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This paper presents a fast and principled approach for solving the visual anomaly detection and segmentation problem. In this setup, we have access to only anomaly-free training data and want to detect and identify anomalies of an arbitrary nature on test data. We propose the application of linear statistical dimensionality reduction techniques on the intermediate features produced by a pretrained DNN on the training data, in order to capture the low-dimensional subspace truly spanned by said features. We show that the \emph{feature reconstruction error} (FRE), which is the $\ell_2$-norm of the difference between the original feature in the high-dimensional space and the pre-image of its low-dimensional reduced embedding, is extremely effective for anomaly detection. Further, using the same feature reconstruction error concept on intermediate convolutional layers, we derive FRE maps that provide pixel-level spatial localization of the anomalies in the image (i.e. segmentation). Experiments using standard anomaly detection datasets and DNN architectures demonstrate that our method matches or exceeds best-in-class quality performance, but at a fraction of the computational and memory cost required by the state of the art. It can be trained and run very efficiently, even on a traditional CPU.