Homogenius

Nothing pisses me off more than seeing a post I'm really interested in, only for it to disappear a second later because Twitter randomly updates its feed. Then it's lost to the ether, replaced by nonsense I don't care about. Is the algorithm just rage bait at this point?

Geometry of Data and Biology ams.org/notices/201510…

In flow matching, a coupling determines how noise and data samples are paired during training. The choice of coupling is important because it influences the geometry of trajectories at inference time. The simplest choice is the independent coupling, where noise and data points are paired arbitrarily. This can lead to curved trajectories as the model averages over many conflicting pairings. However, if we use optimal transport on batches of pairs, this leads to fewer ambiguous intersections that the model must resolve, leading to straighter trajectories at inference time.

SIAM Conference on Mathematics of Data Science (MDS26) hotel information is posted. Reserve now! siam.org/conferences-ev… #SIAMMDS26

This video, created by my dear coauthor @MahdiKahou for our teaching and papers, shows how overparameterized neural networks produce smooth function approximations even in the context of the Runge phenomenon. Some background. Imagine you want to approximate the Runge function en.wikipedia.org/wiki/Runge%27s… using polynomial interpolation at equally spaced points. It is well known that, despite targeting an infinitely differentiable function, such a polynomial approximation produces oscillatory behavior that worsens with the degree of the polynomial. In other words, higher-degree polynomial approximations might not improve accuracy. Instead, approximate the Runge function with a neural network (here, two layers are just to make the example concrete; nothing fundamental depends on it). As you increase the number of parameters well above the 11 training points (in our example, a two-layer neural network with 128 nodes each), you nicely converge to the target, without wild oscillations. Yes, this has much to do with double descent and benign overparameterization, but the main punchline of this post is that neural networks are really very different types of animals than polynomial approximations. And yes, Chebyshev nodes and splines exist, and in this case, they will prevent the oscillations. But that's not the point. Chebyshev nodes and splines still confront Faber’s theorem, which states that for any system of polynomial interpolation nodes, there exists a continuous function whose sequence of interpolating polynomials diverges as the number of nodes grows to infinity. Faber’s theorem does not apply to neural networks because they are not polynomials. The notebook, if you want to check the details, is here: github.com/Mekahou/Notes/… Stay tuned for more on this 👀

@Simmy16242170 I like to call this a plate.

This isnt even correct as a dumb math meme

@saqshum Most are? Atleast for certain hours. But I promise the majority of university libraries are not a significantly better place to work than a coffee shop

universities should open their libraries up to the public. im so sick of working from overpriced cafes with bad music

@Anthony_Bonato You sonofabitch

"What's an easy way to offend a mathematician?" Whiteboards are superior to chalkboards

@Quasilocal Who on earth wants to drink their coffee off of a plate.

take that, topologists

@Quasilocal It’s pretty ambitious, but u could try to add the proof that there exists irrational numbers x and y such that x^y is rational & using the \sqrt{2}^\sqrt{2} trick. I love this one, and if people couldn’t fallow the irrationality of \sqrt{2} problem, they get a related cute one.

Tomorrow I'm going to show some 7th graders how to prove the irrationality of \sqrt{2} -- is this too ambitious?

@quinnswm You don’t see the usefulness of taking an abstract topic like sheafs and giving an informal intro to build some intuition before someone formally tackles it? The author very clearly sets expectations for what the articles goals are. I think it’s silly to say there is NO utility.

With all due respect to people who write this stuff, I find it very difficult to behave this type of pedagogy is in any way useful. Also the definition of sheaf is not that difficult, just go learn the definition!

@miniapeur @andrewgwils Math is a great field but to be marketable to employment you need to either want to be an actuary, or get at the very least a masters. That being said, data science is an exploading field rn particularly topological data analysis and geometric machine learning which needs math!

@andrewgwils What about mathematics?

"Does it still make sense to get a CS degree?" A CS degree has never been primarily about software engineering. It's about core skills, about learning how to think. That never becomes obsolete. But really you should get a physics degree.

@pi0us0 We even have our own wizards!

A friend of me who is studying math said: "It's like creating a magic system in a fantasy world. Only there is no story and no characters and no world, just the magic-system."

@Creative_Math_ @SLCTDMBNTWRKS yeah sorry, I meant Banach spaces in particular, so yeah R^n or at the very least an isomorphic copy of it, not necessarily explicitly the euclidian norm, but an equivalent norm none the less.

@SLCTDMBNTWRKS @AlexSampson95 I think you mean only for R^n In metric spaces you actually need totally bounded + complete, for eg R with the discrete metric is a closed and bounded metric space, but not compact

I never really got over the fact that the rationals, despite being countable, are dense in the reals--an uncountable set

@nadienadianadie One benefit about studying math — there aren’t nearly as many armchair mathematians as there are armchair physicists.

you would be surprised by the number of people who try to talk about quantum physics at a bookstore-cofffeeshop.

@heterophobeV2 @Creative_Math_ Topology is very cool though don’t worry! I find it amazing that topology, as vague as it is, is enough to define the notion of a limit, so w/ very little extra structure, you can actually define derivatives on something as abstract as a (locally convex) topological vector space!

@AlexSampson95 @Creative_Math_ hearing stuff like this has me so excited/scared to learn topology

@RaminNasibov I think it boils down to two paths. (1) AI actively dramatically reduces human workforce and we institute some sort of universal basic income. (2) regulation comes through to keep corporations from massively replacing work forces with AI agents.

If Al will take away all the jobs, who will buy the products that Al will produce?

@VazeKshitij The lab I learned it in was the reason I switched from physics to math 😂. I had completed all the courses but explicitly avoided the labs until the end and they drove me crazy.

@AlexSampson95 i remember the schematics for both via op-amps!

So, not to trigger some buried JEE trauma, but have any of you ever wondered - how can integration and differentiation be done with software?