. We show that, even in this much more general setting, it is still possible to recover the latent clustering exactly using a number of queries that scales only logarithmically with the number of input points. More precisely, we design an algorithm that, given $n$ points to be partitioned into $k$ clusters, uses $O(k^3 \ln k \ln n)$ oracle queries and $\tilde{O}(kn + k^3)$ time to recover the clustering with zero misclassification error. The $O(\cdot)$ notation hides an exponential dependence on the dimensionality of the clusters, which we show to be necessary thus characterizing the query complexity of the problem. Our algorithm is simple, easy to implement, and can also learn the clusters using low-stretch separators, a class of ellipsoids with additional theoretical guarantees. Experiments on large synthetic datasets confirm that we can reconstruct clusterings exactly and efficiently.