Math and code for efficiently computing large convolutions with FFTs.

ConvolutionsConvolutions are ubiquitous in data analysis. For decades, they’ve been used in signal and image processing. More recently, they became an important ingredient in modern neural networks. You’ve probably encountered convolutions if you work with data at all.Mathematically, convolutions are expressed as:

Although discrete convolutions are more common in computational applications, I’ll be working with the continuous form for most of this article, because it is much easier to prove the Convolution Theorem (discussed below) using continuous variables. After that, we will return to the discrete case and implement it in PyTorch using Fourier transforms. Discrete convolutions can be viewed as an approximation of continuous ones, where continuous functions are discretized on a regular grid. So we will not re-prove the Convolution Theorem for the discrete case.

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