Notes

Short expository pieces, written to be checked. Where a claim depends on a theorem the theorem is named, and where a formula has a numerical value I compute the value instead of quoting it.

Effective diffusivity in a periodic potential

How fast does a Langevin sampler actually explore a rough landscape? In one periodic dimension the answer is exact, 1/I₀(β)², which makes it a benchmark a sampler either reproduces or fails.

What the Metropolis correction buys, and what it does not

The unadjusted Langevin algorithm is SGLD without the minibatch noise. On a Gaussian target its stationary law is exact, so the bias can be measured against a formula, and a metastable target then shows that removing the bias does nothing for the harder problem.

Grading a diffusion model when you know the answer

A Gaussian mixture is closed under the variance-preserving forward kernel, so the score is exact and the sampler can be marked on mode coverage rather than judged on whether its output looks plausible.