Near-Optimal Comparison Based Clustering

TL;DR

This paper introduces a near-optimal method for clustering based solely on ordinal comparisons, using a two-step approach with similarity estimation and SDP-based clustering, supported by theoretical guarantees and real data validation.

Contribution

It presents a novel two-step clustering method that recovers planted clusters from passive comparisons with near-optimal sample complexity.

Findings

Exact recovery of planted clusters under certain conditions

Method performs well on real datasets

Theoretical guarantees match empirical results

Abstract

The goal of clustering is to group similar objects into meaningful partitions. This process is well understood when an explicit similarity measure between the objects is given. However, far less is known when this information is not readily available and, instead, one only observes ordinal comparisons such as "object i is more similar to j than to k." In this paper, we tackle this problem using a two-step procedure: we estimate a pairwise similarity matrix from the comparisons before using a clustering method based on semi-definite programming (SDP). We theoretically show that our approach can exactly recover a planted clustering using a near-optimal number of passive comparisons. We empirically validate our theoretical findings and demonstrate the good behaviour of our method on real data.