iFRAME (inhomogeneous Filters Random Field And Maximum Entropy)

Experiment 7: Mixing of sampling

Code and dataset

Experiment description

We evaluate the mixing of the Gibbs sampling on wavelet coefficients by running parallel chains that sample from the fitted sparse FRAME model p(I; B, λ), where B={Bi, i=1,...,n} are the selected wavelets. These chains start from independent white noise images. Let be the image of chain m produced at iteration t, where each iteration is a sweep of the Gibbs sampler with random scan. Let rm,t,i =<, Bi> be the response of the synthesized image to Bi. Let Rt,i=( rm,t,i, m=1,...,) be the dimensional vector. Fix t = 100, let ρk,i be the correlation between vectors Rt,i and Rt+k,i Then ρk=∑ni=1 ρk,i/n measures the average auto-correlation of lag k, and is an indicator of how well the parallel chains are mixing. As a comparison, we also compute ρk for HMC sampler, where each iteration consists of 30 leapfrog steps. Figures plot ρk for k = 1,...,20, and for n = 10, 50, 100, 300, 500, and 700, where each plot corresponds a particular n. It seems that HMC can be easily trapped in the local modes, whereas the Gibbs sampler is less prone to local modes.

Case 1: Tiger

Case 2: Cat