Lie groups are an essential element of modelling in applications such as computer vision and computer graphis (namely handling cameras), and also robotics and machine learning.

They do not constitute an easy subject. A lot of presentation is either overly technical, or glosses over mathematical foundations or properties which enables practitioners to carry out computations on Lie groups by themselves, or understand often unsaid assumptions or conventions in many implementations.

In this blog post, we will be taking a tour of Lie groups, Lie algebras, and useful computational facts.

Last time we had an introductory look at OT, the definition and computation of the Wasserstein distance.

A few years ago, I wrote a solver for this variational formulation of mean-field games, as part of a project. This eventually led to a small C++ library as a toy project, which might or might not be usable.

There's been a lot of noise about large language models (LLMs) and them possibly replacing programmers in the near future -- a perspective shared by many technology enthusiasts but met with a tad more skepticism from researchers and software engineers. OpenAI's release of ChatGPT has generated lots of discussion on the subject. Although complete automation might be far off, some folks have already found some use for Copilot, automating things like rote boilerplate code. However, that does not necessarily translate to correct, let alone efficient, code especially in complex applications and specific domains. An obvious caveat: different language models lead to different results, and the randomness in the inference process can create a lot of variability -- some friends have told me OpenAI playground does generate more reliable code.

Consider once again an optimal control problem (OCP) in continuous time: $$ \begin{equation} \begin{alignedat}{2} &\min_{x, u}{}&& \int_0^T \ell(x(t), u(t)) dt + h(x(T)) \\ &\suchthat&& \dot{x} = f(x, u), \quad t \in [0, T) \end{alignedat} \end{equation} $$

In general nonlinear optimization, we seek to solve problems of the form $$ \begin{equation}\tag{NLP} \begin{aligned} \min_z{} &\ell(z) \ \suchthat &c(z) = 0 \end{aligned} \end{equation} $$ This kind of formulation is often encountered in machine learning, control problems and others where structured, constrained optimization crops up.

The Differential Dynamic Programming (DDP) is a nowadays quite popular optimal control algorithm, which was introduced by Jacobson all the way back in the 1970s. Recently, it has made a resurgence following work by Tassa and Todorov.