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Graphical Linear Algebra

graphicallinearalgebra.net

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Re: Graphical Linear Algebra

#51

Post by chobytes »

spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

Yeah, I have to agree. It's frustrating because CT and these kinds of diagrams can actually be kind of interesting... They are just not nearly as grand or as useful as their proponents would have us believe.

(My spicier opinion is that the machinery of CT is often completely unnecessary for many applications, and actually only serves to obfuscate simple concepts.)

Re: Graphical Linear Algebra

#52

Post by a9h74j »

uglycoyote wrote: post #47
buddhiajuke wrote: post #31

Earlier quoted context omitted.

You’re not wrong. OTOH the other day I heard someone in the Netflix doc about Alberto Nisman’s murder that described a terrorist network as a “matrix”. I grumped as card-carrying pedants are duty-bound to, but then realized that if there’s a 1:1 correspondence between graphs and (adjacency, incidence) matrices, then there’s very little loss of meaning in referring to graphs as “matrices”. Maybe in some communication…

That's funny, I would have just chalked it up to "they are using a different sense of the word matrix", knowing that most dictionaries contain multiple definitions of the same word. But nice work making this mental leap, I have never thought about it! But it's true, you can envision it as a NxN matrix of N terrorists, where the number in cell i,j represent's the strength of the relationship (it's a weighted graph!),…

A few months ago I was imagining having a square matrix representation for fractions/attenuation of viral loads (e.g. per week) being transferred among a large population. I pictured a sum of isolated groups, after a sort, causing this larger matrix to be composed of a set of on-diagonal square submatrices.

Re: Graphical Linear Algebra

#53

Post by 909832 »

wenc wrote: post #45
spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

I often go back to this post by John D Cook Category theory can be very useful, but you don’t use it the same way you use other kinds of math. You can apply optimization theory, for example, by noticing that a problem has a certain form, and therefore a certain algorithm will converge to a solution. Applications of category theory are usually more subtle. You’re not likely to quote some theorem from category theory t…

Here are some other applications. Look at: http://zxcalculus.com/ & https://www.youtube.com/watch?v=iC-KVdB8pf0

Re: Graphical Linear Algebra

#56

Post by 909832 »

chobytes wrote: post #51
spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

Yeah, I have to agree. It's frustrating because CT and these kinds of diagrams can actually be kind of interesting... They are just not nearly as grand or as useful as their proponents would have us believe. (My spicier opinion is that the machinery of CT is often completely unnecessary for many applications, and actually only serves to obfuscate simple concepts.)

ZX-calculus is a rigorous graphical language that has applications in quantum circuits.

Re: Graphical Linear Algebra

#58

Post by uglycoyote »

a9h74j wrote: post #52
uglycoyote wrote: post #47

Earlier quoted context omitted.

That's funny, I would have just chalked it up to "they are using a different sense of the word matrix", knowing that most dictionaries contain multiple definitions of the same word. But nice work making this mental leap, I have never thought about it! But it's true, you can envision it as a NxN matrix of N terrorists, where the number in cell i,j represent's the strength of the relationship (it's a weighted graph!),…

A few months ago I was imagining having a square matrix representation for fractions/attenuation of viral loads (e.g. per week) being transferred among a large population. I pictured a sum of isolated groups, after a sort, causing this larger matrix to be composed of a set of on-diagonal square submatrices.

Yes I would think that you are dealing with sparse matrices with little clusters of nonzero values in a model like this.

One could say that I'm "part of a terrorist matrix", and that my value in the matrix is zero. We are all part of a terrorist matrix. I hope the CIA is not reading this, because they probably are not mathy enough to know what I'm talking about, and they may just take my quoted statement at face value, but in a mathematical sense we probably all actually have some nonzero value in our rows and columns of the terrorist matrix. You know someone, who knows someone, who knows someone, who is a terrorist. It really just depends how many people and how many degrees of separation you want to include in your matrix.

Re: Graphical Linear Algebra

#59

Post by theaeolist »

enriquto wrote: post #19
spekcular wrote: post #5

The short version appears to be that the author has managed to replace standard linear algebra notation with an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. This submission is the latest in a trend of what I would classify as "math-lite" category theory and Haskell articles that reach the HN front page, which purport to explain something int…

> an alternative (but mathematically equivalent) graphical notation, which is harder to do computations and write programs in. But this does need to be the case. Sometimes a graphical notation is a great help. For example, many tensor computations in differential geometry become straightforward using Penrose's graphical notation. Is that the case here? It is difficult to say, as the authors fail to present that notat…

Yes, it is the case here as well. Reducing terms can be tricky because of distributive laws. The graph-like representation makes reduction algorithmically easier.

Re: Graphical Linear Algebra

#60

Post by theaeolist »

impendia wrote: post #32
spekcular wrote: post #17

Earlier quoted context omitted.

I don't believe the exposition is written well (judged as either an exposition of linear algebra or just the graphical calculus the article develops). But this is perhaps a subjective point, and others here have already commented on this in detail. More importantly, I would like to remark that string diagrams are not standard. The vast majority of mathematicians have never read the definition of a string diagram (or…

> When I read articles by category theory boosters, I get the sense (rightly or wrongly) that they think the world revolves around them and they have stumbled onto some deep and fundamental truths. (Professional mathematician here) My sense, from talking to category theory boosters, is not typically that they regard category theory as "deep and fundamental truths". Rather, they often find it a useful way to declutter…

Indeed. A high-level programming language (e.g. Python) does not allow you to do _more_ than assembly but _less_. But it helps you declutter the presentation to such a great extent that realistically you could not write Python program in assembly.
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