The optimal constant in Hölder-Rogers-type uniform continuity assumption satisfies an interesting condition with respect to its index parameter, with some superficial connections to exponential family theory. In this post, I briefly describe this result and its proof.
'Log-Convexity of the \( C^\alpha \) Semi-Norm'
https://hackmd.io/@sp-monte-carlo/S1IAwHtHn
Certain methods and concepts become clearest in the context of other ideas. One interesting version of this is when some new idea can be viewed as an approximation to some other thing, which was perhaps overlooked or viewed as out-of-reach. This post briefly discusses some aspects of this phenomenon.
'The “Approximation to What?” Principle'
https://hackmd.io/@sp-monte-carlo/SkWUiBFHn
A condensation of a few perspectives on the rising topic of diffusion models, focusing less on my usual areas of comfort (i.e. Markov processes and friends) and more on the other practical core, (i.e. denoising operators).
'Denoising-Centric Diffusions'
https://hackmd.io/@sp-monte-carlo/SJWAZBtr2
Various sampling algorithms can (implicitly or explicitly) involve some recursive structure, which enables certain tools for their theoretical analysis. Here is a small write-up of some basic thoughts on the topic.
https://hackmd.io/@sp-monte-carlo/BkHYt7tBn
'Nested Structure in MCMC Algorithms'
A small note on a reinterpretation / strengthening of a classical probabilistic inequality.
'Hoeffding’s Inequality by Convex Ordering'
A proper write-up of the ramblings above, for easier reference: (I can't remember how much, if at all, I edited things)
'Logarithmic Sobolev Inequalities and Logarithmic Lipschitz Regularity'
Adding some details to an earlier post (https://mathstodon.xyz/@sp_monte_carlo/109542712262310989): deriving and describing some near-Gaussian distributions which are relatively { explicit, tractable } in some (mildly) useful ways:
'Isoperimetric Surrogates for the Gaussian'
Some calculations from a train ride: I was interested in cooking up some probability measures in 1D which qualitatively resemble the Gaussian measure (in terms of e.g. concentration, isoperimetry, etc.), but are a bit more { tractable / tangible / etc. } in terms of analytic PDF, CDF, etc.
Taking stock of the past week (with apologies for the dense formatting).
'#BayesComp2023 Recap'
Earlier this year, I learned one of my current favourite 'concise results', i.e. results where you can concisely communicate the substance of the result and why it is true. In this post, I offer a snappy presentation to this effect.
'Univariate Log-Concave Rejection Sampling is Solved'
One of the few things which all Markov chains have in common with one another is the use of a time variable. As such, one can probe the structure of general Markov chains by examining how they look both forwards and backwards in time. This post collects some very brief thoughts on the matter.
'Forwards, Backwards, and Stochastic Formulations of Optimal Transport'
Sam Power
