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### Maximally Divergent Intervals for Anomaly Detection

We present new methods for batch anomaly detection in multivariate time series. Our methods are based on maximizing the Kullback-Leibler divergence between the data distribution within and outside an interval of the time series. An empirical analysis shows the benefits of our algorithms compared to methods that treat each time step independently from each other without optimizing with respect to all possible intervals.

### Robust Variational Autoencoder for Tabular Data with Beta Divergence

We propose a robust variational autoencoder with $\beta$ divergence for tabular data (RTVAE) with mixed categorical and continuous features. Variational autoencoders (VAE) and their variations are popular frameworks for anomaly detection problems. The primary assumption is that we can learn representations for normal patterns via VAEs and any deviation from that can indicate anomalies. However, the training data itself can contain outliers. The source of outliers in training data include the data collection process itself (random noise) or a malicious attacker (data poisoning) who may target to degrade the performance of the machine learning model. In either case, these outliers can disproportionately affect the training process of VAEs and may lead to wrong conclusions about what the normal behavior is. In this work, we derive a novel form of a variational autoencoder for tabular data sets with categorical and continuous features that is robust to outliers in training data. Our results on the anomaly detection application for network traffic datasets demonstrate the effectiveness of our approach.

### Detecting Regions of Maximal Divergence for Spatio-Temporal Anomaly Detection

Automatic detection of anomalies in space- and time-varying measurements is an important tool in several fields, e.g., fraud detection, climate analysis, or healthcare monitoring. We present an algorithm for detecting anomalous regions in multivariate spatio-temporal time-series, which allows for spotting the interesting parts in large amounts of data, including video and text data. In opposition to existing techniques for detecting isolated anomalous data points, we propose the "Maximally Divergent Intervals" (MDI) framework for unsupervised detection of coherent spatial regions and time intervals characterized by a high Kullback-Leibler divergence compared with all other data given. In this regard, we define an unbiased Kullback-Leibler divergence that allows for ranking regions of different size and show how to enable the algorithm to run on large-scale data sets in reasonable time using an interval proposal technique. Experiments on both synthetic and real data from various domains, such as climate analysis, video surveillance, and text forensics, demonstrate that our method is widely applicable and a valuable tool for finding interesting events in different types of data.

### Nonparametric Divergence Estimation with Applications to Machine Learning on Distributions

Low-dimensional embedding, manifold learning, clustering, classification, and anomaly detection are among the most important problems in machine learning. The existing methods usually consider the case when each instance has a fixed, finite-dimensional feature representation. Here we consider a different setting. We assume that each instance corresponds to a continuous probability distribution. These distributions are unknown, but we are given some i.i.d. samples from each distribution. Our goal is to estimate the distances between these distributions and use these distances to perform low-dimensional embedding, clustering/classification, or anomaly detection for the distributions. We present estimation algorithms, describe how to apply them for machine learning tasks on distributions, and show empirical results on synthetic data, real word images, and astronomical data sets.

### KL divergence, JS divergence, and Wasserstein metric in Deep Learning

In this blog, I will continue the discussion of essential probability/statistics concepts by including 3 more concepts, which are widely used in deep learning to measure distances between probabilities distributions.