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A least squares semi-supervised local clustering algorithm based on the ideaof compressed sensing is proposed to extract clusters from a graph with knownadjacency matrix. The algorithm is based on a two-stage approach similar to theone in \cite{LaiMckenzie2020}. However, under a weaker assumption and with lesscomputational complexity than the one in \cite{LaiMckenzie2020}, the algorithmis shown to be able to find a desired cluster with high probability. The ``onecluster at a time" feature of our method distinguishes it from other globalclustering methods. Several numerical experiments are conducted on thesynthetic data such as stochastic block model and real data such as MNIST,political blogs network, AT\&T and YaleB human faces data sets to demonstratethe effectiveness and efficiency of our algorithm.

A Compressed Sensing Based Least Squares Approach to Semi-supervised Local Cluster Extraction