###### tags: `one-offs` `convexity`
# Log-Convexity of the $C^\alpha$ Semi-Norm
**Overview**: In this note, I give a quick proof of a nice convexity result.
## From Moment-Generating Functions to Uniform Continuity Constants
A well-known fact in the context of exponential family theory (and / or statistical mechanics) is that for suitable functions $F$, the cumulant-generating function
\begin{align}
A:\theta\mapsto\log\left(\int\mu\left(\mathrm{d}x\right)\cdot\exp\left(\left\langle \theta,F\left(x\right)\right\rangle \right)\right)
\end{align}
is convex, modulo a few standard conditions. Less well-known (but perhaps indirectly related?) is that the logarithm of the $C^{\alpha}$ seminorm of a function is also convex! That is, fix a function $F$ and define
\begin{align}
H:\alpha\mapsto\sup\left\{ \frac{\mathsf{d}_{\mathcal{Y}}\left(F\left(x\right),F\left(x^{'}\right)\right)}{\mathsf{d}_{\mathcal{X}}\left(x,x^{'}\right)^{\alpha}}:x,x^{'}\in\mathcal{X},x\neq x^{'}\right\} .
\end{align}
It then holds that
\begin{align}
\log H\left(\alpha\right)=\sup\left\{ \log\mathsf{d}_{\mathcal{Y}}\left(F\left(x\right),F\left(x^{'}\right)\right)-\alpha\cdot\log\mathsf{d}_{\mathcal{X}}\left(x,x^{'}\right):x,x^{'}\in\mathcal{X},x\neq x^{'}\right\} ,
\end{align}
which is a supremum of functions which are affine in $\alpha$, and is hence convex in $\alpha$. I'm not sure about whether this has any serious applications, but it seems neat! It would also be rewarding to see whether there is a genuine connection to the result about cumulant-generating functions.

Overview: In this note, I log some basic observations about diffusion-based generative models.

8/14/2023Overview: In this note, I describe some aspects of hierarchical structure in MCMC algorithms, and how they can be of theoretical and practical relevance.

8/9/2023Overview: In this note, I discuss a recurrent question which can be used to generate research questions about methods of all sorts. I then discuss a specific instance of how this question has proved fruitful in the theory of optimisation algorithms. Methods and Approximations A nice story is that when Brad Efron derived the bootstrap, it was done in service of the question “What is the jackknife an approximation to?”. I can't help but agree that there's something quite exciting about research questions which have this same character of ''What is (this existing thing) an approximation to?''. One bonus tilt on this which I appreciate is that there can be multiple levels of approximation, and hence many answers to the same question. One well-known example is gradient descent, which can be viewed as an approximation to the proximal point method, which can then itself be viewed as an approximation to a gradient flow. There are probably even more stops along the way here. In this case, there is even the perspective that from the perspective of mathematical theory, there may be at least as much to be gained by stopping off at the proximal point interpretation, as there is from the gradient flow perspective. My experience is that generalist applied mathematicians get to grips with the gradient flow quickly, but optimisation theorists can squeeze more out of the PPM formulation. There is thus some hint that using this 'intermediate' approximation can be particularly insightful in its own right. It would be interesting to collect more examples with this character.

5/22/2023Overview: In this note, I prove Hoeffding's inequality from the perspectives of martingales and convex ordering. The Basic Construction Let $-\infty<a<x<b<\infty$, and define a random variable $M$ with law $M\left(x;a,b\right)$ by \begin{align} M=\begin{cases} a & \text{w.p. }\frac{b-x}{b-a}\ b & \text{w.p. }\frac{x-a}{b-a}. \end{cases}

5/22/2023
Published on ** HackMD**