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Convolution Is Fancy Multiplication

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Re: Convolution Is Fancy Multiplication

#61

Post by sriku »

rualca wrote: post #28
jaggirs wrote: post #19

Earlier quoted context omitted.

Woha This is cool. It literally is the same, if you use base-infinity digits (not binary, or decimal, but infinity-ary?). How do you multiply two polynomials with a single variable?

> This is cool I don't agree the example has anything to do with convolution. The example is explained with plain old primary school multiplication. 100101 * 23 = (100000+100+1)*23 = 2300000+2300+23 There isn't any fantastic property, only cherry-picked number which works as decimal left-shifts. Convolution is not a number. Convolution is an operator that outputs a function.

(sum_i ai x^i)(sum_j bj x^j) when you expand that, the coefficient of x^k is the sum of aibj such that i+j=k which is discrete convolution.

Set x=10 and you're dealing with decimal representation of numbers.

Re: Convolution Is Fancy Multiplication

#62

Post by alstaxi »

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…

In your example, I guess you are reversing the 2nd signal so if the signals are the opposite. "Adding" or convolving them results in a zero?

i.e. if the signal is a (+)slope /, convolving it with a reverse of that signal \ = 0

In this sense you are leaving the difference between 2 signals.

Re: Convolution Is Fancy Multiplication

#63

Post by sriku »

rualca wrote: post #43
sriku wrote: post #8

Convolution is digit-based multiplication, so why not start with that? 100101 * 23 = 2302323. If you defer the carry over to the end, the digit-based steps we do is convolution. That said, giving lots of examples like those in the article is useful. Next up looking at multiplying two polynomials in a single variable. Edit: changed digit-wise to digit-based to make it clear that I'm not asking to multiply correspondin…

> Convolution is digit-based multiplication, It really isn't, and I don't understand the level of confusion that leads to this assertion. Is it because the elementary school algorithm for multiplication decomposes a number in it's decimal places before multiplying? > so why not start with that? Because it would be wrong, misguided, and it would fail to demonstrate the logic behind convolution. Let's take a step back…

Ref comment https://news.ycombinator.com/item?id=25195046 about this.

Further to that, if you replace x with complex variable z^-1, you get what is usually called the z-transform. Set z to a specific complex root of unity exp(i2pi/N) and you have your discrete FT.

One of the algorithms for multiplying large arbitrary precision numbers uses multiplication of the discrete transform of the digit sequences (in a different base) iirc.

Same thing.

Re: Convolution Is Fancy Multiplication

#64

Post by anonytrary »

f(x)g(x-t)dt

Integrating over t can be visualized as sliding g over f over the range of t. Result is a function of x as t has been integrated out. It all made sense to me when I thought of it as sliding one function over another fixed function.

Re: Convolution Is Fancy Multiplication

#65

Post by dwrodri »

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. While researchers, inventors, startup founders, and all other types of entrepreneurs are pushing the envelope on what's possible, there are literally thousands of creators and educators who are doing their best to bring some of that knowledge to the common man.

My own experience with academia has lead me to believe that there problems that are truly difficult and complex, but there are also problems which are described as difficult because so many people were exposed to the topic through a resource (typically another person) who had little experience (or interest) in effectively teaching the subject.

Re: Convolution Is Fancy Multiplication

#66

Post by sriku »

rualca wrote: post #43
sriku wrote: post #8

Convolution is digit-based multiplication, so why not start with that? 100101 * 23 = 2302323. If you defer the carry over to the end, the digit-based steps we do is convolution. That said, giving lots of examples like those in the article is useful. Next up looking at multiplying two polynomials in a single variable. Edit: changed digit-wise to digit-based to make it clear that I'm not asking to multiply correspondin…

> Convolution is digit-based multiplication, It really isn't, and I don't understand the level of confusion that leads to this assertion. Is it because the elementary school algorithm for multiplication decomposes a number in it's decimal places before multiplying? > so why not start with that? Because it would be wrong, misguided, and it would fail to demonstrate the logic behind convolution. Let's take a step back…

> Convolution is mainly and primarily used as a convenient way to covert signals between time and frequency domain

Btw that isn't true. That operation is called a "transform" as in "Fourier transform" and "z-transform". The property of those transforms is that element-wise multiplication in one domain becomes convolution in the transformed domain. Element-wise multiplication in "time" domain is convolution in "frequency" domain and element-wise multiplication in "frequency" domain is convolution in "time" domain.

Re: Convolution Is Fancy Multiplication

#67

Post by sriku »

jaggirs wrote: post #19
sriku wrote: post #8

Convolution is digit-based multiplication, so why not start with that? 100101 * 23 = 2302323. If you defer the carry over to the end, the digit-based steps we do is convolution. That said, giving lots of examples like those in the article is useful. Next up looking at multiplying two polynomials in a single variable. Edit: changed digit-wise to digit-based to make it clear that I'm not asking to multiply correspondin…

Woha This is cool. It literally is the same, if you use base-infinity digits (not binary, or decimal, but infinity-ary?). How do you multiply two polynomials with a single variable?

Ref reply below - https://news.ycombinator.com/item?id=25195046

Re: Convolution Is Fancy Multiplication

#68

Post by signaru »

working with image processing makes convolution easily intuitive. you can get a "feel" of what the math is doing by looking what's happening with the images. while blurring is a common example (convolve a Gaussian function with the image), edge detection and other image operations are just a matter of changing the convolved signal (or kernel). in a simplified model, blurry photos due to poor lenses also share the same principle (the lens cannot form a perfect point, that imperfect point is effectively what is convolved). the same can be said for audio and other 1d data, but images are more visual.

Re: Convolution Is Fancy Multiplication

#70

Post by parksy »

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…

Definitely agree - I struggled beyond basic integration in High School, and my "why" and "what is it" type questions about calculus were met with annoyed "read the book" replies and young me lost interest in higher maths at a time when my mind was more plastic.

As an adult having these personalities enthusiastically breaking down the fundamentals and using analogies to help cement the concepts intuitively has been a taste of what I imagine having a brilliant and engaging math professor would be like.

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