swNoData

Gibbs sampling was originally designed by Geman and Geman (1984) for drawing updates from the Gibbs distribution, hence the name. However, single-site Gibbs sampling exhibits poor mixing due to the posterior correlation between the pixel labels. Thus it is very slow to converge when the correlation (controlled by the inverse temperature \(\beta\)) is high.

The algorithm of Swendsen and Wang (1987) addresses this problem by forming clusters of neighbouring pixels, then updating all of the labels within a cluster to the same value. When simulating from the prior, such as a Potts model without an external field, this algorithm is very efficient.

The SW function in the PottsUtils package is implemented in a combination of R and C. The swNoData function in bayesImageS is implemented using RcppArmadillo, which gives it a speed advantage. It is worth noting that the intention of bayesImageS is not to replace PottsUtils. Rather, an efficient Swendsen-Wang algorithm is used as a building block for implementations of ABC (Grelaud et al. 2009), path sampling (Gelman and Meng 1998), and the exchange algorithm (Murray, Ghahramani, and MacKay 2006). These other algorithms will be covered in future posts.

There are two things that we want to keep track of in this simulation study: the speed of the algorithm and the distribution of the summary statistic. We will be using system.time(..) to measure both CPU and elapsed (wall clock) time taken for the same number of iterations, for a range of inverse temperatures:

beta <- seq(0,2,by=0.1)
tmMx.PU <- tmMx.bIS <- matrix(nrow=length(beta),ncol=2)
rownames(tmMx.PU) <- rownames(tmMx.bIS) <- beta
colnames(tmMx.PU) <- colnames(tmMx.bIS) <- c("user","elapsed")

iter <- 600
burn <- 100
samp.PU <- samp.bIS <- matrix(nrow=length(beta),ncol=iter-burn)

The distribution of pixel labels can be summarised by the sufficient statistic of the Potts model:

where \(i \sim \ell \in \mathscr{N}\) are all of the pairs of neighbours in the lattice (ie. the cliques) and \(\delta(u,v)\) is 1 if \(u = v\) and 0 otherwise (the Kronecker delta function). swNoData returns this automatically, but with SW we will need to use the function sufficientStat to calculate the sufficient statistic for the labels.

library(bayesImageS)

mask <- matrix(1,50,50)
neigh <- getNeighbors(mask, c(2,2,0,0))
block <- getBlocks(mask, 2)
edges <- getEdges(mask, c(2,2,0,0))

n <- sum(mask)
k <- 2
bcrit <- log(1 + sqrt(k))
maxSS <- nrow(edges)

for (i in 1:length(beta)) {
  if (requireNamespace("PottsUtils", quietly = TRUE)) {
    tm <- system.time(result <- PottsUtils::SW(iter,n,k,edges,beta=beta[i]))
    tmMx.PU[i,"user"] <- tm["user.self"]
    tmMx.PU[i,"elapsed"] <- tm["elapsed"]
    res <- sufficientStat(result, neigh, block, k)
    samp.PU[i,] <- res$sum[(burn+1):iter]
    print(paste("PottsUtils::SW",beta[i],tm["elapsed"],median(samp.PU[i,])))
   } else {
      print("PottsUtils::SW unavailable on this platform.")
   }

  
  # bayesImageS
  tm <- system.time(result <- swNoData(beta[i],k,neigh,block,iter))
  tmMx.bIS[i,"user"] <- tm["user.self"]
  tmMx.bIS[i,"elapsed"] <- tm["elapsed"]
  samp.bIS[i,] <- result$sum[(burn+1):iter]
  print(paste("bayesImageS::swNoData",beta[i],tm["elapsed"],median(samp.bIS[i,])))
}

Here is the comparison of elapsed times between the two algorithms (in seconds):

boxplot(tmMx.PU[,"elapsed"],tmMx.bIS[,"elapsed"],ylab="seconds elapsed",names=c("SW","swNoData"))

On average, swNoData using RcppArmadillo (Eddelbuettel and Sanderson 2014) is seven times faster than SW.

s_z <- c(samp.PU,samp.bIS)
s_x <- rep(beta,times=iter-burn)
s_a <- rep(1:2,each=length(beta)*(iter-burn))
s.frame <- data.frame(s_z,c(s_x,s_x),s_a)
names(s.frame) <- c("stat","beta","alg")
s.frame$alg <- factor(s_a,labels=c("SW","swNoData"))
  if (requireNamespace("lattice", quietly = TRUE)) {
    lattice::xyplot(stat ~ beta | alg, data=s.frame)
  }

plot(c(s_x,s_x),s_z,pch=s_a,xlab=expression(beta),ylab=expression(S(z)))
abline(v=bcrit,col="red")

The overlap between the two distributions is almost complete, although it is a bit tricky to verify that statistically. The relationship between \(beta\) and \(S(z)\) is nonlinear and heteroskedastic.

s.frame$beta <- factor(c(s_x,s_x))
  if (requireNamespace("PottsUtils", quietly = TRUE)) {
    s.fit <- aov(stat ~ alg + beta, data=s.frame)
    summary(s.fit)
    TukeyHSD(s.fit,which="alg")
  }

References

Eddelbuettel, Dirk, and Conrad Sanderson. 2014. “RcppArmadillo: Accelerating R with High-Performance C++ Linear Algebra.” Comput. Stat. Data Anal. 71: 1054–63. https://doi.org/10.1016/j.csda.2013.02.005.

Gelman, Andrew, and Xiao-Li Meng. 1998. “Simulating Normalizing Constants: From Importance Sampling to Bridge Sampling to Path Sampling.” Statist. Sci. 13 (2): 163–85. https://doi.org/10.1214/ss/1028905934.

Geman, Stuart, and Donald Geman. 1984. “Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images.” IEEE Trans. PAMI 6: 721–41.

Grelaud, Aude, Christian P. Robert, Jean-Michel Marin, François Rodolphe, and Jean-François Taly. 2009. “ABC Likelihood-Free Methods for Model Choice in Gibbs Random Fields.” Bayesian Analysis 4 (2): 317–36. https://doi.org/10.1214/09-BA412.

Murray, Iain, Zoubin Ghahramani, and David J. C. MacKay. 2006. “MCMC for Doubly-Intractable Distributions.” In Proc. \(22^{nd}\) Conf. UAI, 359–66. Arlington, VA: AUAI Press.

Swendsen, Robert H., and Jian-Sheng Wang. 1987. “Nonuniversal Critical Dynamics in Monte Carlo Simulations.” Phys. Rev. Lett. 58: 86–88. https://doi.org/10.1103/PhysRevLett.58.86.

swNoData

Matt Moores

2021-04-11

References