Title: Spatial Entropy as an Inductive Bias for Vision Transformers. Abstract: Recent work showed that the attention maps of Vision Transformers (VTs), when trained with self-supervision, can contain a semantic segmentation structure which does not spontaneously emerge when training is supervised. In this paper, we explicitly encourage the emergence of this spatial clustering as a form of training regularization, this way including a self-supervised pretext task into the standard supervised learning. In more detail, we exploit the assumption that, in a given image, objects usually correspond to few connected regions, and we propose a spatial formulation of the information entropy to quantify this object-based inductive bias. By minimizing the proposed spatial entropy, we include an additional self-supervised signal during training. Using extensive experiments, we show that the proposed regularization is beneficial with different training scenarios, datasets, downstream tasks and VT architectures. The code will be available upon acceptance.