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This paper proposes a new defense against neural network backdooring attacks that are maliciously trained to mispredict in the presence of attacker-chosen triggers. Our defense is based on the intuition that the feature extraction layers of a backdoored network embed new features to detect the presence of a trigger and the subsequent classification layers learn to mispredict when triggers are detected. Therefore, to detect backdoors, the proposed defense uses two synergistic anomaly detectors trained on clean validation data: the first is a novelty detector that checks for anomalous features, while the second detects anomalous mappings from features to outputs by comparing with a separate classifier trained on validation data. The approach is evaluated on a wide range of backdoored networks (with multiple variations of triggers) that successfully evade state-of-the-art defenses. Additionally, we evaluate the robustness of our approach on imperceptible perturbations, scalability on large-scale datasets, and effectiveness under domain shift. This paper also shows that the defense can be further improved using data augmentation.