Long Convolutions via Polynomial Multiplication
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Long Convolutions via Polynomial Multiplication
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Re: Long Convolutions via Polynomial Multiplication
#2Also known as Z-transform in digital signal processing.
Re: Long Convolutions via Polynomial Multiplication
#3Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
Re: Long Convolutions via Polynomial Multiplication
#4[dead]
Re: Long Convolutions via Polynomial Multiplication
#5Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
I think I'm beginning to wrap my head around the way modern, "deep" state-space models (e.g Mamba, S4, etc.) leverage polynomial multiplication to speed up very long convolutions.
I'm curious if there are other methods for approximating long convolutions that are well-known or widely-used, outside of overlap-add and overlap-save? I'm in the audio field and interested in learning long _FIR_ filters to describe the resonances of physical objects, like instruments, or rooms. Block-coding, or fixed-frame size approaches reign supreme, of course, but have their own issues in terms of windowing artifacts, etc.
I'm definitely aware that multiplication in the (complex) frequency domain is equivalent to convolution in the time domain and that, because of the fast-fourier transform, this can yield increased efficiency. However, this still results in storing a lot of gradient information that my intuition tells me (possibly incorrectly) is full of redundancy and waste.
Stateful, IIR, or auto-regressive approaches are _one_ obvious answer, but this changes the game in terms of training and inference parallelization.
A couple ideas I've considered, but have not yet tried, or looked too deeply into:
- First performing PCA in the complex frequency domain, reducing the point-wise multiplication that must occur. Without some additional normalization up-front, it's likely this would be equivalent to downsampling/low-pass filtering and performing the convolution there. The learnable filter bank would live in the PCA space, reducing the overall number of learned parameters.
- A Compressed Sensing inspired approach, where we perform a sparse, sub-sampled random set of points from both signals and recover the full result based on the assumption that both convolver and convolvee? are sparse in the fourier domain. This one is pretty half-baked.
I'd love to hear about papers you've read, or thoughts you've had about this problem.
Re: Long Convolutions via Polynomial Multiplication
#6Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
I don't understand this reasoning. Depending on the target (GPU/DSP/FPGA), going to frequency domain for convolution made some amount of sense when FFT primitives were highly optimized relative to conventional conv or matmul implementations. But now we're like 10 years into the software/hardware arms race and the conv/matmul kernels are just as highly optimized. In addition the hardware has adapted too.
> using Tensor Cores for FFT
Why would I do this when I could just directly use tensor cores for matmul...? We have MMA, WMMA, WGMMA, etc and they all target tensor cores explicitly.
Re: Long Convolutions via Polynomial Multiplication
#7Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
I think I'm beginning to wrap my head around the way modern, "deep" state-space models (e.g Mamba, S4, etc.) leverage polynomial multiplication to speed up very long convolutions. I'm curious if there are other methods for approximating long convolutions that are well-known or widely-used, outside of overlap-add and overlap-save? I'm in the audio field and interested in learning long _FIR_ filters to describe the res…
The gradient information in backroom can be computed similarly to forwards I think. Certainly the FFT blocks are linear and so now it's a question about the multiplication which is pretty compact.
Re: Long Convolutions via Polynomial Multiplication
#8Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
I don't understand this reasoning. Depending on the target (GPU/DSP/FPGA), going to frequency domain for convolution made some amount of sense when FFT primitives were highly optimized relative to conventional conv or matmul implementations. But now we're like 10 years into the software/hardware arms race and the conv/matmul kernels are just as highly optimized. In addition the hardware has adapted too. > using Tenso…
The time complexity of a large matrix multiplication is still much higher than using fourier, for large matrices it has superior performance.
Re: Long Convolutions via Polynomial Multiplication
#9Earlier quoted context omitted.
I don't understand this reasoning. Depending on the target (GPU/DSP/FPGA), going to frequency domain for convolution made some amount of sense when FFT primitives were highly optimized relative to conventional conv or matmul implementations. But now we're like 10 years into the software/hardware arms race and the conv/matmul kernels are just as highly optimized. In addition the hardware has adapted too. > using Tenso…
The time complexity of a large matrix multiplication is still much higher than using fourier, for large matrices it has superior performance.
Exactly this. I was thinking exactly like GP but I've been doing a large amount of benchmarking on this and the FFT rapidly overcomes direct convolution with cudnn, cublas, cutlass... I think I'd seen a recent Phd thesis exploring and confirming this recently. NlogN beats N^2 or N^3 quickly, even with tensor cores. At some point complexity overcomes even the bestest hardware optimizations.
And the longer the convolution the more matrices look tall-skinny (less optimized). Also you have to duplicate a lot of data to make your matrix toeplitz/circulant and fit into matmul kernels which convolution is a special case...
Re: Long Convolutions via Polynomial Multiplication
#10Talking long convolutions, on parallel architectures, through FFT there's a lot of performance to gain from Overlap-and-Add or Overlap-and-Save schemes, especially on GPUs with cuFFTDx which brings a whole new set of device primitives to the table and looking at (or getting inspiration from) tcFFT which allows using Tensor Cores for FFT and actually increases throughput on lots of convolution workloads.
I think I'm beginning to wrap my head around the way modern, "deep" state-space models (e.g Mamba, S4, etc.) leverage polynomial multiplication to speed up very long convolutions. I'm curious if there are other methods for approximating long convolutions that are well-known or widely-used, outside of overlap-add and overlap-save? I'm in the audio field and interested in learning long _FIR_ filters to describe the res…
The convolution by FFT overlap and save can have very low intermediate storage (none on GPU with cuFFTDx for example). And most of the time, the IFFT doesn't have to happen right away, lots processing can still be performed in the frequential domain.
Having each of 18k CUDA cores of a L40s perform small 128-points FFTs and with very little sync or overlap manage long filters... is pretty efficient by itself.
There's a lot happening in the HPC world on low-rank (what you're intuiting with PCA), sparse and tiled operations. I have a hard time applying all this to 'simple' signal processing and most of it lacks nicer APIs.
I've seen lots of interesting things with 'irregular FFT' codes and working on reducing either the storage space necessary for FFT intermediate results, sometimes through multi-resolution tricks.
Look up Capon filters and adaptative filtering in general, there's a whole world of tricks there too. You might need a whole lot of SVDs and matrix inversions there...
Bust mostly if you're on a GPU there's a wealth of parallelism to exploit and work-around the 'memory-bound' limits of FFT-based convolution. This thesis https://theses.hal.science/tel-04542844 had some discussion and numbers on the topic. Not complete but inspiring.