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Comment by akrymski | Hacker News Reader

I'm really interested in training NNs in the frequency domain:

- a ton of data (JPEGs, MPEGs, MFCCs) exists in the compressed form of FFT-ed features

- FFTs are hardware encoded/decoded & optimised

- FFTs are great at separating signal from noise, leading to a huge reduction in input features (eg JPEG encoded) as compared to raw pixel values

- convolutions in the time domain are just multiplications in the frequency domain

- converting to log polar coordinates turns them into additions

- distance between two log polar coordinates could be the loss function

However, I've not had much luck getting a frequency domain network like this to converge. Anyone tried something similar with success?

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The choice of activation function isn't entirely clear to me, but I think it's definitely possible to make a network that operates entirely in the frequency domain. It would probably be pretty easy to start experimenting with such a thing with the nice complex number and FFT support in PyTorch 1.8. :)

Like you said, there's already a significant connection between convolutional networks and the Fourier domain (the convolution theorem).

Tangentially, I've recently worked on a project that focused on implementing convolution in the Fourier domain, and how that allows one to control useful properties of convolutions (like "smoothness" and orthogonality).

I made a demonstration of convolution in the Fourier domain in PyTorch, which you might find interesting: https://nbviewer.jupyter.org/github/locuslab/orthogonal-conv...

More generally, you could look here for more code and the corresponding paper: https://github.com/locuslab/orthogonal-convolutions

I don't see why processing JPEG coefficients directly as input features shouldn't have similar accuracy to a much larger NN using raw pixel data.

In particular I'm interested in the efficiency gains from avoiding convolutions and the possibility of running a compressed frequency-domain NN on CPUs.

I have trained GANs on raw JPEG coefficients with moderate success as a pet project. Read the raw DCT coefficients without decompressing and train in this space. JPEG-decompress the output of the net to reconstruct an image in the pixel space. There are few papers doing similar things for supervised learning tasks iirc

Using a standard loss function like MSE?

Yeah it kinda works when you feed JPEG coefficients into a typical time-domain CNN, but mathematically it seems that if you're using frequencies as inputs, your convolutions should become simple multiplications. Am I wrong?

Yep, you can successfully do it without convolutions. Here is a pointer if you want to dig deeper: https://eng.uber.com/neural-networks-jpeg/ (there's prior art to that, but this one is well written)

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