I am reading a paper on Markov Chain Monte Carlo sampling via diffusion type models:
https://arxiv.org/abs/1503.03585
Herein, a claim is made (see section 2.1) - namely that we can convert any ("suitable") data distribution $\pi$ into a Gaussian (mean zero, identity covariance matrix) by repeated application of diffusion kernels. More specifically, the paper uses transitions $$x_t \sim N(\sqrt{(1-\beta_t)}x_{t-1}, \beta_t I).$$ and claims that repeated applications of such transitions result in a standard Gaussian distribution.
I have three questions:
- What is the specific continuous time diffusion process on which this result is based? I know, for example, that a linear SDE of the type $$\mathrm{d}X_t = -AX(t)\mathrm{d}t + \sigma \mathrm{d}W_t$$ results in a steady state distribution which is Gaussian. But I don't know whether this type of process is referred to by the author, nor do I know about how $\beta_t$ would relate to $A$ and $\sigma$
- Further, their transitions seem to come from a time-inhomogenous process (as $\beta_t$ may depend on $t$). Hence, in contrast to the candidate process I gave in $1)$ I might need a continuous, time-inhomogeneous diffusion process with a Gaussian as a steady state distribution. I don't know any results for limiting distributions in the time-inhomogenous case
- Assume that I know the answer to 1) or 2), that is, I got a continuous time diffusion process which underlies the claim of the authors. Then still I don't know about how to derive from the process the discretization-based sampling used by the authors.