  The transformer model is known to be computationally demanding, and
prohibitively costly for long sequences, as the self-attention module uses a
quadratic time and space complexity with respect to sequence length. Many
researchers have focused on designing new forms of self-attention or
introducing new parameters to overcome this limitation, however a large portion
of them prohibits the model to inherit weights from large pretrained models. In
this work, the transformer's inefficiency has been taken care of from another
perspective. We propose Fourier Transformer, a simple yet effective approach by
progressively removing redundancies in hidden sequence using the ready-made
Fast Fourier Transform (FFT) operator to perform Discrete Cosine Transformation
(DCT). Fourier Transformer is able to significantly reduce computational costs
while retain the ability to inherit from various large pretrained models.
Experiments show that our model achieves state-of-the-art performances among
all transformer-based models on the long-range modeling benchmark LRA with
significant improvement in both speed and space. For generative seq-to-seq
tasks including CNN/DailyMail and ELI5, by inheriting the BART weights our
model outperforms the standard BART and other efficient models. \footnote{Our
code is publicly available at
\url{https://github.com/LUMIA-Group/FourierTransformer}}
