### Efficient anomaly detection using bipartite k-NN graphs

Learning minimum volume sets of an underlying nominal distribution is a very effective approach to anomaly detection. Several approaches to learning minimum volume sets have been proposed in the literature, including the K-point nearest neighbor graph (K-kNNG) algorithm based on the geometric entropy minimization (GEM) principle [4]. The K-kNNG detector, while possessing several desirable characteristics, suffers from high computation complexity, and in [4] a simpler heuristic approximation, the leave-one-out kNNG (L1O-kNNG) was proposed. In this paper, we propose a novel bipartite k-nearest neighbor graph (BP-kNNG) anomaly detection scheme for estimating minimum volume sets. Our bipartite estimator retains all the desirable theoretical properties of the K-kNNG, while being computationally simpler than the K-kNNG and the surrogate L1O-kNNG detectors. We show that BP-kNNG is asymptotically consistent in recovering the p-value of each test point. Experimental results are given that illustrate the superior performance of BP-kNNG as compared to the L1O-kNNG and other state of the art anomaly detection schemes.

### Geometric entropy minimization (GEM) for anomaly detection and localization

We introduce a novel adaptive nonparametric anomaly detection approach, called GEM, that is based on the minimal covering properties of K-point entropic graphs when constructed on N training samples from a nominal probability distribution. Suchgraphs have the property that as N their span recovers the entropy minimizing set that supports at least ρ K/N(100)% of the mass of the Lebesgue part of the distribution. When a test sample falls outside of the entropy minimizing set an anomaly can be declared at a statistical level of significance α 1 ρ. A method for implementing this nonparametric anomaly detector is proposed that approximates this minimum entropy set by the influence region of a K-point entropic graph built on the training data. By implementing an incremental leave-one-out k-nearest neighbor graph on resampled subsets of the training data GEM can efficiently detect outliers at a given level of significance and compute their empirical p-values. We illustrate GEM for several simulated and real data sets in high dimensional feature spaces.

### Learning Efficient Anomaly Detectors from $K$-NN Graphs

We propose a non-parametric anomaly detection algorithm for high dimensional data. We score each datapoint by its average $K$-NN distance, and rank them accordingly. We then train limited complexity models to imitate these scores based on the max-margin learning-to-rank framework. A test-point is declared as an anomaly at $\alpha$-false alarm level if the predicted score is in the $\alpha$-percentile. The resulting anomaly detector is shown to be asymptotically optimal in that for any false alarm rate $\alpha$, its decision region converges to the $\alpha$-percentile minimum volume level set of the unknown underlying density. In addition, we test both the statistical performance and computational efficiency of our algorithm on a number of synthetic and real-data experiments. Our results demonstrate the superiority of our algorithm over existing $K$-NN based anomaly detection algorithms, with significant computational savings.

### Latent Laplacian Maximum Entropy Discrimination for Detection of High-Utility Anomalies

Data-driven anomaly detection methods suffer from the drawback of detecting all instances that are statistically rare, irrespective of whether the detected instances have real-world significance or not. In this paper, we are interested in the problem of specifically detecting anomalous instances that are known to have high real-world utility, while ignoring the low-utility statistically anomalous instances. To this end, we propose a novel method called Latent Laplacian Maximum Entropy Discrimination (LatLapMED) as a potential solution. This method uses the EM algorithm to simultaneously incorporate the Geometric Entropy Minimization principle for identifying statistical anomalies, and the Maximum Entropy Discrimination principle to incorporate utility labels, in order to detect high-utility anomalies. We apply our method in both simulated and real datasets to demonstrate that it has superior performance over existing alternatives that independently pre-process with unsupervised anomaly detection algorithms before classifying.

### A Rank-SVM Approach to Anomaly Detection

We propose a novel non-parametric adaptive anomaly detection algorithm for high dimensional data based on rank-SVM. Data points are first ranked based on scores derived from nearest neighbor graphs on n-point nominal data. We then train a rank-SVM using this ranked data. A test-point is declared as an anomaly at alpha-false alarm level if the predicted score is in the alpha-percentile. The resulting anomaly detector is shown to be asymptotically optimal and adaptive in that for any false alarm rate alpha, its decision region converges to the alpha-percentile level set of the unknown underlying density. In addition we illustrate through a number of synthetic and real-data experiments both the statistical performance and computational efficiency of our anomaly detector.