R/vbgmm.R

In keyuan/ccube: ccube

# Variational Bayesian Gaussian mixture model (VB-GMM)
# X: N x D data matrix
# init: k (1 x 1) or label (1 x n, 1<=label(i)<=k) or center (d x k)
# Reference: Pattern Recognition and Machine Learning by Christopher M. Bishop (P.474)
# Part of the implementation is based on the Matlab code vbgm.m from Michael Chen and R code from Yue Li
# Matlab code: http://www.mathworks.com/matlabcentral/fileexchange/35362-variational-bayesian-inference-for-gaussian-mixture-model



############ Sort the model  ############
# Sort model paramters in increasing order of averaged means
# of d variables
sort_components_gmm <- function(model) {

	idx <- order(apply(model$m, 2, mean))

	model$m <- model$m[, idx]

	model$R <- model$R[, idx]

	model$logR <- model$logR[, idx]

	model$alpha <- model$alpha[idx]

	model$kappa <- model$kappa[idx]

	model$v <- model$v[idx]

	model$M <- model$M[,,idx]

	model
}


#' Variational Bayesian Gaussian mixture model (VB-SMM)
#' @param data N x D data matrix
#' @param init k (1 x 1) or label (1 x n, 1<=label(i)<=k) or center (d x k)
#' @param prior prior parameters
#' @param tol VBEM convergence threshold
#' @param maxiter VBEM maximum iteration
#' @param verbose show progress
#' @return a list containing model parameters
#' @export
vbgmm <- function(data, init=2, prior, tol=1e-20, maxiter=1e3, verbose=FALSE) {

	data <- as.matrix(data)

	n <- nrow(data)
	d <- ncol(data)

	X <- t(data) # Work with D by N for convenience

	message(sprintf("Running VB-GMM on a %d-by-%d data ...\n", n, d))

	if(missing(prior)) {

		if(length(init) == 1) {

			 # more general prior with equal alpha

		  k <-  init
		  ranged <- range(data)
			prior <- list(
				alpha = rep(1e-3,k),
				kappa = 1,
				m = t(as.matrix(seq(ranged[1], ranged[2], length.out = k))),
				v = d,
				M = var(data)/k # M = inv(W)
			)
		}
	} else {

		stopifnot(
			all(names(prior) %in% c("alpha","kappa","m","v","M")) &
			all(sapply(prior, is.numeric)) & nrow(prior$m) == d &
			ncol(prior$m) == 1 &
			nrow(prior$M) == d & ncol(prior$M) == d)
	}

	# lower variational bound (objective function)
	L <- rep(-Inf, maxiter)
	converged <- FALSE
	t <- 1

	model <- list()

	model$R <- initialization_gmm(X, init) # initialize responsibility R

	while(!converged & t < maxiter) {

		t <- t + 1
		model <- vmax(X, model, prior)
		model <- vexp(X, model)
		L[t] <- vbound(X, model, prior)/n
		converged <- abs(L[t] - L[t-1]) < tol * abs(L[t])

		if(verbose) message(sprintf("VB-EM-%d: L = %.6f", t, L[t]))
	}

	L <- L[2:t]

	model <- sort_components_gmm(model)

	if (init == 1) {
	  label <- rep(1, n)
	  model$Epi <- 1
	} else {
	  label <- rep(0, n)
	  label <- apply(model$R, 1, which.max)
	  nk <- apply(model$R, 2, sum)
	  Epi <- (model$alpha + nk) / (k*prior$alpha + n)
	  model$Epi <- Epi/sum(Epi)

	}


	if(converged) message(sprintf("Converged in %d steps.\n", t-1)) else
	warnings(sprintf("Not converged in %d steps.\n", maxiter))

	list(label=label, R=model$R, mu=model$m, full.model=model, L=L)
}


############ Initialization of responsibility (intialization) ############
initialization_gmm <- function(X, init) {

	d <- nrow(X)

	n <- ncol(X)

	stopifnot(length(init) %in% c(1, n) ||
		(nrow(init) == d  & ncol(init) == k))

	if(length(init) == 1) { # init = k gaussian components

		k <- init

		res <- kmeans(t(X), k)
		label <- res$cluster
		normalMean <- t(res$centers)

		R <- as.matrix(Matrix::sparseMatrix(1:n, label, x=1))

	} else {

		if(length(init) == n) { # initialize with labels

			label <- init
			k <- max(label)
			R <- as.matrix(Matrix::sparseMatrix(1:n, label, x=1))

		} else {

			if(!is.null(dim(init))) {

				if(nrow(init) == d  & ncol(init) == k) { # initialize with centers

					k <- ncol(init)
					m <- init
					label <- apply(bsxfun.se("-", crossprod(m, X),
						as.matrix(dot.ext(m,m,1)/2)), 2, which.max)
					R <- as.matrix(Matrix::sparseMatrix(1:n, label, x=1))

				} else stop(message("Invalid init."))
			}
		}
	}

	R
}


############ Variational-Maximimization ############
vmax <- function(X, model, prior) {

	alpha0 <- prior$alpha
	kappa0 <- prior$kappa
	m0 <- prior$m;
	v0 <- prior$v;
	M0 <- prior$M;
	R <- model$R;

	nk <- apply(R, 2, sum) # 10.51
	alpha <- alpha0 + nk # 10.58
	nxbar <- X %*% R
	kappa <- kappa0 + nk # 10.60
	m <- bsxfun.se("*", bsxfun.se("+", nxbar, kappa0 * m0), 1/kappa) # 10.61
	v <- v0 + nk # 10.63 (NB: no 1 in the matlab code)

	d <- nrow(m)
	k <- ncol(m)

	M <- array(0, c(d, d, k))
	sqrtR <- sqrt(R)

	xbar <- bsxfun.se("*", nxbar, 1/nk) # 10.52
	xbarm0 <- bsxfun.se("-", xbar, m0)
	w <- (kappa0 * nk) * (1/(kappa0 + nk))

	for(i in 1:k) {

		Xs <- bsxfun.se("*", bsxfun.se("-", X, xbar[,i]), t(sqrtR[,i]))

		xbarm0i <- xbarm0[,i]

		# 10.62
		M[,,i] <- M0 + Matrix::tcrossprod(Xs, Xs) + w[i] * Matrix::tcrossprod(xbarm0i, xbarm0i)
	}

	model$alpha <- alpha
	model$kappa <- kappa
	model$m <- m
	model$v <- v
	model$M <- M # Whishart: M = inv(W)


	model
}


############ Variational-Expectation ############
vexp <- function(X, model) {

	alpha <- model$alpha	# Dirichlet
	kappa <- model$kappa	# Gaussian
	m <- model$m			# Gasusian
	v <- model$v			# Whishart
	M <- model$M			# Whishart: inv(W) = V'*V

	n <- ncol(X)
	d <- nrow(m)
	k <- ncol(m)

	logW <- array(0, dim=c(1,k))
	EQ <- array(0, dim=c(n,k))

	for(i in 1:k) {

		U <- chol(M[,,i])

		logW[i] <- -2 * sum(log(diag(U)))

		Q <- solve(t(U), bsxfun.se("-", X, m[,i]))

		EQ[,i] <- d/kappa[i] + v[i] * dot.ext(Q,Q,1)	# 10.64
	}

	vd <- bsxfun.se("-", matrix(rep(v+1, d),nrow=d,byrow=T), as.matrix(1:d))/2

	ElogLambda <- colSums(digamma(vd)) + d*log(2) + logW	# 10.65
	Elogpi <- digamma(alpha) - digamma(sum(alpha))			# 10.66
	logRho <- (bsxfun.se("-", EQ, 2*Elogpi + ElogLambda - d*log(2*pi)))/(-2) # 10.46

	# ke: add bound to avoid numerical issue
	mapper <- logRho < -700
	logRho <- mapper * -700 + (!mapper) * logRho

	logR <- bsxfun.se("-", logRho, logsumexp(logRho, 1))	# 10.49
	R <- exp(logR)

	model$logR <- logR
	model$R <- R



	model
}



############ Variational-(lower)-Bound Evaluation ############
vbound <- function(X, model, prior) {

	alpha0 <- prior$alpha
	kappa0 <- prior$kappa
	m0 <- prior$m
	v0 <- prior$v
	M0 <- prior$M

	alpha <- model$alpha	# Dirichlet
	kappa <- model$kappa	# Gaussian
	m <- model$m			# Gasusian
	v <- model$v			# Whishart
	M <- model$M			# Whishart: inv(W) = V'*V
	R <- model$R
	logR <- model$logR

	d <- nrow(m)
	k <- ncol(m)
	nk <- colSums(R)									# 10.51

	Elogpi <- digamma(alpha) - digamma(sum(alpha))		# 10.66

	Epz = pracma::dot(nk, Elogpi)								# 10.72
	Eqz = pracma::dot(as.numeric(R), as.numeric(logR))			# 10.75
	# logCalpha0 = lgamma(k * alpha0) - k * lgamma(alpha0) # for scalar alpha0
	logCalpha0 = lgamma(sum(alpha0)) - sum(lgamma(alpha0))
	# Eppi <- logCalpha0+(alpha0-1)*sum(Elogpi) # for scalar alpha0
	Eppi <- logCalpha0+pracma::dot(alpha0-1, Elogpi)			# 10.73
	logCalpha <- lgamma(sum(alpha))-sum(lgamma(alpha))
	Eqpi = pracma::dot(alpha-1, Elogpi) + logCalpha				# 10.76

	# part of 10.70
	L <- Epz - Eqz + Eppi - Eqpi

	U0 <- chol(M0)
	sqrtR <- sqrt(R)
	xbar <- bsxfun.se("*", X %*% R, 1/nk)				# 10.52

	logW <- array(0, dim = c(1, k))
	trSW <- array(0, dim = c(1, k))
	trM0W <- array(0, dim = c(1, k))
	xbarmWxbarm <- array(0, dim = c(1, k))
	mm0Wmm0 <- array(0, dim = c(1, k))

	for(i in 1:k) {

		U <- chol(M[,,i])
		logW[i] <- -2 * sum(log(diag(U)))

		Xs <- bsxfun.se("*", bsxfun.se("-", X, as.matrix(xbar[,i,drop=F])), t(sqrtR[,i,drop=F]))
		V <- chol(tcrossprod(Xs, Xs)/nk[i])
		Q <- solve(U, V)
		# equivalent to tr(SW)=trace(S/M)
		trSW[i] <- pracma::dot(as.numeric(Q), as.numeric(Q))
		Q <- solve(U, U0)
		trM0W[i] <- pracma::dot(as.numeric(Q), as.numeric(Q))

		q <- solve(t(U), xbar[,i,drop=F]-m[,i,drop=F])
		xbarmWxbarm[i] = pracma::dot(q, q)
		#q <- solve(t(U), m[,i,drop=F]-m0)
		q <- solve(t(U), m[,i,drop=F]-m0[,i,drop=F]) #ke: allow multivariate prior
		mm0Wmm0[i] <- pracma::dot(q, q)
	}

	vd <- bsxfun.se("-", matrix(rep(v+1, d),nrow=d,byrow=T), as.matrix(1:d))/2
	ElogLambda <- colSums(digamma(vd)) + d*log(2) + logW	# 10.65

	# first half of 10.74
	Epmu <- sum(d*log(kappa0/(2*pi))+ElogLambda-d*kappa0/kappa-kappa0*(v*mm0Wmm0))/2
	logB0 <- v0*sum(log(diag(U0)))-0.5*v0*d*log(2)-logmvgamma(0.5*v0,d) # B.79
	# second half of 10.74
	EpLambda <- k*logB0+0.5*(v0-d-1)*sum(ElogLambda)-0.5*pracma::dot(v,trM0W)


	Eqmu <- 0.5*sum(ElogLambda+d*log(kappa/(2*pi)))-0.5*d*k	# 10.77 (1/2)
	logB <- (-v) * (logW+d*log(2))/2 - logmvgamma(0.5*v, d)	# B.79
	EqLambda <- 0.5*sum((v-d-1)*ElogLambda-v*d)+sum(logB) 	# 10.77 (2/2)

	EpX <- 0.5*pracma::dot(nk, ElogLambda-d/kappa-v*trSW-v*xbarmWxbarm-d*log(2*pi))	# 10.71

	L <- L+Epmu-Eqmu+EpLambda-EqLambda+EpX	# 10.70

	L
}