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Title: On the Relationship between Self-Attention and Convolutional Layers

Paper content:
Recent trends of incorporating attention mechanisms in vision have led researchers to reconsider the supremacy of convolutional layers as a primary building block.
Beyond helping CNNs to handle long-range dependencies, Ramachandran et al. (2019) showed that attention can completely replace convolution and achieve state-of-the-art performance on vision tasks.
This raises the question: do learned attention layers operate similarly to convolutional layers?
This work provides evidence that attention layers can perform convolution and, indeed, they often learn to do so in practice.
Specifically, we prove that a multi-head self-attention layer with sufficient number of heads is at least as expressive as any convolutional layer.
Our numerical experiments then show that self-attention layers attend to pixel-grid patterns similarly to CNN layers, corroborating our analysis.
Our code is publicly available.
Recent advances in Natural Language Processing (NLP) are largely attributed to the rise of the transformer (Vaswani et al., 2017) .
Pre-trained to solve an unsupervised task on large corpora of text, transformer-based architectures, such as GPT-2 (Radford et al., 2018) , BERT (Devlin et al., 2018) and Transformer-XL , seem to possess the capacity to learn the underlying structure of text and, as a consequence, to learn representations that generalize across tasks.
The key difference between transformers and previous methods, such as recurrent neural networks (Hochreiter & Schmidhuber, 1997) and convolutional neural networks (CNN), is that the former can simultaneously attend to every word of their input sequence.
This is made possible thanks to the attention mechanism-originally introduced in Neural Machine Translation to better handle long-range dependencies (Bahdanau et al., 2015) .
With self-attention in particular, the similarity of two words in a sequence is captured by an attention score measuring the distance of their representations.
The representation of each word is then updated based on those words whose attention score is highest.
Inspired by its capacity to learn meaningful inter-dependencies between words, researchers have recently considered utilizing self-attention in vision tasks.
Self-attention was first added to CNN by either using channel-based attention (Hu et al., 2018) or non-local relationships across the image (Wang et al., 2018) .
More recently, augmented CNNs by replacing some convolutional layers with self-attention layers, leading to improvements on image classification and object detection tasks.
Interestingly, Ramachandran et al. (2019) noticed that, even though state-of-the art results are reached when attention and convolutional features are combined, under same computation and model size constraints, self-attention-only architectures also reach competitive image classification accuracy.
These findings raise the question, do self-attention layers process images in a similar manner to convolutional layers?
From a theoretical perspective, one could argue that transfomers have the capacity to simulate any function-including a CNN.
Indeed, Pérez et al. (2019) showed that a multilayer attention-based architecture with additive positional encodings is Turing complete under some strong theoretical assumptions, such as unbounded precision arithmetic.
Unfortunately, universality results do not reveal how a machine solves a task, only that it has the capacity to do so.
Thus, the question of how self-attention layers actually process images remains open.
We showed that self-attention layers applied to images can express any convolutional layer (given sufficiently many heads) and that fully-attentional models learn to combine local behavior (similar to convolution) and global attention based on input content.
More generally, fully-attentional models seem to learn a generalization of CNNs where the kernel pattern is learned at the same time as the filters-similar to deformable convolutions (Dai et al., 2017; Zampieri, 2019) .
Interesting directions for future work include translating existing insights from the rich CNNs literature back to transformers on various data modalities, including images, text and time series.

assistant: text: A self-attention layer can perform convolution and often learns to do so in practice.