Maths to Share

Monte Carlo integration in high dimensions

Plain Monte Carlo often fails in high dimensional problems such as estimating the volume of a high dimensional ball. Importance sampling is a powerful variance reduction tool.

by michaelnhowes 7th Aug 20247th Aug 2024

Uniformly sampling orthogonal matrices

An $n \times n$ matrix $M \in \mathbb{R}^{n \times n}$ is orthogonal if $M^\top M = I$ . The set of all $n \times n$ orthogonal matrices is a compact group written as $O_n$ . The uniform distribution on $O_n$ is called Haar measure. There are many ways to generate random…

by michaelnhowes 9th Mar 2024

Auxiliary variables sampler

The auxiliary variables sampler is a general Markov chain Monte Carlo (MCMC) technique for sampling from probability distributions with unknown normalizing constants [1, Section 3.1]. Specifically, suppose we have $n$ functions $f_i : \mathcal{X} \to (0,\infty)$ and we want to sample from the probability distribution $P(x) \propto \prod_{i=1}^n f_i(x).$ That is $latex…

by michaelnhowes 9th Feb 2024

Fibonacci series

In “Probabilizing Fibonacci Numbers” Persi Diaconis recalls asking Ron Graham to help him make his undergraduate number theory class more fun. Persi asks about the chance a Fibonacci number is even and whether infinitely many Fibonacci numbers are prime. After answering “1/3” and “no one knows” to Persi’s questions, Ron suggests giving the undergrads the…

by michaelnhowes 17th Nov 2023

The fast Fourier transform as a matrix factorization

The discrete Fourier transform (DFT) is a mapping from $\mathbb{C}^n$ to $\mathbb{C}^n$ . Given a vector $u \in \mathbb{C}^n$ , the discrete Fourier transform of $u$ is another vector $U \in \mathbb{C}^n$ given by, $\displaystyle{U_k = \sum_{j=0}^{n-1} \exp\left(-2 \pi i kj/m\right)u_j}.$ If we let $\omega_n = \exp\left(-2 \pi i /m\right)$ ,…

by michaelnhowes 11th Aug 2023

Doing Calvin’s homework

Growing up, my siblings and I would read a lot of Bill Watterson’s Calvin and Hobbes. I have fond memories of spending hours reading and re-reading our grandparents collection during the school holidays. The comic strip is witty, heartwarming and beautifully drawn. Recently, I came across this strip online Seeing Calvin take this test took…

by michaelnhowes 6th May 20236th Aug 2023

Bernoulli’s inequality and probability

Suppose we have independent events $E_1,E_2,\ldots,E_m$ , each of which occur with probability $1-\varepsilon$ . The event that all of the $E_i$ occur is $E = \bigcap_{i=1}^m E_i$ . By using independence we can calculate the probability of $E$ , $P(E) = P\left(\bigcap_{i=1}^m E_i\right) = \prod_{i=1}^m P(E_i) = (1-\varepsilon)^m.$ We could also get…

by michaelnhowes 27th Apr 202327th Apr 2023

Solving a system of equations vs inverting a matrix

I used to have trouble understanding why inverting an $n \times n$ matrix required the same order of operations as solving an $n \times n$ system of linear equations. Specifically, if $A$ is an $n\times n$ invertible matrix and $b$ is a length $n$ vector, then computing $A^{-1}$ …

by michaelnhowes 16th Apr 2023

Braids and the Yang-Baxter equation

I recently gave a talk on the Yang-Baxter equation. The focus of the talk was to state the connection between the Yang-Baxter equation and the braid relation. This connection comes from a system of interacting particles. In this post, I’ll go over part of my talk. You can access the full set of notes here.…

by michaelnhowes 12th Mar 2023

Concavity of the squared sum of square roots

The $p$ -norm of a vector $x\in \mathbb{R}^n$ is defined to be: $\displaystyle{\Vert x \Vert_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}}.$ If $p \ge 1$ , then the $p$ -norm is convex. When $0<p<1$ , this function is not convex and actually concave when all the entries of $x$ are non-negative. On a recent exam…

by michaelnhowes 13th Feb 202325th Feb 2023

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