Asymptotic Analysis of a Thresholding Method for Sparse Models with Application to Network Delay Detection

This paper explores a stochastic model of noisy observations with a sparse true signal structure. Such models arise in a wide range of applications, including signal processing, anomaly detection, and performance monitoring in telecommunication networks. As a motivating example, we consider round-trip time (RTT) data, which characterize the transit time of network packets, where rare, anomalously large values correspond to localized network congestion or failures. The focus is on the asymptotic properties of the mean-square risk associated with thresholding procedures. Upper bounds are obtained for the mean-square risk when using the theoretically optimal threshold. In addition, a central limit theorem and a strong law of large numbers are established for the empirical risk estimate. The results provide a theoretical basis for assessing the effectiveness of thresholding methods in localizing rare anomalous components in noisy data.