- Proof. Also logic, axiom, finitary proof.
- Infinity. Also limit, continuity, calculus, analysis.
- Set. Also number, function, more abstract objects.
- Symmetry. Also group, geometry, isomorphism.
- Probability. Also statistics, sampling, conditionals.
- Convexity. Also linearization, optimization.
That is what I used to think, but now I have concluded that convexity is at the core of most real-world applications of math.
In particular, convexity is crucial to the math of artifical intelligence AI.
Another crucial math idea in AI is:
The manifold hypothesis posits that many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space.[1][2][3][4] As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, linked to the local coordinate system of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features.Another is the various scaling and power laws:
In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to the change raised to a constant exponent: one quantity varies as a power of another. The change is independent of the initial size of those quantities.With AI becoming more important, I expect the math of AI to also become more important.
See, see, see, Roger,
ReplyDeleteWhat happens when none connects with you at your blog, hell, let alone me but also the regular commentators you've managed to gather, by now? Hmmm?
Y'all are mathematician, BTW. Rich. [In Berkeley, Princeton, et al. and et cetera.]
Oh, by the by, I did like this post by you about... ahem... maths.
I mean, it's good to know how even maths people imagine what they are doing in their waking hours, so that they could tell the rest of us as to the question:
[reproduced from a decades old memory, an ``apology,'' IIRC]:
Quote:
``What do you do when you go to office?''
Unquote.
---
BTW, no, convexity wasn't a surprise. Not to me, anyway. But no influence, whatsoever of some Berkeley PhD returning to India via TIFR Pune and all.
A mild curiosity, OTOH, was the conspicuous absence of of SDIC. Regimes, if you will. (Or, nonlinear couplings.) But anyway, expected. They all retired, after their best attempts in the mid-1970's through the mid-1980's. Then, realizing (rather like in the JPBTI manner) that they were Berkeley graduates after all, they left the vein alone. Regardless of richness. In that California Gold Rush. Enforced, these days, by either party in your nation / country / whatever.
Anyway,
Best,
--Ajit