Comment by jacobolus | Hacker News Reader

Comment by jacobolus | Hacker News Reader

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You can learn all of these ideas on your own (by e.g. going through textbooks and doing a significant proportion of the exercises), but the guidance of a course / expert is pretty helpful.

To understand computational complexity, take a course with a title like “theory of computation” or similar. https://en.wikipedia.org/wiki/Computational_complexity_theor...

To understand linear maps, tensor products, etc., take a course (or 2–3 courses) in linear algebra. To understand various matrix decompositions, take a course in numerical linear algebra. https://en.wikipedia.org/wiki/Tensor_product https://en.wikipedia.org/wiki/Cholesky_decomposition

To understand the FFT and convolutions, take a course in signal processing, maybe after a course in ordinary differential equations. https://en.wikipedia.org/wiki/Fast_Fourier_transform https://en.wikipedia.org/wiki/Convolution

To understand the theory of polynomial rings, take a course in abstract algebra. https://en.wikipedia.org/wiki/Polynomial_ring

To understand numerical approximations and error propagation, take a course in numerical analysis. https://en.wikipedia.org/wiki/Numerical_analysis

Courses in discrete math, algorithms, and complex analysis would also be helpful.

Thank you so much! I'll definitely be looking into courses in these subjects. I can learn a language on my own, but theory is something that I personally need someone to help walk me through.

Note that really understanding and grasping all of that can easily take some years, as a full-time student, when you are taking all these courses.

>but the guidance of a course / expert is pretty helpful.

From my peers I get the feeling that this is underappreciated in the tech world where many successful developers skipped college.

Well it’s two different things. This paper is, in fact, useless in the tech world, since the algorithm is not practical. The paper does, however, has value in advancing academic CS as an intellectual pursuit.

> From my peers I get the feeling that this is underappreciated in the tech world where many successful developers skipped college.

I agree that most developers probably underestimate the value of a guidance from a course/expert, and I feel that is largely do to the shear amount of resources available for learning programming.

I have found that as I get into more and more advanced topics the amount of resources available has decreased and has made me want guidance a lot more. Published papers really aren't a great way to learn advanced topics unless you are already an expert in that field.

Mathematics gets hit harder by this than computer science does. I think this is because it is incredibly interconnected and in each topic is very deep; there has been lots of time for the cutting edge to grow.

This is something I really appreciated about some of my well-taught graduate classes (of course, "well-taught" is a big caveat). In undergrad there's often a sense that the professor is supposed to teach you the material. In advanced topics, though, I often found it most useful if the professor actually skipped most of the details, and served as more of a tour guide of the landscape of material for me. Once I know that some specific topic can be learned from a specific tutorial, book chapter, etc., then I can just go learn it on my own. But what's hard is figuring out how to make your way through this sequence entirely DIY, i.e. knowing what to read in what order, whether you're on the right track, where to look if you get stuck, and having someone to answer big-picture questions about how things fit together, why something that seems obvious to me isn't true, etc., etc. I've found some of the best courses helped with that.

A good textbook can also serve this role, but I've found good textbooks rarer than well-taught graduate classes. A lot of textbooks don't really seem to be designed for self-teaching the topic by reading it sequentially cover-to-cover. Maybe because they're often intended to be used in courses, they often have too much material, presented in a somewhat haphazard way, with an expectation that a course instructor will pick and choose parts and supplement it with lectures.

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