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I major math and cs. and i'm interested in ai/machine learning/data mining.

so i want to know what math subjects are used in frontier of these technology.

especially, high mathematical tool, like topology,abstract algebra and geometry, can be used computer science?

and what interests are suitable for me to write master degeree?

thank you for reading

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Even though this is not really a research question, because my work is a mix of machine learning and applied math, I thought it may be worthwhile mentioning a few useful connections here --- ostensibly, the same advice applies to almost any other discipline that makes use of mathematics!

  1. All the math that you need for doing statistics will come in handy in one way or the other for machine learning.

  2. Linear algebra --- without fail, the most commonly needed subject within machine learning; some critical parts are closer to numerical linear algebra such as eigenvector decompositions, singular vector decompositions, sparse linear systems, etc.

  3. Functional analysis --- no surprise here! Harmonic analysis, kernel functions, representation theorems

  4. Probability theory --- concentration inequalities, stochastic processes, etc.

  5. Information theory

  6. Convex analysis -- mostly to build the fundamentals for doing optimization, as most of the heavy algorithmic work that happens in ML is based on optimization.

  7. Combinatorics -- a lot of data are represented as graphs, spectral graph theory is very useful, but more and more combinatorial methods and combinatorial mathematical models are of great interest (e.g., submodularity)

  8. Economics / Game theory

  9. Algebraic geometry -- a little bit so far, but mostly the "convex algebraic geometry" and "sum of squares" part of the game

  10. Differential geometry -- again, mostly for models and work related to optimization over special manifolds.

  11. Topology --- most notably, check out the subfield "topological data analysis"

  12. Metric geometry -- a fundamental part of ML is the idea of "similarity" or "dissimilarity", so studying metric spaces, their interrelations, embeddings, etc. can be quite interesting.

  13. and many more, ...

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  • $\begingroup$ This answer is a work in progress; I'll improve it with references to books and papers to make it more useful, as soon as I get a chance $\endgroup$
    – Suvrit
    Commented Jul 22, 2014 at 16:47
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    $\begingroup$ Maybe it is worth mentioning the statistical manifolds stuff of Amari, also called information geometry. Viewing families of probability distributions as Riemannian manifolds has helped clarify some statistical concepts. I specifically remember papers analyzing the EM-type algorithms from the information-geometry point of view. $\endgroup$
    – Matthias Wendt
    Commented Jul 22, 2014 at 18:12
  • $\begingroup$ Regarding 1, I'd say that measure theory would mainly be a distraction in a ML context. In general, mathematical topics that lend themselves to computer implementation are most helpful. For example, in terms of functional analysis, an existence proof that some function can be approximated in some basis is much more useful if it is constructive. This amplifies Suvrit's comment that numerical linear algebra is often more useful than abstract linear algebra. $\endgroup$
    – R Hahn
    Commented Jul 22, 2014 at 18:55
  • $\begingroup$ @RHahn: actually you'd be surprised; knowledge of measure theory is useful in ML research; see e.g., arxiv.org/abs/1202.6504, arxiv.org/abs/0907.5309, amongst many others. $\endgroup$
    – Suvrit
    Commented Jul 22, 2014 at 19:19

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