Anomalous cluster detection

We consider the problem of anomalous cluster detection (ACD) on a graph under the elevated mean Gaussian model, where each node is associated with a feature. Under the null hypothesis, features are i.i.d. standard Gaussian, while under the alternative, there is an unknown connected cluster of nodes whose features are i.i.d. Gaussian with positive mean and unit variance instead. For this problem the GLRT scan statistic is usually adopted; however there are very few practical algorithms that target arbitrarily connected clusters. We formulate this problem as an integer program (IP) in terms of indicator variables, and characterize the connectivity of a cluster by a linear matrix inequality (LMI) constraint. We then propose a convex relaxation of the IP together with a rounding scheme, leading to a completely convex formulation for computing the scan statistic over arbitrarily connected clusters. Synthetic and real experiments justify our idea.