Vision Transformers provably learn spatial structure

Vision Transformers (ViTs) have recently achieved comparable or superior performance to Convolutional neural networks (CNNs) in computer vision. This empirical breakthrough is even more remarkable since ViTs discards spatial information by mixing patch embeddings and positional encodings and do not embed any visual inductive bias (e.g.\ spatial locality). Yet, recent work showed that while minimizing their training loss, ViTs specifically learn spatially delocalized patterns. This raises a central question: how do ViTs learn this pattern by solely minimizing their training loss using gradient-based methods from \emph{random initialization}? We propose a structured classification dataset and a simplified ViT model to provide preliminary theoretical justification of this phenomenon. Our model relies on a simplified attention mechanism --the positional attention mechanism-- where the attention matrix solely depends on the positional encodings. While the problem admits multiple solutions that generalize, we show that our model implicitly learns the spatial structure of the dataset while generalizing. We finally prove that learning the structure helps to sample-efficiently transfer to downstream datasets that share the same structure as the pre-training one but with different features. We empirically verify that ViTs using only the positional attention mechanism perform similarly to the original one on CIFAR-10/100, SVHN and ImageNet.