3Blue1Brown

**Shared November 18, 2022**

Discrete convolutions, from probability, to image processing and FFTs.
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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
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