Discrete convolutions, from probability to image processing and FFTs.
Video on the continuous case: https://youtu.be/IaSGqQa5O-M
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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.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: Emanuele Vezzoli
Vietnamese: lkhphuc

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https://github.com/ManimCommunity/manim/

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Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/

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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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