I think you're right that the inelegant part is how AI seems to just consist of endless loops of multiplication. I say this as a graphics programmer who realized years ago that all those beautiful images were just lots of MxNs, and AI takes this to a whole new level. When I was in college they told us most of computing resources were used doing Linear Programming. I wonder when that crossed over to graphics or AI (or some networking operation like SSL)?

What could any complex phenomenon possibly be other than small “mundane” components combined together in a variety of ways and in immense quantities?

All such things are like this.

For me, this is fascinating, mind-boggling, non-sensical, and unsurprising, all at once.

But I wouldn’t call it inelegant.

> When I was in college they told us most of computing resources were used doing Linear Programming.

I seriously doubt that was ever true, except perhaps for a very brief time in the 1950s or 60s.

Linear programming is an incredibly niche application of computing used so infrequently that I've never seen it utilised anywhere despite being a consultant that has visited hundreds of varied customers including big business.

It's like Wolfram Mathematica. I learned to use it in University, I became proficient at it, and I've used it about once a decade "in industry" because most jobs are targeted at the median worker. The median worker is practically speaking innumerate, unable to read a graph, understand a curve fit, or if they do, their knowledge won't extend to confidence intervals or non-linear fits such as log-log graphs.

Teachers that are exposed to the same curriculum year after year, seeing the same topic over and over assume that industry must be the same as their lived experience. I've lost count of the number of papers I've seen about Voronoi diagrams or Delaunay triangulations, neither of which I've ever seen applied anywhere outside of a tertiary education setting. I mean, seriously, who uses this stuff!?

In the networking course in my computer science degree I had to use matrix exponentiation to calculate the maximum throughput of an arbitrary network topology. If I were to even suggest something like this at any customer, even those spending millions on their core network infrastructure, I would be either laughed at openly, or their staff would gape at me in wide-eyed horror and back away slowly.

The first two results from Google with "Voronoi astro" gave two different uses than the one I knew about (sampling fibre bundles): https://galaxyproject.org/news/2025-06-11-voronoi-astronomy/ https://arxiv.org/abs/2511.14697

Astronomy is pure research and is performed almost exclusively by academics.

I’m not saying these things have zero utility, it’s just that they’re used far less frequently in industry than academics imagine.

And astronomy tends to throw up technology that becomes widely used (WiFi being the obvious example) or becomes of "interest" to governments. I expect that AMR code will be used/ported to nuclear simulations if it proves to be useful. Do I expect it to be used in a CRUD app? Obviously not, but use by most software shops isn't a measure of importance.

3d modelers would like to have a word with you.

Part of the reason why linear programming does need t get used as often is that there are no industry standard software implementation that is not atrociously priced. Same deal with Mathematica.

3D modelling is mostly linear algebra, not linear programming, which is an entirely different set of algorithms.

Oh I mentioned it in the context of mesh geometries, tesselations.

I have not only used linear programming in the industry, I have also had to write my own solver because the existing ones (even commercial) were to slow. (This was possible only because I only cares about a very approximate solution)

The triangulations you mention are important in the current group I'm working in.

I'm curious to hear what you specifically use these algorithms for!

PS: My point is not that these things are never used, they clearly are, I'm saying that the majority of CPU cycles globally goes towards "idle", then pushing pixels around with simple bitblt-like algorithms for 2D graphics, then whatever it is that browsers do on the inside, then operating system internals, and then specialised and more interesting algorithms like Linear Programming are a vanishingly small slice of whatever is left of that pie chart.

If the O(n^3) schoolbook multiplication were the best that could be done, then I'd totally agree that "it's simply the nature of matrices to have a bulky multiplication process". Yet there's a whole series of algorithms (from the Strassen algorithm onward) that use ever-more-clever ways to recursively batch things up and decrease the asymptotic complexity, most of which aren't remotely practical. And for all I know, it could go on forever down to O(n^(2+ε)). Overall, I hate not being able to get a straight answer for "how hard is it, really".

For anyone interested, there is a introductory survey of the current lower bound at: https://en.wikipedia.org/wiki/Computational_complexity_of_ma...

> sends the pair (x, y) to the pair (−x, y)

I know linear algebra, but this part seems profoundly unclear. What does "send" mean? Following with different examples in 2 by 2 notation only makes it worse. It seems like you're changing referents constantly.

Let me try.

In US schools during K-12, we generally learn functions in two ways:

1. 2-d line chart with an x-axis and y-axis, like classic temperature over time, history of stock price, etc. Classic independent variable is on the horizontal axis, dependent variable is on the vertical axis. And even people who forgotten almost all math can instantly understand the graphics displayed when they're watching CNBC or a TV weather report.

2. We also think of functions like little machines that do things for us. E.g., y = f(x) means that f() is like a black box. We give the black box input 'x'; then the black box f() returns output 'y'. (Obviously very relevant to the life of programmers.)

But one of 3blue-1brown's excellent videos finally showed me at least a few more ways of thinking of functions. This is where a function acts as a map from what "thing" to another thing (technical from a Domain X to a Co-Domain Y).

So if we think of NVIDIA stock price over time (Interpretation 1) as a graph, it's not just a picture that goes up and to the right. It's mapping each point in time on the x-axis to a price on the y-axis, sure! But we can see it as mapping let's say x=November 21, 2025 to y=$178/share. Of course, interpretation 2 might say that the black box of the function takes in "November 21, 2025" as input and returns "$178" as output.

But what what I call Interpretation 3 does is that it maps from the domain of Time to the output Co-domain of NVDA Stock Price.

This is a 1D to 1D mapping. aka, both x and y are scalar values. In the language that jamespropp used, we send the value "November 21, 2025" to the value "$178".

But we need not restrict ourselves to a 1-dimensional input domain (time) and a 1-dimensional output domain (price).

We could map from a 2-d Domain X to another 2-d Co-Domain Y. For example X could be 2-d geographical coordinates. And Y could be 2-d wind vector.

So we would feed input of say location (5,4) as input. and our 2Dto2D function would output wind vector (North by 2mph, East by 7mph).

Thanks for pointing this out. I’ll work on this passage tomorrow.

I’ve updated this passage. Let me know if the new version is clearer.

> that, for all x and y between −1 and 1, changes the sign of x

Perfect! Overall, it's much better (though I really just meant that "send" was vague, which "change" improves)!

Maybe the problem is that matrices are too general.

You can have very beautiful algorithms when you assume the matrices involved have a certain structure. You can even have that A*B == B*A, if A and B have a certain structure.

Ignore me then because I agree with you. :) He sounds like someone who upon first hearing jazz to complain it was ugly.

The inelegance to me isn't in the definition of the operation, but that it's doing a huge amount of brute-force work to mix every part of the input with every other part, when the answer really only depends on a tiny fraction of the input. If we somehow "just knew" what parts to look at, we could get the answer much more efficiently.

Of course that doesn't really make any sense at the matrix level. And (from what I understand) techniques like MoE move in that direction. So the criticism doesn't really make sense anymore, except in that brains are still much much more efficient than LLMs so we know that we could do better.

I think 4x4 matrices for 3D transforms (esp homogenous coordinates) are very elegant. I think the intended critique is that the huge n*m matrices used in ML are not elegant - but the point is made poorly by pointing out properties of general matrices. In ML matrices are just "data", or "weights". There are no interesting properties to these matrices. In a way a Neumann (https://en.wikipedia.org/wiki/Von_Neumann%27s_elephant) Elephant. Now, this might just be what it is needed for ML to work and deal with messy real world data! But mathematically it is not elegant.

I think it conflates the map and the territory.

Linear transformations are a beautiful thing, but matrices are an ugly representation that nevertheless is a convenient one when we actually want to compute.

Elegant territory. Inelegant, brute-force, number crunching map.