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How can we detect anomalies: that is, samples that significantly differ from a given set of high-dimensional data, such as images or sensor data? This is a practical problem with numerous applications and is also relevant to the goal of making learning algorithms more robust to unexpected inputs. Autoencoders are a popular approach, partly due to their simplicity and their ability to perform dimension reduction. However, the anomaly scoring function is not adaptive to the natural variation in reconstruction error across the range of normal samples, which hinders their ability to detect real anomalies. In this paper, we empirically demonstrate the importance of local adaptivity for anomaly scoring in experiments with real data. We then propose our novel Adaptive Reconstruction Error-based Scoring approach, which adapts its scoring based on the local behaviour of reconstruction error over the latent space. We show that this improves anomaly detection performance over relevant baselines in a wide variety of benchmark datasets.