Title: Attention: Self-Expression Is All You Need. Abstract: Transformer models have achieved significant improvements in performance for various learning tasks in natural language processing and computer vision. Much of their success is attributed to the use of attention layers that capture long-range interactions among data tokens (such as words and image patches) via attention coefficients that are global and adapted to the input data at test time. In this paper we study the principles behind attention and its connections with prior art. Specifically, we show that attention builds upon a long history of prior work on manifold learning and image processing, including methods such as kernel-based regression, non-local means, locally linear embedding, subspace clustering and sparse coding. Notably, we show that self-attention is closely related to the notion of self-expressiveness in subspace clustering, wherein data points to be clustered are expressed as linear combinations of other points with global coefficients that are adapted to the data and capture long-range interactions among data points. We also show that heuristics in sparse self-attention can be studied in a more principled manner using prior literature on sparse coding and sparse subspace clustering. We thus conclude that the key innovations of attention mechanisms relative to prior art are the use of many learnable parameters, and multiple heads and layers.