Non‐negative residual matrix factorization: problem definition, fast solutions, and applications

Matrix factorization is a very powerful tool to find graph patterns, e.g. communities, anomalies, etc. A recent trend is to improve the usability of the discovered graph patterns, by encoding some interpretation-friendly properties (e.g., non-negativity, sparseness, etc) in the factorization. Most, if not all, of these methods are tailored for the task of community detection.We propose NrMF, a non-negative residual matrix factorization framework, aiming to improve the interpretation for graph anomaly detection. We present two optimization formations and their corresponding optimization solutions. Our method can naturally capture abnormal behaviors on graphs. We further generalize it to admit sparse constrains in the residual matrix. The effectiveness and efficiency of the proposed algorithms are analyzed, showing that our algorithm (i) leads to a local optima; and (ii) scales to large graphs. The experimental results on several data sets validate its effectiveness as well as efficiency. © 2011 Wiley Periodicals, Inc. Statistical Analysis and Data Mining 5: 3–15, 2012, © 2012 Wiley Periodicals, Inc. (Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. It is continuing through participation in the Anomaly Detection at Multiple Scales (ADAMS) program sponsored by the U.S. Defense Advanced Research Projects Agency (DARPA) under Agreement Number W911NF-11-C-0200. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.)