> To play devil's advocate, probabilistic reasoning (probability theory, stochastic processes, Bayesian reasoning, graphical models, variational inference) might be equally if not more important.
For intuition, particularly if you care about vision applications, I think one field of math which is severely underrated by the community is group theory. Trying to understand methods which largely proceed by divining structure without first trying to understand symmetry has to be a challenge.
I'm biased; my training was as a mineralogist and crystallographer! But the serious point here is that much of the value of math is as a source of intuition and useful metaphor. Facility with notation is pretty secondary.
Can you talk about the use of group theory for computer vision or crystallography a bit? I'm familiar with the math but I'm not familiar with group theory's applications in those areas. That sounds pretty interesting. Is it primarily group theory, or does it so venture into abstract algebra more generally?
For crystallography, the use of group theory in part originates in X-ray crystallography [1], where the goal is to take 2D projections of a repeating 3D structure (crystal), and use that along with other rules that you know to re-infer what the 3D structure is.
Repeating structures have symmetries, so seeing the symmetries in your diffraction pattern inform you of the possible symmetries (and hence possible arrangements) in your crystal. Group theory is the study of symmetry.
By the way, this is also how the structure of DNA was inferred [2], although not from a crystal.
> use that along with other rules that you know to re-infer what the 3D structure is
Great answer, thank you :-) Saved me a bunch of typing to explain it less well than you just did.
It's worth adding, for this crowd, that another way of thinking about the "other rules" you allude to is as a system of constraints; you can then set this up as an optimization problem (find the set of atomic positions minimizing reconstruction error under the set of symmetry constraints implied by the space group – so that means that solving crystal structures and machine learning are functionally isomorphic problems.
I thought the work on the structure of DNA used Fourier analysis more than group theory.
I know harmonic analysis in general combines the two, but I'm sure Crick and Watson could have done their work without knowing the definition of a group.
And by Crick and Watson you mean Crick, Watson, Franklin and Wilkins, right? It's fairly clear all four deserve at least partial authorship by modern standards. James Watson was a piece of work.
Crick was absolutely certainly familiar with the crystallographic space groups; he was the student of Lawrence Bragg (https://en.wikipedia.org/wiki/Lawrence_Bragg), who is the youngest ever Nobel laureate in physics – winning it with his father for more or less inventing X-ray crystallography. It's mostly 19th-century mathematics, after all.
For intuition, particularly if you care about vision applications, I think one field of math which is severely underrated by the community is group theory. Trying to understand methods which largely proceed by divining structure without first trying to understand symmetry has to be a challenge.
I'm biased; my training was as a mineralogist and crystallographer! But the serious point here is that much of the value of math is as a source of intuition and useful metaphor. Facility with notation is pretty secondary.