# Useful Math

Published:

I have spent the last several months doing applied math, culminating in a submission of a paper to a robotics conference (although culminating might be the wrong word, since I’m still working on the project).

Unfortunately the review process is double-blind so I can’t talk about that specifically, but I’m more interested in going over the math I ended up using (not expositing on it, just making a list, more or less). This is meant to be a moderate amount of empirical evidence for which pieces of math are actually useful, and which aren’t (of course, the lack of appearance on this list doesn’t imply uselessness, but should be taken as Bayesian evidence against usefulness).

I’ll start with the stuff that I actually used in the paper, then stuff that helped me formulate the ideas in the paper, then stuff that I’ve used in other work that hasn’t yet come to fruition. These will be labelled I, II, and III below. Let me know if you think something should be in III that isn’t [in other words, you think there’s a piece of math that is useful but not listed here, preferably with the application you have in mind], or if you have better links to any of the topics below.

I. Ideas used directly

Differential equations: Lyapunov functions, linear differential equations, Poincaré return map, exponential stability, Ito calculus

Linear algebra: matrix exponential, trace, determinantCholesky decomposition, plus general matrix manipulation and familiarity with eigenvalues and quadratic forms

Multivariable calculus: partial derivative, full derivative, gradient, Hessian, Matrix calculus, Taylor expansion

Inequalities: Jensen’s inequality, testing critical points

Optimization: (non-convex) function minimization

III. Other useful ideas

Function Approximation: variational approximation, neural networks

Graph Theory: random walks and relation to Markov Chains, Perron-Frobenius Theorem, combinatorial linear algebra, graphical models (Bayesian networks, Markov random fields, factor graphs)

Miscellaneous: Kullback-Leibler divergence, Riccati equation, homogeneity / dimensional analysis, AM-GM, induced maps (in a general algebraic sense, not just the homotopy sense; unfortunately I have no good link for this one)

Probability: Bayes’ rule, Dirichlet process, Beta and Bernoulli processes, details balance and Markov Chain Monte Carlo

Spectral analysis: Fourier transform, windowing, aliasing, wavelets, Pontryagin duality

Optimization: quasiconvexity

Categories:

It seems to me that something from graph theory ought to be in here. For instance, random graph traversal (with PageRank as an application example). Perhaps I’m missing your intent, though, or missing how graph theory was already captured by one of the above categories.

### jsteinhardt

Good point. Graph theory is definitely relevant, e.g.:

-analyzing networks (e.g. random walks) -combinatorial linear algebra -graphical models

Am I missing something there?