  Multi-scale Vision Transformer (ViT) has emerged as a powerful backbone for
computer vision tasks, while the self-attention computation in Transformer
scales quadratically w.r.t. the input patch number. Thus, existing solutions
commonly employ down-sampling operations (e.g., average pooling) over
keys/values to dramatically reduce the computational cost. In this work, we
argue that such over-aggressive down-sampling design is not invertible and
inevitably causes information dropping especially for high-frequency components
in objects (e.g., texture details). Motivated by the wavelet theory, we
construct a new Wavelet Vision Transformer (\textbf{Wave-ViT}) that formulates
the invertible down-sampling with wavelet transforms and self-attention
learning in a unified way. This proposal enables self-attention learning with
lossless down-sampling over keys/values, facilitating the pursuing of a better
efficiency-vs-accuracy trade-off. Furthermore, inverse wavelet transforms are
leveraged to strengthen self-attention outputs by aggregating local contexts
with enlarged receptive field. We validate the superiority of Wave-ViT through
extensive experiments over multiple vision tasks (e.g., image recognition,
object detection and instance segmentation). Its performances surpass
state-of-the-art ViT backbones with comparable FLOPs. Source code is available
at \url{https://github.com/YehLi/ImageNetModel}.
