Network function virtualization (NFV) is a promising network paradigm that enables the design and implementation of novel network services with lower cost, increased agility, and faster time-to-value. However, network anomalies caused by software malfunction, hardware failure, mis-configuration, or cyber attacks can greatly degrade the performance of NFV networks. A few matrix decomposition-based methods have shown their effectiveness in finding the existence of network-wide anomalies. However, a little attention has been paid to multiple anomalies detection and anomaly devices localization. To bridge this gap, in this paper, we propose a matrix differential decomposition (MDD)-based anomaly detection and localization algorithm for NFV networks. First, an NFV network prototype is built to investigate the property of NFV networks, and the effectiveness of traditional anomaly detection methods is evaluated. Second, we detail the MDD-based Anomaly DEtection and Localization (MADEL) algorithm. Finally, a series of experiments are conducted on three different NFV networks to evaluate the performance of the proposed algorithm. Experimental results show that the MADEL algorithm could effectively detect and localize different types of network anomalies.