But what is a convolution?

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3Blue1Brown 

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Shared November 18, 2022

Discrete convolutions, from probability, to image processing and FFTs. Help fund future projects: www.patreon.com/3blue1brown​ Special thanks to these supporters: 3b1b.co/lessons/convolutions#thanks An equally valuable form of support is to simply share the videos. ------------------ Other videos I referenced Live lecture on image convolutions for the MIT Julia lab https://youtu.be/8rrHTtUzyZA Lecture on Discrete Fourier Transforms https://youtu.be/g8RkArhtCc4 Reducible video on FFTs https://youtu.be/h7apO7q16V0 Veritasium video on FFTs https://youtu.be/nmgFG7PUHfo A small correction for the integer multiplication algorithm mentioned at the end. A “straightforward” application of FFT results in a runtime of O(N * log(n) log(log(n)) ). That log(log(n)) term is tiny, but it is only recently in 2019, Harvey and van der Hoeven found an algorithm that removed that log(log(n)) term. Another small correction at 17:00. I describe O(N^2) as meaning "the number of operations needed scales with N^2". However, this is technically what Theta(N^2) would mean. O(N^2) would mean that the number of operations needed is at most constant times N^2, in particular, it includes algorithms whose runtimes don't actually have any N^2 term, but which are bounded by it. The distinction doesn't matter in this case, since there is an explicit N^2 term. These animations are largely made using a custom python library, manim. See the FAQ comments here: www.3blue1brown.com/faq#manim github.com/3b1b/manim github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: github.com/3b1b/videos/ Music by Vincent Rubinetti. www.vincentrubinetti.com/ Download the music on Bandcamp: vincerubinetti.bandcamp.com/album/the-music-of-3bl… Stream the music on Spotify: open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u Timestamps 0:00 - Where do convolutions show up? 2:07 - Add two random variables 6:28 - A simple example 7:25 - Moving averages 8:32 - Image processing 13:42 - Measuring runtime 14:40 - Polynomial multiplication 18:10 - Speeding up with FFTs 21:22 - Concluding thoughts ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: 3b1b.co/subscribe Various social media stuffs: Website: www.3blue1brown.com/ Twitter: twitter.com/3blue1brown Reddit: www.reddit.com/r/3blue1brown Instagram: www.instagram.com/3blue1brown Patreon: patreon.com/3blue1brown Facebook: www.facebook.com/3blue1brown


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