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Has anyone ever come across this way of visualizing the convolution of two functions f and g: Look at the graph of the two-variable function f(x)g(y), and consider diagonal slices over the line x + y = k. The area of those slices represents (f * g)(k).

397,338 Aufrufe • vor 3 Jahren •via X (Twitter)

9 Kommentare

Profilbild von Grant Sanderson
Grant Sandersonvor 3 Jahren

I doubt this is original, but I haven't seen it anywhere. I'm searching to see if there's a good source to cite here. If you've come across it, feel free to share links with me!

Profilbild von Nick
Nickvor 3 Jahren

Yes, any binary operation can be visualized this way. Addition in the integers is a graph over a lattice, and in the continuum, kernels of integral operators like the fourier transform, green's functions, are seen as graphs over the plane rather than matrices. Just order the vars

Profilbild von Smarter Every Day
Smarter Every Dayvor 3 Jahren

It’s pretty. A great way to visualize a hike. I can imagine you doing a slice on a strange meandering path, and then “unrolling” it.

Profilbild von Mathoma
Mathomavor 3 Jahren

See 18:00 or so here:

Profilbild von Chris Billington
Chris Billingtonvor 3 Jahren

I tried

Profilbild von Craig Gidney
Craig Gidneyvor 3 Jahren

Anecdotally, this is how I've always visualized it and I assumed it was the "typical" way to view it. I do have a couple of old diagrams I drew of it, e.g. at there's

Profilbild von Tanuj Alapati
Tanuj Alapativor 3 Jahren

Pretty sure my probability textbook showed this method for when you have two rvs X and Y, and you want to calculate a pdf of Z := X+Y by integrating along a diagonal. Super intuitive

Profilbild von Artur Chakhvadze
Artur Chakhvadzevor 3 Jahren

I wonder whether this leads to some interesting generalisations of convolution for alternative couplings (e.g. optimal transport)

Profilbild von Luluberlu
Luluberluvor 3 Jahren

Je me le suis toujours représenté comme ça jpense pas que c’est nouveau

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