Homogenius

Math PhD Student FSU
Infinite Dimensional Riemannian Geometry
Montana — Vegas — Tallahassee

[Homogenius @AlexSampson95]

Where do yall primarily keep your reading notes/lecture notes. Do you type or handwrite? I have been doing a lot of stuff in obsidian the last year and it’s been very convenient. But I’d also like to hear other people’s systems.

[Homogenius @AlexSampson95]

@LimYenKheng https://t.co/ooHm3y6VYK

This gives a decent style guide on how to hand write fraktur.

[Nicole Feng @nicolefeng_]

If you’re interested in generalized signed distance functions but didn’t make it to our SIGGRAPH talk, I’ve recorded my presentation:

https://t.co/CVxLjNMCzk

Look for official code releases in the coming weeks!

[Geometry Collective @GeomCollective]

Announcing the Geometry Processing Worldwide Discord Server: https://t.co/Y98TGXk03K, hosted by the CMU @GeomCollective!

Anyone and everyone is invited to come discuss all things related to geometry, computation, + more! #GPWW

[Keenan Crane @keenanisalive]

Giving a talk in the @Stanford SCIEN seminar this Wednesday (1/3) at 4:30pm:

https://t.co/EgDLBwv2YU

The topic is “normal coordinates”: a shape representation little-used outside of mathematics—but which turns out to have nice applications in geometry processing & learning.

[Homogenius @AlexSampson95]

@RedCollie1 As soon as I see something with the word “vortex” thrown in, my quack alarms get primed and ready.

[Homogenius @AlexSampson95]

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?

[Steven Strogatz @stevenstrogatz]

Geometry of Data and Biology https://t.co/l0n0oS1bIW

[Alec Helbling @alec_helbling]

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 @TheSIAMNews]

SIAM Conference on Mathematics of Data Science (MDS26) hotel information is posted. Reserve now! https://t.co/yszAMFkix8
#SIAMMDS26

[Jesús Fernández-Villaverde @JesusFerna7026]

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

https://t.co/oKwNLE4PBH

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:

https://t.co/vwQjaYLbre

Stay tuned for more on this 👀

[Homogenius @AlexSampson95]

@Simmy16242170 I like to call this a plate.

[Homogenius @AlexSampson95]

@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

[Homogenius @AlexSampson95]

@Anthony_Bonato You sonofabitch

[Homogenius @AlexSampson95]

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

[Homogenius @AlexSampson95]

@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.

[Homogenius @AlexSampson95]

@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.

[Homogenius @AlexSampson95]

@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!

[Homogenius @AlexSampson95]

@pi0us0 We even have our own wizards!

[Homogenius @AlexSampson95]

@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.