Correlation Clustering with Noisy Input

Abstract

Correlation clustering is a type of clustering that uses a basic form of input data: For every pair of data items, the input specifies whether they are similar (belonging to the same cluster) or dissimilar (belonging to different clusters). This information may be inconsistent, and the goal is to find a clustering (partition of the vertices) that disagrees with as few pieces of information as possible.Correlation clustering is APX-hard for worst-case inputs. We study the following semi-random noisy model to generate the input: start from an arbitrary partition of the vertices into clusters. Then, for each pair of vertices, the similarity information is corrupted (noisy) independently with probability p. Finally, an adversary generates the input by choosing similarity/dissimilarity information arbitrarily for each corrupted pair of vertices.In this model, our algorithm produces a clustering with cost at most 1 + O(n −1/6 ) times the cost of the optimal clustering, as long as p ≤ 1/2 − n −1/3 . Moreover, if all clusters have size at least 1 c1 √ n then we can exactly reconstruct the planted clustering. If the noise p is small, that is, p ≤ n −δ /60, then we can exactly reconstruct all clusters of the planted clustering that have size at least 3150/δ, and provide a certificate (witness) proving that those clusters are in any optimal clustering.Among other techniques, we use the natural semidefinite programming relaxation followed by an interesting rounding phase. The analysis uses SDP duality and spectral properties of random matrices.

Membership representation for detecting block-diagonal structure in low-rank or sparse subspace clustering

“…A membership matrix M is a symmetric matrix whose elements are either one or zero, which can be transformed into a block-diagonal matrix by permuting the same indices of rows and columns. This type of matrix frequently appears in correlation clustering [3,16,20]. For ease of explanation, we will assume that the rows and columns of M are aligned as…”

“…Note that the first two conditions also imply M ≤ 11 T . A similar relaxation technique has been used in correlation clustering [16,20]. Based on this relaxation, (9) becomes a convex problem which can be efficiently solved either by a semidefinite programming (SDP) or an augmented Lagrangian method (ALM) [14].…”

“…Our motivation is that this detection problem can be formulated as another self-expressive system with a semidefinite constraint. Our formulation represents a latent matrix by a Hadamard product of the latent matrix itself and a membership matrix, which frequently appears in correlation clustering [3,16,20]. Handling a membership matrix, however, is a difficult problem, because it is a discrete matrix with a special structure.…”

“…This work has been largely inspired by the recent studies in correlation clustering [3,9,16,20], and doubly stochastic normalization [22,27,28] in spectral clustering. The remainder of this paper is organized as follows: We present a brief review of SSC and LRR, and present the main idea of MR in Section 2.…”

Improved Theoretical and Practical Guarantees for Chromatic Correlation Clustering

“…One can easily notice that the correlation clustering problem is the special case of this problem when |L| = 1, and thus, it immediately follows that the chromatic correlation clustering problem is NP-hard [6]. The study of correlation clustering under complete graphs has been given a considerable attention in the past (see, e.g., [6,13,2,30,15,8]). Indeed, in many applications, the underlying graph is assumed to be complete as missing edges can be interpreted as negative.…”