Many radiological studies can reveal the presence of several co-existing abnormalities, each one represented by a distinct visual pattern. In this article we address the problem of learning a distance metric for plain radiographs that captures a notion of "radiological similarity": two chest radiographs are considered to be similar if they share similar abnormalities. Deep convolutional neural networks (DCNs) are used to learn a low-dimensional embedding for the radiographs that is equipped with the desired metric. Two loss functions are proposed to deal with multi-labelled images and potentially noisy labels. We report on a large-scale study involving over 745,000 chest radiographs whose labels were automatically extracted from free-text radiological reports through a natural language processing system. Using 4,500 validated exams, we demonstrate that the methodology performs satisfactorily on clustering and image retrieval tasks.