Compressed Anomaly Detection with Multiple Mixed Observations
We consider a collection of independent random variables that are identically distributed, except for a small subset which follows a different, anomalous distribution. We study the problem of detecting which random variables in the collection are governed by the anomalous distribution. Recent work proposes to solve this problem by conducting hypothesis tests based on mixed observations (e.g., linear combinations) of the random variables. Recognizing the connection between taking mixed observations and compressed sensing, we view the problem as recovering the “support” (index set) of the anomalous random variables from multiple measurement vectors (MMVs). Many algorithms have been developed for recovering jointly sparse signals and their support from MMVs. We establish the theoretical and empirical effectiveness of these algorithms in detecting anomalies. We also extend the LASSO algorithm to an MMV version for our purpose. Further, we perform experiments on synthetic data, consisting of samples from the random variables, to explore the trade-off between the number of mixed observations per sample and the number of samples required to detect anomalies.
The initial research for this effort was conducted at the Research Collaboration Workshop for Women in Data Science and Mathematics, July 17–21 held at ICERM. Funding for the workshop was provided by ICERM, AWM, and DIMACS (NSF grant CCF-1144502). SL was supported by NSF CAREER grant CCF− 1149225. DN was partially supported by the Alfred P. Sloan Foundation, NSF CAREER #1348721, and NSF BIGDATA #1740325. JQ was supported by NSF DMS-1818374.
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