  Since their introduction the Trasformer architectures emerged as the
dominating architectures for both natural language processing and, more
recently, computer vision applications. An intrinsic limitation of this family
of "fully-attentive" architectures arises from the computation of the
dot-product attention, which grows both in memory consumption and number of
operations as $O(n^2)$ where $n$ stands for the input sequence length, thus
limiting the applications that require modeling very long sequences. Several
approaches have been proposed so far in the literature to mitigate this issue,
with varying degrees of success. Our idea takes inspiration from the world of
lossy data compression (such as the JPEG algorithm) to derive an approximation
of the attention module by leveraging the properties of the Discrete Cosine
Transform. An extensive section of experiments shows that our method takes up
less memory for the same performance, while also drastically reducing inference
time. This makes it particularly suitable in real-time contexts on embedded
platforms. Moreover, we assume that the results of our research might serve as
a starting point for a broader family of deep neural models with reduced memory
footprint. The implementation will be made publicly available at
https://github.com/cscribano/DCT-Former-Public
