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Graphical Linear Algebra

graphicallinearalgebra.net

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Re: Graphical Linear Algebra

#71

Post by edflsafoiewq »

tel wrote: post #70
edflsafoiewq wrote: post #8

The usual notation for a linear transformation X' = aX + bY Y' = cX + dY minimizes irrelevant information. But the diagram for one seems to bring it to the fore: a diagram is basically a system of fully parenthesized, unsimplified expressions eg X' = (3X + (2(X+4Y) - 6X)) + Y ... In fact, it contains even more information than that, since it also tells you if, in (X+Y)+(X+Y), you are supposed to compute (X+Y) once an…

Depends on what you mean by relevant information. GLA exposes the many compositional natures of linear transformations. The fact that there is a compact matrix representation is a wonderful treat.

Can you be more concrete?

Re: Graphical Linear Algebra

#73

Post by da39a3ee »

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…

> 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,

That's pretty arrogant and rude. This is some mathematicians' PhDs and research, not a blog put in front of overpaid programmers to provide them with category-theory porn. Admittedly, the dumbed down language of the blog is misleading and makes it very hard to read.

https://dl.acm.org/doi/pdf/10.1145/3290338

https://www.sciencedirect.com/science/article/pii/S089054011...

Fabio Zanasi. 2015. Interacting Hopf Algebras: the theory of linear systems. Ph.D. Dissertation. Ecole Normale Supérieure de Lyon

Re: Graphical Linear Algebra

#74

Post by to11mtm »

dmch-1 wrote: post #34
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…

These types of notations have been around for a long time. In fact they fist started to study geometric objects (such as knots and braids) with algebraic tools. The graphical notation does the opposite - represents algebraic objects by geometric ones. These may have some benefit for human eyes, but for machines they are quite useless indeed. Incidentally, I partially agree with you on category theory, especially on c…

> In abstract maths, however, category theory has more significance than pure language. It can serve more or less like an algebraic tool. There are abstract mathematicians who see deeper meaning in it too.

One of the things that has made me very successful as a problem solver and software developer has been the ability to apply category theory abstractly to projects and work that comes in. IOW, Using category theory well in day-to-day software engineering isn't about which monad to pick, but which abstraction (or series of abstractions) are the best fit to a problem.

Re: Graphical Linear Algebra

#75

Post by da39a3ee »

_f9cu wrote: post #15
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 really wish this would stop > Getting excited about category theory is like getting excited about matrix notation – useful, sure, but not where the meat is Patience. These kind of work is not worth for verbal readers, but for visual learners and visual creators this is very educative. It's a work of visual learner and these kind of work will not stop. > replace standard linear algebra notation with an alternative…

Good visual explanations are useful to all learners. I'm sorry, but the concept of "visual learner" when used in contradistinction to "verbal learner" is just "lacks attention span to concentrate on something hard".

Re: Graphical Linear Algebra

#76

Post by guerrilla »

da39a3ee wrote: post #72

I would like to read this but I really need a version without the verbiage and humour. I'm really not convinced it's necessary to try to make these things so "accessible".

Then read the paper that it's based on which is linked to by the blog and my other comments.

Re: Graphical Linear Algebra

#77

Post by guerrilla »

da39a3ee wrote: post #75
_f9cu wrote: post #15

Earlier quoted context omitted.

> I really wish this would stop > Getting excited about category theory is like getting excited about matrix notation – useful, sure, but not where the meat is Patience. These kind of work is not worth for verbal readers, but for visual learners and visual creators this is very educative. It's a work of visual learner and these kind of work will not stop. > replace standard linear algebra notation with an alternative…

Good visual explanations are useful to all learners. I'm sorry, but the concept of "visual learner" when used in contradistinction to "verbal learner" is just "lacks attention span to concentrate on something hard".

God forbid we show any kindness to people with attention deficits.

Re: Graphical Linear Algebra

#78

Post by Hercuros »

stablenode wrote: post #36
graphlinalg wrote: post #23

Earlier quoted context omitted.

It's interesting to suddenly get all these hits.. I haven't touched the blog in a long time. I'm sorry that you didn't find it well-written -- it wasn't written with you in mind. Originally I wanted to write about my research in a way that was understandable to a lay person, but I quickly abandoned that and went for the mythical "second year undergrad" level. You have pretty strong thoughts about what is "useful math…

"Are you the high priest and decider of the usefulness of mathematics? To be honest, it almost sounds like some category theorist was super mean to you..." Interesting and totally not ad hominem response... I believe Kevin Buzzard (an actual mathematician) had a few words about this last year: https://youtu.be/Dp-mQ3HxgDE?t=1039

Note that Kevin Buzzard is talking in the context of convincing "mainstream" mathematicians to use a proof assistant. He specifically clarifies somewhere (not sure where in this video, but I've seen other videos where he clarifies it) that when he talks about "proper mathematics" he is doing so in kind of a tongue-in-cheek, British way and does not mean to suggest that other kinds of mathematics are not proper (don't remember the exact wording).

In the context of that video, his comments about category theorists and type theorists make a lot of sense: type theory people and (perhaps to a more limited extent) category theory people tend to be more easily sold on using a proof assistant since the kind of mathematics they do translate more easily into current proof assistants than, say, analysis or topology.

I definitely do not think that Kevin Buzzard is suggesting that type theorists and category theorists are not doing interesting and/or useful work (after all, the proof assistant he is advocating for is based on type theory). At best, he is making a sociological observation that there is a gap between the type theory/category theory community and "mainstream" mathematicians making the widespread adoption of proof assistants in mathematics more difficult.

Re: Graphical Linear Algebra

#79

Post by hansvm »

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…

With the caveat that I learned this material the old-fashioned way and probably can't adequately put myself in the shoes of a new learner, this feels overly pessimistic. A few general points come to mind:

(1) The human mind is pretty good at visual pattern recognition. For the same reason that people advocate for a big status monitor that every developer can see for your live service (since you can quickly spot anomalies your monitoring might miss) and for the same reason visualizing a data set is one of the first things you ought to do on your way to understanding it, creating a diagrammatic representation of linear algebra seems helpful for immediately getting a sense of where something feels "off" or for quickly grasping the intuition of a problem.

(2) The right notation (pictures in this case -- not that the idea I'm about to highlight couldn't be presented textually) can help highlight the portions of a problem that matter. Thinking about matrices as boxes of elements that you can individually manipulate is IMO actively harmful to understanding them and puts you in a situation where you miss the forest for the trees.

(2a) The matrix wiring diagram presented here naturally extends to higher order tensors, and for the life of me I can't find the source right now, but it was precisely that representation that made tensors click for me and gave me a foothold into other material about them.

(3) I think the current consensus is that multiple representations of a problem (via some mechanism -- call it "magic") are actively beneficial for learning a topic. From that point of view alone a diagrammatic representation doesn't seem especially bad.

Re: Graphical Linear Algebra

#80

Post by tel »

edflsafoiewq wrote: post #71
tel wrote: post #70

Earlier quoted context omitted.

Depends on what you mean by relevant information. GLA exposes the many compositional natures of linear transformations. The fact that there is a compact matrix representation is a wonderful treat.

Can you be more concrete?

A lot of category theory is just about composition, how different structures compose and how compositions of those structures continue to be composable.

So in this case, GLA builds the theory of linear transformations from trivial/simple pieces and their various ways of being composed. It also discusses the mechanisms for proof and reduction (you noted how GLA is clear about whether a sub-computation is reused or not, it'll also be clear about how those two choices are equivalent).

So it's really not studying "just" linear transformation (which are, in finite cases, summarized by a matrix of numbers/field elements) but also the theory of their construction, manipulation, simplification. It gives you a rich language for talking about how two linear combinations might be related to one another, something that's more challenging to access from a matrix.

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