Robust Correlation Clustering

In this paper, we introduce and study the Robust-Correlation-Clustering problem: given a graph G = (V,E) where every edge is either labeled + or - (denoting similar or dissimilar pairs of vertices), and a parameter m, the goal is to delete a set D of m vertices, and partition the remaining vertices V D into clusters to minimize the cost of the clustering, which is the sum of the number of + edges with end-points in different clusters and the number of - edges with end-points in the same cluster. This generalizes the classical Correlation-Clustering problem which is the special case when m = 0. Correlation clustering is useful when we have (only) qualitative information about the similarity or dissimilarity of pairs of points, and Robust-Correlation-Clustering equips this model with the capability to handle noise in datasets. In this work, we present a constant-factor bi-criteria algorithm for Robust-Correlation-Clustering on complete graphs (where our solution is O(1)-approximate w.r.t the cost while however discarding O(1) m points as outliers), and also complement this by showing that no finite approximation is possible if we do not violate the outlier budget. Our algorithm is very simple in that it first does a simple LP-based pre-processing to delete O(m) vertices, and subsequently runs a particular Correlation-Clustering algorithm ACNAlg [Ailon et al., 2005] on the residual instance. We then consider general graphs, and show (O(log n), O(log^2 n)) bi-criteria algorithms while also showing a hardness of alpha_MC on both the cost and the outlier violation, where alpha_MC is the lower bound for the Minimum-Multicut problem

Online Correlation Clustering

We study the online clustering problem where data items arrive in an online fashion. The algorithm maintains a clustering of data items into similarity classes. Upon arrival of v, the relation between v and previously arrived items is revealed, so that for each u we are told whether v is similar to u. The algorithm can create a new cluster for v and merge existing clusters. When the objective is to minimize disagreements between the clustering and the input, we prove that a natural greedy algorithm is O(n)-competitive, and this is optimal. When the objective is to maximize agreements between the clustering and the input, we prove that the greedy algorithm is .5-competitive; that no online algorithm can be better than .834-competitive; we prove that it is possible to get better than 1/2, by exhibiting a randomized algorithm with competitive ratio .5+c for a small positive fixed constant c.Comment: 12 pages, 1 figur

Correlation Clustering with Low-Rank Matrices

Correlation clustering is a technique for aggregating data based on qualitative information about which pairs of objects are labeled 'similar' or 'dissimilar.' Because the optimization problem is NP-hard, much of the previous literature focuses on finding approximation algorithms. In this paper we explore how to solve the correlation clustering objective exactly when the data to be clustered can be represented by a low-rank matrix. We prove in particular that correlation clustering can be solved in polynomial time when the underlying matrix is positive semidefinite with small constant rank, but that the task remains NP-hard in the presence of even one negative eigenvalue. Based on our theoretical results, we develop an algorithm for efficiently "solving" low-rank positive semidefinite correlation clustering by employing a procedure for zonotope vertex enumeration. We demonstrate the effectiveness and speed of our algorithm by using it to solve several clustering problems on both synthetic and real-world data

Correlation Clustering with Same-Cluster Queries Bounded by Optimal Cost

Several clustering frameworks with interactive (semi-supervised) queries have been studied in the past. Recently, clustering with same-cluster queries has become popular. An algorithm in this setting has access to an oracle with full knowledge of an optimal clustering, and the algorithm can ask the oracle queries of the form, "Does the optimal clustering put vertices u and v in the same cluster?" Due to its simplicity, this querying model can easily be implemented in real crowd-sourcing platforms and has attracted a lot of recent work. In this paper, we study the popular correlation clustering problem (Bansal et al., 2002) under the same-cluster querying framework. Given a complete graph G=(V,E) with positive and negative edge labels, correlation clustering objective aims to compute a graph clustering that minimizes the total number of disagreements, that is the negative intra-cluster edges and positive inter-cluster edges. In a recent work, Ailon et al. (2018b) provided an approximation algorithm for correlation clustering that approximates the correlation clustering objective within (1+epsilon) with O((k^{14} log{n} log{k})/epsilon^6) queries when the number of clusters, k, is fixed. For many applications, k is not fixed and can grow with |V|. Moreover, the dependency of k^14 on query complexity renders the algorithm impractical even for datasets with small values of k. In this paper, we take a different approach. Let C_{OPT} be the number of disagreements made by the optimal clustering. We present algorithms for correlation clustering whose error and query bounds are parameterized by C_{OPT} rather than by the number of clusters. Indeed, a good clustering must have small C_{OPT}. Specifically, we present an efficient algorithm that recovers an exact optimal clustering using at most 2C_{OPT} queries and an efficient algorithm that outputs a 2-approximation using at most C_{OPT} queries. In addition, we show under a plausible complexity assumption, there does not exist any polynomial time algorithm that has an approximation ratio better than 1+alpha for an absolute constant alpha > 0 with o(C_{OPT}) queries. Therefore, our first algorithm achieves the optimal query bound within a factor of 2. We extensively evaluate our methods on several synthetic and real-world datasets using real crowd-sourced oracles. Moreover, we compare our approach against known correlation clustering algorithms that do not perform querying. In all cases, our algorithms exhibit superior performance

Correlation Clustering with Adaptive Similarity Queries

In correlation clustering, we are given $n$ objects together with a binary similarity score between each pair of them. The goal is to partition the objects into clusters so to minimise the disagreements with the scores. In this work we investigate correlation clustering as an active learning problem: each similarity score can be learned by making a query, and the goal is to minimise both the disagreements and the total number of queries. On the one hand, we describe simple active learning algorithms, which provably achieve an almost optimal trade-off while giving cluster recovery guarantees, and we test them on different datasets. On the other hand, we prove information-theoretical bounds on the number of queries necessary to guarantee a prescribed disagreement bound. These results give a rich characterization of the trade-off between queries and clustering error