  Vision transformers (ViTs) have pushed the state-of-the-art for various
visual recognition tasks by patch-wise image tokenization followed by
self-attention. However, the employment of self-attention modules results in a
quadratic complexity in both computation and memory usage. Various attempts on
approximating the self-attention computation with linear complexity have been
made in Natural Language Processing. However, an in-depth analysis in this work
shows that they are either theoretically flawed or empirically ineffective for
visual recognition. We further identify that their limitations are rooted in
keeping the softmax self-attention during approximations. Specifically,
conventional self-attention is computed by normalizing the scaled dot-product
between token feature vectors. Keeping this softmax operation challenges any
subsequent linearization efforts. Based on this insight, for the first time, a
softmax-free transformer or SOFT is proposed. To remove softmax in
self-attention, Gaussian kernel function is used to replace the dot-product
similarity without further normalization. This enables a full self-attention
matrix to be approximated via a low-rank matrix decomposition. The robustness
of the approximation is achieved by calculating its Moore-Penrose inverse using
a Newton-Raphson method. Extensive experiments on ImageNet show that our SOFT
significantly improves the computational efficiency of existing ViT variants.
Crucially, with a linear complexity, much longer token sequences are permitted
in SOFT, resulting in superior trade-off between accuracy and complexity.
