Tutorials

Technical tutorials I’ve written on topics related to my research interests.

Symmetry, invariance, and the kernel trick

When do symmetric optimization problems have symmetric solutions? We often can identify invariant subspaces containing optimal solutions, then restricting the problem domain to these subspaces. Finally I show that the kernel “trick” can be derived by applying a restriction to an invariant subspace.

Diffusion models from scratch

Diffusion models have been very successful in text-to-image generation, and generally excel in sampling from multimodal distributions. This tutorial aims to give a gentle introduction to diffusion models, with a running example to illustrate how to build, train and sample from a simple diffusion model from scratch.

Sum-of-squares optimization

Sum of squares relaxation is a powerful paradigm for solving polynomial optimization problems. This tutorial covers the numerical and computational aspects of solving sum of squares problems with a number example problems. Code examples for solving these problems are provided using the respective software packages in the Macaulay, Matlab, Julia and Python languages.