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Convolution Is Fancy Multiplication

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Re: Convolution Is Fancy Multiplication

#51

Post by pmiller2 »

rectang wrote: post #25
nielsbot wrote: post #21

Earlier quoted context omitted.

depends on the audience I think

Math entries in Wikipedia generally favor completeness and correctness over clarity, which makes them of limited use for many users. In order to explain something simply, you usually have to lie a little bit — but I speculate that those little lies bother the contributors to mathematical Wikipedia articles a great deal. So they correct those little lies, making the articles more accurate but less useful for many of u…

In this instance, though, you don't really have to lie, but a lot of the hard work is fobbed off to Fubini's theorem.

Re: Convolution Is Fancy Multiplication

#52

Post by Tomminn »

adamnemecek wrote: post #41
Tomminn wrote: post #34

Earlier quoted context omitted.

The type signature of the outer product is not correct. We are mapping two functions to a function in the same domain. Convolution is neither a valid example of an inner product or an outer product. No basic geometric operation on vectors has the correct type signature and axioms for convolution to be interpreted as a generalized "x". What we'd be looking for is a billinear mapping of vectors to vectors, and complex…

Ok, exterior product.

The exterior product also exists in a different vector space to its factors.

Re: Convolution Is Fancy Multiplication

#53

Post by salty_biscuits »

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…

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?

Re: Convolution Is Fancy Multiplication

#56

Post by kazinator »

Tomminn wrote: post #31
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…

I don't think you can really call it a generalized dot product, because it doesn't map to a scalar. The inner product is the well accepted definition of a generalized dot product, and convolution does not follow the axioms that an inner product must follow.

Yes, thank you; you are right. The integral which is involved at the kernel of it is like a dot product, not the entire convolution itself.

Re: Convolution Is Fancy Multiplication

#57

Post by knappa »

rualca wrote: post #42
phaker wrote: post #11

And multiplication is a fancy convolution (with carries). Fast convolution algorithms use FFT and run in O(n log n) time, which is why fast multiplication algorithms are based on FFT and why everyone was looking for O(n log n) one, which was found like a year ago. (Though from what i can tell in "Faster Convolutions" section the article claims you can easily build O(n log n) multiplication using FFT which doesn't wor…

> And multiplication is a fancy convolution (with carries). It really isn't. Just because you see different digits being multiplied that doesn't mean you're convolving functions.

Let me show you some things: https://en.wikipedia.org/wiki/Multiplication_algorithm#Fouri... https://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strasse...

Re: Convolution Is Fancy Multiplication

#60

Post by dasudasu »

salty_biscuits wrote: post #53
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…

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.

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