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Convolution Is Fancy Multiplication

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Re: Convolution Is Fancy Multiplication

#71

Post by Barrin92 »

dwrodri wrote: post #65

I'm a firm believer that resources which raise the bar for math and computing education are huge catalysts for innovation. The fact that resources like BetterExplained, Paul's Math Notes, 3Blue1Brown's YouTube Channel, Ben Eater's YouTube channel, and the entire body of high quality MOOCs are just available for free over the Internet is probably one of my favorite accomplishments of the human race in the 21st century…

It's a good goal from just a humanistic standpoint and because learning should be recreational but from an innovation standpoint it's honestly likely overrated.

Innovation is the result of very particular institutions and resources being coordinated and people networking and clustering tightly, which is why, despite incredibly dispersion of knowledge over the internet, innovation and VC money actually still is so geographically concentrated.

I think these online learning resources are good but the expectations should be correct, they're not tools to kickstart innovation, which is much more tacit and reliant on close interaction between people rather than knowledge.

Re: Convolution Is Fancy Multiplication

#72

Post by ReactiveJelly »

dasudasu wrote: post #60
salty_biscuits wrote: post #53

Earlier quoted context omitted.

I might be wrong (so happy to be corrected by a native speaker) but I liked when I heard that the German word for convolution is "faltung" which translates to "folding". This is a much nicer word to metaphorically understand the operation. Maybe it still confuses the hell out of German undergrads though?

The etymology is the same in English. To convolve : to roll together : writhe. Convolution : a form or shape that is folded in curved or tortuous windings.

That confuses the hell out of me.

I first encountered it as a young gamedev hobbyist, doing image post-processing in OpenGL.

One kind of kernel makes images blurry, one (attempts to) make them sharp.

It doesn't bring folding to mind at all. And in FP, "fold" means "reduce".

How much of dyscalculia is just one person thinking an analogy is completely obvious and the other person thinking it makes no sense?

Re: Convolution Is Fancy Multiplication

#73

Post by salty_biscuits »

ReactiveJelly wrote: post #72
dasudasu wrote: post #60

Earlier quoted context omitted.

The etymology is the same in English. To convolve : to roll together : writhe. Convolution : a form or shape that is folded in curved or tortuous windings.

That confuses the hell out of me. I first encountered it as a young gamedev hobbyist, doing image post-processing in OpenGL. One kind of kernel makes images blurry, one (attempts to) make them sharp. It doesn't bring folding to mind at all. And in FP, "fold" means "reduce". How much of dyscalculia is just one person thinking an analogy is completely obvious and the other person thinking it makes no sense?

Probably due to the really idiosyncratic way I think about things and from the post it is not at all clear what I mean, sorry for the imprecision! I think of cross correlation of 1D signals with a kernel as sliding one of the functions over the other [0]. In my head I imagine drawing the signal and kernel next to each other one a big piece of paper and literally folding it over and sliding it back over the signal to intuit the convolution (with an imaginary see through piece of paper). I really don't think that was obvious from what I posted, so definitely not dyscalculia!

https://commons.wikimedia.org/wiki/File:Cross_correlation_an...

Re: Convolution Is Fancy Multiplication

#74

Post by [deleted] »

kazinator wrote: post #12

Convolution is a correlation with a reversed signal. Correlation is a generalized dot product: multiplying corresponding pairs of values from two signals, and then adding the factors together. The result is zero if the signals are orthogonal (like the dot product of two vectors in 2D or 3D that are at 90 degrees). The intuition behind the reversed signal comes from processing in the time domain. There are application…

[deleted]

Re: Convolution Is Fancy Multiplication

#75

Post by jonahx »

spekcular wrote: post #16

I also enjoy Terence Tao's explanation [0]: > I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad first approximation for this effect). I found this to be…

"if one thinks of functions as fuzzy versions of points"

Can anyone elaborate on this?

Re: Convolution Is Fancy Multiplication

#76

Post by syntaxing »

My diffeq professor explained it as the fifth form of arithmetic (first four being addition, subtraction, multiplication, and division). This fifth form is unique because you sweep two functions relative to time and each other. To be honest, I still don't fully grasp the concept and I just use it as a mathematical tool. I need a 3blue1brown video to explain this to me so I can have an "aha!" moment like I had in his linear algebra videos.

Re: Convolution Is Fancy Multiplication

#77

Post by sinfulprogeny »

dwrodri wrote: post #65

I'm a firm believer that resources which raise the bar for math and computing education are huge catalysts for innovation. The fact that resources like BetterExplained, Paul's Math Notes, 3Blue1Brown's YouTube Channel, Ben Eater's YouTube channel, and the entire body of high quality MOOCs are just available for free over the Internet is probably one of my favorite accomplishments of the human race in the 21st century…

I wholeheartedly agree on the point about decentralizing and modernizing education, and thank you for reminding me of this. We still have a massive bias towards academic institutions to work against, but the fact that knowledge is so easily accessible, and of such a high quality, is pretty amazing.

Re: Convolution Is Fancy Multiplication

#78

Post by peppery »

jonahx wrote: post #75
spekcular wrote: post #16

I also enjoy Terence Tao's explanation [0]: > I remember as a graduate student that Ingrid Daubechies frequently referred to convolution by a bump function as "blurring" - its effect on images is similar to what a short-sighted person experiences when taking off his or her glasses (and, indeed, if one works through the geometric optics, convolution is not a bad first approximation for this effect). I found this to be…

"if one thinks of functions as fuzzy versions of points" Can anyone elaborate on this?

Tao is likely inviting us (just as many physical/probabilistic laws do) to view any arbitrary function as relatively "thicker"/"fuzzier" than an infinitely-thin, infinitely-tall spike function at a certain value: the Dirac delta function (https://en.wikipedia.org/wiki/Dirac_delta_function). If you convolve ≡ integrate this Dirac delta function (located at some value x) against any function g(t), by construction the integral is zero everywhere except at t=x, so the result is an infinitely thin slice of g at x, exactly g(x) (the 'sifting property,' https://math.stackexchange.com/questions/1015498/convolution...). Now imagine you begin to thicken/fuzz the spike; now you begin to accumulate the behavior of g(t) not just exactly at x, but also at points nearby, getting a schmeared representation of g. Deforming our spike to an arbitrary function of interest in this way gives an arbitrary convolution schmear.

Re: Convolution Is Fancy Multiplication

#79

Post by mathnmusic »

Barrin92 wrote: post #71
dwrodri wrote: post #65

I'm a firm believer that resources which raise the bar for math and computing education are huge catalysts for innovation. The fact that resources like BetterExplained, Paul's Math Notes, 3Blue1Brown's YouTube Channel, Ben Eater's YouTube channel, and the entire body of high quality MOOCs are just available for free over the Internet is probably one of my favorite accomplishments of the human race in the 21st century…

It's a good goal from just a humanistic standpoint and because learning should be recreational but from an innovation standpoint it's honestly likely overrated. Innovation is the result of very particular institutions and resources being coordinated and people networking and clustering tightly, which is why, despite incredibly dispersion of knowledge over the internet, innovation and VC money actually still is so geo…

Internet has some amazing learning resources these days. Not just articles like these and videos, but also interactive explorables. Even my 7yo was able to understand basic aspects of game theory and complex systems with the help of explorables made by NickyCase, MeltingAsphalt etc. Most recently, I taught myself to not just solve 4x4 Rubik's cube but understand its connections to ideas in abstract algebra such as commutators. I feel extreme gratitude for all the people who create such enlightening material and offer it freely, without asking for a paid subscription to their Udemy course.

It inspired me enough that I started creating a DMOZ-style catalog of learning resources as a FOSS project: https://github.com/learn-awesome/learn

Re: Convolution Is Fancy Multiplication

#80

Post by enriquto »

sinfulprogeny wrote: post #77
dwrodri wrote: post #65

I'm a firm believer that resources which raise the bar for math and computing education are huge catalysts for innovation. The fact that resources like BetterExplained, Paul's Math Notes, 3Blue1Brown's YouTube Channel, Ben Eater's YouTube channel, and the entire body of high quality MOOCs are just available for free over the Internet is probably one of my favorite accomplishments of the human race in the 21st century…

I wholeheartedly agree on the point about decentralizing and modernizing education, and thank you for reminding me of this. We still have a massive bias towards academic institutions to work against, but the fact that knowledge is so easily accessible, and of such a high quality, is pretty amazing.

Why do you think academic institutions are something to work against to? I'm in academia and here everybody is enthralled and really happy about the explosion of high-quality, freely available learning resources.

The only kind of complain that I have ever heard (and very rarely so) is because some of this material is supported by videos in a platform with shitty terms of service.

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