It's 50 pages and cites this other paper in the same repo:
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation
The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.
It's 50 pages and cites this other paper in the same repo:
OpenAI. An explicit power saving for the exact discrete Fourier transform.
Here's a random excerpt:
8.3 The middle transform and the final permutation The factor QFt in (35) can be computed from a cyclic convolution and two pointwise phase multiplications. The chirp identity below performs the frequency change in Q without applying Q as a separate permutation of the array. The second identity shows how the retained source permutation R cancels when computing a convolution. Here ∗ denotes cyclic convolution on the product of the coordinate groups and a dot denotes coordinatewise multiplication.
Sure, I don’t see what’s horrible about this? It is a lot to read, sure, but it doesn’t seem unreasonably advanced