Correlation Clustering with Noisy Input

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 ale similar (belonging to the same cluster) or dissimilar (belonging to different clusters) This lamination 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 Colleration clustering is APX-hard for worst-case inputs We study the following semi-random noisy model to generate the input stall, 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 similality/dissimilarity information arbitrarily for each corrupted pair of vertices In this model, out algorithm produces a. clustering with cost at most 1 + O(n(-1/6)) tones the cost of the optimal clustering, as long as p <= 1/2 71- n(-1/3) Moreover, if all clusters have size at least(1) c(1)root n then we can exactly reconstruct the planted clustering If the noise p is small, that p <= n(-delta)/60, then we can exactly reconstruct all clusters of the planted clustering that have size at least 3150/delta, and provide a certificate (witness) proving that those clusters file in any optimal clustering Among other techniques, we use the natural semi-definite programming relaxation followed by an ink-nesting rounding phase The analysis uses SDP duality and spectral properties of random mattices.