Title: A Functional Perspective on Multi-Layer Out-of-Distribution Detection. Abstract: A crucial component for implementing reliable classifiers is detecting examples far from the reference (training) distribution, referred to as out-of-distribution (OOD) samples. A key feature of OOD detection is to exploit the network by extracting statistical patterns and relationships through the pre-trained multi-layer classifier. Despite achieving solid results, state-of-the-art methods require either additional OOD examples, expensive computation of gradients, or are tightened to a particular architecture, limiting their applications. This work adopts an original approach based on a functional view of the network that exploits the sample's trajectories through the various layers and their statistical dependencies. In this new framework, OOD detection translates into detecting samples whose trajectories differ from the typical behavior characterized by the training set. Our method significantly decreases the OOD detection error of classifiers trained on ImageNet and outperforms the state-of-the-art methods on average AUROC and TNR at 95% TPR. We demonstrate that the functional signature left by a sample in a network carries relevant information for OOD detection. 