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Video anomaly detection can be used in the transportation domain to identify unusual patterns such as traffic violations, accidents, unsafe driver behavior, street crime, and other suspicious activities. A common class of approaches relies on object tracking and trajectory analysis. Very recently, sparse reconstruction techniques have been employed in video anomaly detection. The fundamental underlying assumption of these methods is that any new feature representation of a normal/anomalous event can be approximately modeled as a (sparse) linear combination prelabeled feature representations (of previously observed events) in a training dictionary. Sparsity can be a powerful prior on model coefficients but challenges remain in the detection of anomalies involving multiple objects and the ability of the linear sparsity model to effectively allow for class separation. The proposed research addresses both these issues. First, we develop a new joint sparsity model for anomaly detection that enables the detection of joint anomalies involving multiple objects. This extension is highly nontrivial since it leads to a new simultaneous sparsity problem that we solve using a greedy pursuit technique. Second, we introduce nonlinearity into, that is, kernelize. The linear sparsity model to enable superior class separability and hence anomaly detection. We extensively test on several real world video datasets involving both single and multiple object anomalies. Results show marked improvements in detection of anomalies in both supervised and unsupervised scenarios when using the proposed sparsity models.