tl;dr: Being a homomorphism from a multiplicative structure into an additive structure isn't enough to grant it the logarithm title.
Although logarithms are certainly ubiquitous in mathematics, I don't think that the mappings that the article's author identifies as logarithms are appropriately viewed as such.
I can't endorse viewing dimension as a logarithm. It appears superficially logarithm-like because we typically (and somewhat unfortunately) write the direct sum of n copies of a vector space V as V^n rather than nV. Writing nV, we simply get the dimension identity dim(nV) = n dim(V). Writing nV instead of V^n also conveniently frees up V^n for the tensor product of n copies of V, with corresponding dimension identity dim(V^n) = dim(V)^n. So I don't think there's any "multiplicative-to-additive" business going on here at all.
Also, I don't think it's advisable to view the p-adic valuation ord_p as a logarithm, even though it's a homomorphisms from the multiplicative group of the rational or p-adic field into the additive group of the rational field. In fact, in many number theoretic contexts, the ratio log_p/ord_p is of particular interest.
I think a good rule of thumb for viewing a mapping as some kind of logarithm is that it has to have some relation with the Taylor expansion of log(1 + x) around x=0. Being a homomorphism from a multiplicative structure into an additive structure isn't enough to get the logarithm title.
I think the OpenAI model that resolved the Unit Distance Problem would be capable of solving a significant proportion of mathematics PhD thesis problems.
> Now if I know anything about math for the sake of math, and academics, these are the same people that lament the idea of intelligent people going to the finance sector or any other trade they just happen not to respect as much
IME a vastly more common sentiment among mathematicians regarding mathematical talent leaving the nest to apply their skills in other fields is that those other fields are lucky to get them!
We're very fortunate to have had some very eminent mathematicians backfill the OpenAI proof with history, context, and a literature review [1]. Ideas behind the proof seem to have been "in the air". Indeed, looked at certain point of view, the OpenAI construction can be viewed as a high-dimensional generalization of a known low-dimensional one. In this vein see the remarks of Gowers, Sawin and Tsimerman in [1]. Are LLMs capable of "true leap[s] in understanding"? I have absolutely no idea. But LLMs keep surprising me.
To me, the most interesting feature of the OpenAI solution of the Unit Distance (Erdös) Problem is that the solution - using deep algebraic number theory as a source of extremal combinatorial/geometric constructions - is much more interesting than the problem’s elementary statement might lead one to expect.
Writing off Erdös’s problems as random, useless, or meaningless dismisses his mathematical intuition, second-to-none, and strikes me as somewhat uncharitable.
Finally, I agree that AI threatens mathematical training by rendering an entire class of acolyte-level research problems solvable by prompt. But the Unit Distance Problem is not of this class.
Not sure about these books as a self-study curriculum — their unifying theme seems to be that they require a reasonable level of mathematical maturity going in. But, they absolutely comprise an excellent “greatest hits” list of math books in the most influential subdisciplines. You’re guaranteed to learn a tonne if you study any one of these books.
I don't buy the narrative that the article is promoting.
I think the machine learning community was largely over overfitophobia by 2019 and people were routinely using overparametrized models capable of interpolating their training data while still generalizing well.
The Belkin et al. paper wasn't heresy. The authors were making a technical point - that certain theories of generalization are incompatible with this interpolation phenomenon.
The lottery ticket hypothesis paper's demonstration of the ubiquity of "winning tickets" - sparse parameter configurations that generalize - is striking, but these "winning tickets" aren't the solutions found by stochastic gradient descent (SGD) algorithms in practice. In the interpolating regime, the minima found by SGD are simple in a different sense perhaps more closely related to generalization. In the case of logistic regression, they are maximum margin classifiers; see https://arxiv.org/pdf/1710.10345.
The article points out some cool papers, but the narrative of plucky researchers bucking orthodoxy in 2019 doesn't track for me.
To the left of the "detailed spaceship" I think I see a distortion pattern reminiscent of a cloaked Klingon bird of prey moving to the right. Or I'm just hallucinating patterns in nebular noise.
Two schools of thought here. One posits that models need to have a strict "symbolic" representation of the world explicitly built in by their designers before they will be able to approach human levels of ability, adaptability and reliability. The other thinks that models approaching human levels of ability, adaptability, and reliability will constitute evidence for the emergence of strict "symbolic" representations.
TLDR: Browser vendors made Shadow DOM for themselves.
Browser implementors use Shadow DOM extensively under the hood for built-in HTML elements with internal structure like range inputs, audio and video controls, etc. These elements absolutely need to work everywhere and be consistent, so extreme encapsulation and fixed api for styling them is an absolute must.
The Shadow DOM API is the browsers exposing, to developers, a foundational piece of functionality.
If you’re thinking about whether Shadow DOM is appropriate for your use case, consider how/why the vendors use it —- when an element’s API needs to be totally locked down to guarantee it works in contexts they have no control over. Conversely, if your potential use case is scoped to a single project, the encapsulation imposed (necessarily!) by Shadow DOM is probably overkill.
Web components are a decent way to make reusable UI, but if they don’t have strong encapsulation needs, you might avoid Shadow DOM.
Could you give more details about what precisely you mean by interpolation and generalization? The commonplace use of “generalization” in the machine learning textbooks I’ve been studying is model performance (whatever metric is deemed relevant) on new data from the training distribution. In particular, it’s meaningful when you’re modeling p(y|x) and not the generative distribution p(x,y).
I’m reading a winking, ironic acknowledgement from the authors that the mathematical definition of individual utility may not map perfectly onto the psychology of a patron of the arts.
Are there any GPU emulators you can use to run simple CUDA programs on a commodity laptops, just to get comfortable with the mechanics, the toolchain, etc.?
As a university professor, I admit with some shame that accessibility issues for specific problem types is not on my radar. “Innovation” isn’t the main culprit here.
Fortunately, my university has a good accessibility center that takes care of accommodation issues (large print versions of tests, etc.). I just send them my tests and they take care of it. It’s a great service, and absolutely crucial because I simply don’t have the time to customize assessments. I assume they would get in touch if they were unable to retrofit accessibility onto an assessment, but that hasn’t happened in my fifteen years of employment.