LASSO regularization is a popular regression tool to enhance the prediction accuracy of statistical models by performing variable selection through the $\ell_1$ penalty, initially formulated for the linear model and its variants. In this paper, the territory of LASSO is extended to two-layer ReLU neural networks, a fashionable and powerful nonlinear regression model. Specifically, given a neural network whose output $y$ depends only on a small subset of input $\boldsymbol{x}$, denoted by $\mathcal{S}^{\star}$, we prove that the LASSO estimator can stably reconstruct the neural network and identify $\mathcal{S}^{\star}$ when the number of samples scales logarithmically with the input dimension. This challenging regime has been well understood for linear models while barely studied for neural networks. Our theory lies in an extended Restricted Isometry Property (RIP)-based analysis framework for two-layer ReLU neural networks, which may be of independent interest to other LASSO or neural network settings. Based on the result, we advocate a neural network-based variable selection method. Experiments on simulated and real-world datasets show promising performance of the variable selection approach compared with existing techniques.

The existence of an anomaly detection method that is optimal for all domains is a myth. Thus, there exists a plethora of anomaly detection methods which increases every year for a wide variety of domains. But a strength can also be a weakness; given this massive library of methods, how can one select the best method for their application? Current literature is focused on creating new anomaly detection methods or large frameworks for experimenting with multiple methods at the same time. However, and especially as the literature continues to expand, an extensive evaluation of every anomaly detection method is simply not feasible. To reduce this evaluation burden, we present guidelines to intelligently choose the optimal anomaly detection methods based on the characteristics the time series displays such as seasonality, trend, level change concept drift, and missing time steps. We provide a comprehensive experimental validation and survey of twelve anomaly detection methods over different time series characteristics to form guidelines based on several metrics: the AUC (Area Under the Curve), windowed F-score, and Numenta Anomaly Benchmark (NAB) scoring model. Applying our methodologies can save time and effort by surfacing the most promising anomaly detection methods instead of experimenting extensively with a rapidly expanding library of anomaly detection methods, especially in an online setting.