Notes

Assorted notes that I have written.
Author

Yuxi Liu

Over the years I have written some notes. Too short for publication, and not exactly fit for blog posting, but still might be useful to someone, so I collect them here.

Expository notes

Visually Deriving the Wigner Rotation by Spacetime Diagrams. Obsoleted by my post. This is a term paper I wrote in 2018 for Theoretical Physics. The date in the pdf is the recompilation date, not the time when I actually wrote the paper.

Hyperbolic dynamical systems, chaos, and Smale’s horseshoe: a guided tour. This is the companion paper to a presentation I gave for the course Introduction to Ergodic theory in 2019.

An Overview of Information Geometry. This is the term paper for Advanced Differential Geometry. and since I really did not like lectures, I asked to do something to substitute for the mandatory attendance. I was asked to write a term paper, which produced this. Information geometry is a weird thing. The premise is beautiful, but the books are terribly confusing, and what little I have managed to understand seems disappointing. It feels like the whole field is overpromising and underdelivering.

Handout for honours seminar talk on AIXI. A presentation handout for AIXI. For my undergraduate thesis, I was going to be advised by Marcus Hutter, but he left ANU just before the start of semester, and I had to scramble for another advisor. Still, I found AIXI worth knowing, so for my mandatory short talk, I gave a presentation. I managed to compress the essentials to two pages, perfect for handing out on double-sided printed sheets.

Literature review

Literature Review of In-Context Learning of Simple Function Classes with source code available. A literature review on 223 papers that cited [2208.01066] What Can Transformers Learn In-Context? A Case Study of Simple Function Classes as of 2024-04-18, categorized into 5 classes. There is an appendix on how to do literature review with the help of large language models. I wrote this for a research group that was extending this paper. I don’t know what has come of it, nor have I heard from the group again. I tried uploading it to arXiv, but it was rejected for not being sufficiently academic.

Undergraduate thesis

Beyond expectations, but within limits – the theory of coherent risk measures.

For a quick summary, see my seminar presentation.

My undergraduate thesis written in 2019 at ANU, on the topic of coherent risk measures. The first chapter is a readable introduction to risk measures in general (as in, why we might need to use more than the mean and the variance). The rest of it is very dry and I imagine it is of only interest to specialists. The centerpiece of the thesis is a straightforward proof of the central limit theorem for CVaR, which is a slight generalization of expectation. Like the central limit theorem, this theorem states that the sample CVaR converges to the true CVaR like

\[ \frac{\text{sample CVaR}_\alpha - \text{true CVaR}_\alpha}{\sqrt N} \xrightarrow{d} \mathcal N(0, \sigma^2(\alpha)) \]

where \(\sigma^2(\alpha)\) has a certain expression. As soon as I have calculated it myself, thinking that I had finally made a new discovery, I found it published before in the literature. Still my expression is simpler than the previous publications, so I believe it is still worth something after all.

Corrections

When I was not yet mathematically mature, I used to study textbooks carefully, checking every letter through a brain-filter. I no longer do this, but while I was doing this, I created some erratas. Perhaps those will be of use to some people.

It is a rather odd thing that errata are hard to share. I would have thought there ought to be some kind of Wikipedia for errata, where people just post errata for textbooks. The lack of such an Error-pedia seems to require an economic explanation, as it can just use MediaWiki, the same technology powering Wikipedia.