- Introduction
- Rank 1
- Rank K
- Tree structured data
- Run multiple factors
- SVD initialization
ebpower
Matthew Stephens
2025-03-09
Last updated: 2025-03-16
Checks: 7 0
Knit directory: misc/analysis/
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# load the ebnm library and define an ebnm_binormal function
library("ebnm")
dbinormal = function (x,s,s0,lambda,log=TRUE){
pi0 = 0.5
pi1 = 0.5
s2 = s^2
s02 = s0^2
l0 = dnorm(x,0,sqrt(lambda^2 * s02 + s2),log=TRUE)
l1 = dnorm(x,lambda,sqrt(lambda^2 * s02 + s2),log=TRUE)
logsum = log(pi0*exp(l0) + pi1*exp(l1))
m = pmax(l0,l1)
logsum = m + log(pi0*exp(l0-m) + pi1*exp(l1-m))
if (log) return(sum(logsum))
else return(exp(sum(logsum)))
}
ebnm_binormal = function(x,s){
x = drop(x)
s0 = 0.01
lambda = optimize(function(lambda){-dbinormal(x,s,s0,lambda,log=TRUE)},
lower = 0, upper = max(x))$minimum
g = ashr::normalmix(pi=c(0.5,0.5), mean=c(0,lambda), sd=c(lambda * s0,lambda * s0))
postmean = ashr::postmean(g,ashr::set_data(x,s))
postsd = ashr::postsd(g,ashr::set_data(x,s))
return(list(g = g, posterior = data.frame(mean=postmean,sd=postsd)))
}Introduction
I want to implement an EB version of the power update for a symmetric data matrix S. I haven’t implemented an update to the residual error yet
Rank 1
# This is just the regular power update, used for initialization
power_update_r1 = function(S,v){
newv = drop(S %*% v)
if(!all(newv==0))
v = newv/sqrt(sum(newv^2))
return(v)
}
eb_power_update_r1 = function(S,v,ebnm_fn,sigma){
newv = drop(S %*% v)
if(!all(newv==0))
v = ebnm_fn(newv, sigma)$posterior$mean
if(!all(v==0))
v = v/sqrt(sum(v^2))
return(v)
}Simulate data under a tree model (with very small errors)
set.seed(1)
n = 40
x = cbind(c(rep(1,n),rep(0,n)), c(rep(0,n),rep(1,n)), c(rep(1,n/2),rep(0,3*n/2)), c(rep(0,n/2), rep(1,n/2), rep(0,n)), c(rep(0,n),rep(1,n/2),rep(0,n/2)), c(rep(0,3*n/2),rep(1,n/2)))
E = matrix(0.01*rexp(2*n*2*n),nrow=2*n)
E = E+t(E) #symmetric errors
S = x %*% diag(c(1,1,1,1,1,1)) %*% t(x) + E
image(S)
Check the first PC. It is essentially the constant vector.
pc1 = cbind(svd(S)$v[,1])
plot(-pc1,ylim=c(0,0.16))
If one uses a completely random initialization then the eb method will often (but not always) zero things out completely on the first iteration:
set.seed(1)
v = cbind(rnorm(2*n))
v= v/sqrt(sum(v^2))
sigma = sqrt(mean(S^2))
v = eb_power_update_r1(S,v,ebnm_point_laplace,sigma)
plot(v,ylim=c(-0.25,0.25))
We can help avoid this here by doing a single iteration of the power method before moving to EB (because the matrix is very low rank the single iteration moves us quickly to the right subspace):
set.seed(1)
v = cbind(rnorm(2*n))
v= v/sqrt(sum(v^2))
sigma = sqrt(mean(S^2))
v = power_update_r1(S,v)
plot(v,ylim=c(-0.25,0.25))
for(i in 1:100){
v = eb_power_update_r1(S,v,ebnm_point_laplace,sigma)
}
plot(v,ylim=c(-0.25,0.25))
However, if we use a different random init, it can find the first split of the tree (here a 0-1 split rather than +-1 split).
set.seed(4)
v = cbind(rnorm(2*n))
v= v/sqrt(sum(v^2))
sigma = sqrt(mean(S^2))
v = power_update_r1(S,v)
plot(v,ylim=c(-0.25,0.25))
for(i in 1:100){
v = eb_power_update_r1(S,v,ebnm_point_laplace,sigma)
}
plot(v,ylim=c(-0.25,0.25))
abline(h=0)
If we initialize to the first PC, it finds something close to the first PC (a bit more split).
v = -pc1
sigma = sqrt(mean(S^2))
for(i in 1:100){
v = eb_power_update_r1(S,v,ebnm_point_laplace,sigma)
}
plot(v, ylim=c(-0.25,0.25))
If we initialize to the (positive version of) the first PC, point-exponential performs similarly (we use the negative because the first PC here is negative).
v = -pc1
sigma = sqrt(mean(S^2))
for(i in 1:100){
v = eb_power_update_r1(S,v,ebnm_point_exponential,sigma)
}
plot(v, ylim=c(-0.25,0.25))
Try further running binormal - it levels things out.
for(i in 1:100){
v = eb_power_update_r1(S,v,ebnm_binormal,sigma)
}
plot(v, ylim=c(-0.25,0.25))
Rank K
# model is S \sim VDV' + E with eb prior on V
eb_power_update = function(S,v,d,ebnm_fn){
K = ncol(v)
sigma2=mean((S-v %*% diag(d,nrow=length(d)) %*% t(v))^2)
for(k in 1:K){
U = v[,-k,drop=FALSE]
D = diag(d[-k],nrow=length(d[-k]))
newv = (S %*% v[,k,drop=FALSE] - U %*% D %*% t(U) %*% v[,k,drop=FALSE] )
if(!all(newv==0)){
fit.ebnm = ebnm_fn(newv,sqrt(sigma2))
newv = fit.ebnm$posterior$mean
if(!all(newv==0)){
newv = newv/sqrt(sum(newv^2 + fit.ebnm$posterior$sd^2))
}
}
v[,k] = newv
d[k] = t(v[,k]) %*% S %*% v[,k] - t(v[,k]) %*% U %*% D %*% t(U) %*% v[,k]
}
return(list(v=v,d=d))
}
#helper function
compute_sqerr = function(S,fit){
sum((S-fit$v %*% diag(fit$d,nrow=length(fit$d)) %*% t(fit$v))^2)
}
# a random initialization
random_init = function(S,K,nonneg = FALSE){
n = nrow(S)
v = matrix(nrow=n,ncol=K)
for(k in 1:K){
v[,k] = cbind(rnorm(n)) # initialize v
if(nonneg)
v[,k] = pmax(v[,k],0)
v[,k] = v[,k]/sum(v[,k]^2)
}
d = rep(1e-8,K)
return(list(v=v,d=d))
}Tree structured data
Simulate data under a tree model (with very small errors)
set.seed(1)
n = 40
x = cbind(c(rep(1,n),rep(0,n)), c(rep(0,n),rep(1,n)), c(rep(1,n/2),rep(0,3*n/2)), c(rep(0,n/2), rep(1,n/2), rep(0,n)), c(rep(0,n),rep(1,n/2),rep(0,n/2)), c(rep(0,3*n/2),rep(1,n/2)))
E = matrix(0.01*rexp(2*n*2*n),nrow=2*n)
E = E+t(E) #symmetric errors
S = x %*% diag(c(1,1,1,1,1,1)) %*% t(x) + E
image(S)
Run multiple factors
Here I run with K=9 and point-exponential. It finds a rank 4 solution, essentially zeroing out the other 5. One can compare this with non-negative without the EB approach here.
set.seed(2)
fit = random_init(S,9,nonneg=TRUE)
err = rep(0,10)
err[1] = sum((S-fit$v %*% diag(fit$d) %*% t(fit$v))^2)
for(i in 2:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_point_exponential)
err[i] = sum((S-fit$v %*% diag(fit$d) %*% t(fit$v))^2)
}
plot(err)
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
Here I try the generalized binary prior. So far I’m finding this does not work well, especially if started from random starting point. I debugged and found that what happens is that generally the v are bounded away from 0. So the gb prior puts all its weight on the non-null normal component and does not shrink anything. (Is it worth using a laplace for the non-null component?) The point exponential does not have that problem - it shrinks the smallest values towards 0, and eventually gets to a point where everything is 0. It seems clear that using the gb prior from random initialization is not going to work.
set.seed(2)
fit = random_init(S,9, nonneg=TRUE)
err = rep(0,10)
err[1] = sum((S-fit$v %*% diag(fit$d) %*% t(fit$v))^2)
for(i in 2:10){
fit = eb_power_update(S,fit$v,fit$d,ebnm_generalized_binary)
err[i] = sum((S-fit$v %*% diag(fit$d) %*% t(fit$v))^2)
}
plot(err)
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
Here we try initializing GB with point-exponential. It only changes the fit very little.
set.seed(2)
fit = random_init(S,9,nonneg=TRUE)
for(i in 1:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_point_exponential)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
for(i in 1:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_generalized_binary)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
| Version | Author | Date |
|---|---|---|
| 9006efb | Matthew Stephens | 2025-03-15 |
The binormal prior provides a more bimodal solution. (Note: although not seen here, I have seen issues with it including a factor that puts two non-neighboring populations together.)
for(i in 1:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_binormal)
err[i] = compute_sqerr(S,fit)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
| Version | Author | Date |
|---|---|---|
| 7c17b31 | Matthew Stephens | 2025-03-15 |
for(i in 1:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_binormal)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:9){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
SVD initialization
I’m going to try initializing with SVD, then running point-laplace, then running GB, similar to the strategy in the GBCD paper. (Note: here I initialize with the svd values for d; an alternative is to set these to be very small and just use the v from svd to initialize.)
set.seed(2)
S.svd = svd(S)
fit = list(v=S.svd$u[,1:4],d=S.svd$d[1:4]) #rep(1e-8,4)) #init d to be very small
err = rep(0,10)
err[1] = compute_sqerr(S,fit)
for(i in 2:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_point_laplace)
err[i] = compute_sqerr(S,fit)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:4){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
| Version | Author | Date |
|---|---|---|
| 7c17b31 | Matthew Stephens | 2025-03-15 |
split_v = function(v){
v = cbind(pmax(v,0),pmax(-v,0))
}
fit$v = split_v(fit$v)
fit$d= rep(fit$d/2,2)
for(i in 2:100){
fit = eb_power_update(S,fit$v,fit$d,ebnm_point_exponential)
err[i] = compute_sqerr(S,fit)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:8){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
fit.pe = fit # save for later use
for(i in 2:200){
fit = eb_power_update(S,fit$v,fit$d,ebnm_generalized_binary)
err[i] = compute_sqerr(S,fit)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
| Version | Author | Date |
|---|---|---|
| 7c17b31 | Matthew Stephens | 2025-03-15 |
for(i in 1:8){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
Try binormal instead, initialized from point-exp. This works better.
fit = fit.pe
for(i in 2:200){
fit = eb_power_update(S,fit$v,fit$d,ebnm_binormal)
err[i] = compute_sqerr(S,fit)
}
par(mfcol=c(3,3),mai=rep(0.3,4))
for(i in 1:8){plot(fit$v[,i],main=paste0(trunc(fit$d[i])))}
sessionInfo()R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.3.1
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Chicago
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ebnm_1.1-2
loaded via a namespace (and not attached):
[1] sass_0.4.9 generics_0.1.3 ashr_2.2-63 stringi_1.8.4
[5] lattice_0.22-6 digest_0.6.37 magrittr_2.0.3 evaluate_1.0.3
[9] grid_4.4.2 fastmap_1.2.0 rprojroot_2.0.4 workflowr_1.7.1
[13] jsonlite_1.8.9 Matrix_1.7-2 whisker_0.4.1 mixsqp_0.3-54
[17] promises_1.3.2 scales_1.3.0 truncnorm_1.0-9 invgamma_1.1
[21] jquerylib_0.1.4 cli_3.6.3 rlang_1.1.5 deconvolveR_1.2-1
[25] munsell_0.5.1 splines_4.4.2 cachem_1.1.0 yaml_2.3.10
[29] tools_4.4.2 SQUAREM_2021.1 dplyr_1.1.4 colorspace_2.1-1
[33] ggplot2_3.5.1 httpuv_1.6.15 vctrs_0.6.5 R6_2.5.1
[37] lifecycle_1.0.4 git2r_0.35.0 stringr_1.5.1 fs_1.6.5
[41] trust_0.1-8 irlba_2.3.5.1 pkgconfig_2.0.3 pillar_1.10.1
[45] bslib_0.9.0 later_1.4.1 gtable_0.3.6 glue_1.8.0
[49] Rcpp_1.0.14 xfun_0.50 tibble_3.2.1 tidyselect_1.2.1
[53] rstudioapi_0.17.1 knitr_1.49 htmltools_0.5.8.1 rmarkdown_2.29
[57] compiler_4.4.2 horseshoe_0.2.0