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Rmd 0192295 Annie Xie 2025-06-24 Add further exploration of point laplace splitting init

Introduction

In this analysis, I further explore the point-Laplace plus splitting initialization strategy for symEBcovMF in the tree setting.

Packages and Functions

library(ebnm)
library(pheatmap)
library(ggplot2)
source('code/visualization_functions.R')
source('code/symebcovmf_functions.R')
compute_crossprod_similarity <- function(est, truth){
  K_est <- ncol(est)
  K_truth <- ncol(truth)
  n <- nrow(est)
  
  #if estimates don't have same number of columns, try padding the estimate with zeros and make cosine similarity zero
  if (K_est < K_truth){
    est <- cbind(est, matrix(rep(0, n*(K_truth-K_est)), nrow = n))
  }
  
  if (K_est > K_truth){
    truth <- cbind(truth, matrix(rep(0, n*(K_est - K_truth)), nrow = n))
  }
  
  #normalize est and truth
  norms_est <- apply(est, 2, function(x){sqrt(sum(x^2))})
  norms_est[norms_est == 0] <- Inf
  
  norms_truth <- apply(truth, 2, function(x){sqrt(sum(x^2))})
  norms_truth[norms_truth == 0] <- Inf
  
  est_normalized <- t(t(est)/norms_est)
  truth_normalized <- t(t(truth)/norms_truth)
  
  #compute matrix of cosine similarities
  cosine_sim_matrix <- abs(crossprod(est_normalized, truth_normalized))
  
  assignment_problem <- lpSolve::lp.assign(t(cosine_sim_matrix), direction = "max")
  return((1/K_truth)*assignment_problem$objval)
}

Backfit Functions

optimize_factor <- function(R, ebnm_fn, maxiter, tol, v_init, lambda_k, R2k, n, KL){
  R2 <- R2k - lambda_k^2
  resid_s2 <- estimate_resid_s2(n = n, R2 = R2)
  rank_one_KL <- 0
  curr_elbo <- -Inf
  obj_diff <- Inf
  fitted_g_k <- NULL
  iter <- 1
  vec_elbo_full <- NULL
  v <- v_init
  
  while((iter <= maxiter) && (obj_diff > tol)){
    # update l; power iteration step
    v.old <- v
    x <- R %*% v
    e <- ebnm_fn(x = x, s = sqrt(resid_s2), g_init = fitted_g_k)
    scaling_factor <- sqrt(sum(e$posterior$mean^2) + sum(e$posterior$sd^2))
    if (scaling_factor == 0){ # check if scaling factor is zero
      scaling_factor <- Inf
      v <- e$posterior$mean/scaling_factor
      print('Warning: scaling factor is zero')
      break
    }
    v <- e$posterior$mean/scaling_factor
    
    # update lambda and R2
    lambda_k.old <- lambda_k
    lambda_k <- max(as.numeric(t(v) %*% R %*% v), 0)
    R2 <- R2k - lambda_k^2
    
    #store estimate for g
    fitted_g_k.old <- fitted_g_k
    fitted_g_k <- e$fitted_g
    
    # store KL
    rank_one_KL.old <- rank_one_KL
    rank_one_KL <- as.numeric(e$log_likelihood) +
      - normal_means_loglik(x, sqrt(resid_s2), e$posterior$mean, e$posterior$mean^2 + e$posterior$sd^2)
    
    # update resid_s2
    resid_s2.old <- resid_s2
    resid_s2 <- estimate_resid_s2(n = n, R2 = R2) # this goes negative?????
    
    # check convergence - maybe change to rank-one obj function
    curr_elbo.old <- curr_elbo
    curr_elbo <- compute_elbo(resid_s2 = resid_s2,
                              n = n,
                              KL = c(KL, rank_one_KL),
                              R2 = R2)
    if (iter > 1){
      obj_diff <- curr_elbo - curr_elbo.old
    }
    if (obj_diff < 0){ # check if convergence_val < 0
      v <- v.old
      resid_s2 <- resid_s2.old
      rank_one_KL <- rank_one_KL.old
      lambda_k <- lambda_k.old
      curr_elbo <- curr_elbo.old
      fitted_g_k <- fitted_g_k.old
      print(paste('elbo decreased by', abs(obj_diff)))
      break
    }
    vec_elbo_full <- c(vec_elbo_full, curr_elbo)
    iter <- iter + 1
  }
  return(list(v = v, lambda_k = lambda_k, resid_s2 = resid_s2, curr_elbo = curr_elbo, vec_elbo_full = vec_elbo_full, fitted_g_k = fitted_g_k, rank_one_KL = rank_one_KL))
}
#nullcheck function
nullcheck_factors <- function(sym_ebcovmf_obj, L2_tol = 10^(-8)){
  null_lambda_idx <- which(sym_ebcovmf_obj$lambda == 0)
  factor_L2_norms <- apply(sym_ebcovmf_obj$L_pm, 2, function(v){sqrt(sum(v^2))})
  null_factor_idx <- which(factor_L2_norms < L2_tol)
  null_idx <- unique(c(null_lambda_idx, null_factor_idx))
  
  keep_idx <- setdiff(c(1:length(sym_ebcovmf_obj$lambda)), null_idx)
  
  if (length(keep_idx) < length(sym_ebcovmf_obj$lambda)){
    #remove factors
    sym_ebcovmf_obj$L_pm <- sym_ebcovmf_obj$L_pm[,keep_idx]
    sym_ebcovmf_obj$lambda <- sym_ebcovmf_obj$lambda[keep_idx]
    sym_ebcovmf_obj$KL <- sym_ebcovmf_obj$KL[keep_idx]
    sym_ebcovmf_obj$fitted_gs <- sym_ebcovmf_obj$fitted_gs[keep_idx]
  }
  
  #shouldn't need to recompute objective function or other things
  return(sym_ebcovmf_obj)
}
sym_ebcovmf_backfit <- function(S, sym_ebcovmf_obj, ebnm_fn, backfit_maxiter = 100, backfit_tol = 10^(-8), optim_maxiter= 500, optim_tol = 10^(-8)){
  K <- length(sym_ebcovmf_obj$lambda)
  iter <- 1
  obj_diff <- Inf
  sym_ebcovmf_obj$backfit_vec_elbo_full <- NULL
  sym_ebcovmf_obj$backfit_iter_elbo_vec <- NULL
  
  # refit lambda
  sym_ebcovmf_obj <- refit_lambda(S, sym_ebcovmf_obj, maxiter = 25)
  
  while((iter <= backfit_maxiter) && (obj_diff > backfit_tol)){
    # print(iter)
    obj_old <- sym_ebcovmf_obj$elbo
    # loop through each factor
    for (k in 1:K){
      # print(k)
      # compute residual matrix
      R <- S - tcrossprod(sym_ebcovmf_obj$L_pm[,-k] %*% diag(sqrt(sym_ebcovmf_obj$lambda[-k]), ncol = (K-1)))
      R2k <- compute_R2(S, sym_ebcovmf_obj$L_pm[,-k], sym_ebcovmf_obj$lambda[-k], (K-1)) #this is right but I have one instance where the values don't match what I expect
      
      # optimize factor
      factor_proposed <- optimize_factor(R, ebnm_fn, optim_maxiter, optim_tol, sym_ebcovmf_obj$L_pm[,k], sym_ebcovmf_obj$lambda[k], R2k, sym_ebcovmf_obj$n, sym_ebcovmf_obj$KL[-k])
      
      # update object
      sym_ebcovmf_obj$L_pm[,k] <- factor_proposed$v
      sym_ebcovmf_obj$KL[k] <- factor_proposed$rank_one_KL
      sym_ebcovmf_obj$lambda[k] <- factor_proposed$lambda_k
      sym_ebcovmf_obj$resid_s2 <- factor_proposed$resid_s2
      sym_ebcovmf_obj$fitted_gs[[k]] <- factor_proposed$fitted_g_k
      sym_ebcovmf_obj$elbo <- factor_proposed$curr_elbo
      sym_ebcovmf_obj$backfit_vec_elbo_full <- c(sym_ebcovmf_obj$backfit_vec_elbo_full, factor_proposed$vec_elbo_full)
      
      #print(sym_ebcovmf_obj$elbo)
      sym_ebcovmf_obj <- refit_lambda(S, sym_ebcovmf_obj) # add refitting step?
      #print(sym_ebcovmf_obj$elbo)
    }
    sym_ebcovmf_obj$backfit_iter_elbo_vec <- c(sym_ebcovmf_obj$backfit_iter_elbo_vec, sym_ebcovmf_obj$elbo)
    
    iter <- iter + 1
    obj_diff <- abs(sym_ebcovmf_obj$elbo - obj_old)
    # need to add check if it is negative?
  }
  # nullcheck
  sym_ebcovmf_obj <- nullcheck_factors(sym_ebcovmf_obj)
  return(sym_ebcovmf_obj)
}

Data Generation

To test this procedure, I will apply it to the tree-structured dataset.

sim_4pops <- function(args) {
  set.seed(args$seed)
  
  n <- sum(args$pop_sizes)
  p <- args$n_genes
  
  FF <- matrix(rnorm(7 * p, sd = rep(args$branch_sds, each = p)), ncol = 7)
  # if (args$constrain_F) {
  #   FF_svd <- svd(FF)
  #   FF <- FF_svd$u
  #   FF <- t(t(FF) * branch_sds * sqrt(p))
  # }
  
  LL <- matrix(0, nrow = n, ncol = 7)
  LL[, 1] <- 1
  LL[, 2] <- rep(c(1, 1, 0, 0), times = args$pop_sizes)
  LL[, 3] <- rep(c(0, 0, 1, 1), times = args$pop_sizes)
  LL[, 4] <- rep(c(1, 0, 0, 0), times = args$pop_sizes)
  LL[, 5] <- rep(c(0, 1, 0, 0), times = args$pop_sizes)
  LL[, 6] <- rep(c(0, 0, 1, 0), times = args$pop_sizes)
  LL[, 7] <- rep(c(0, 0, 0, 1), times = args$pop_sizes)
  
  E <- matrix(rnorm(n * p, sd = args$indiv_sd), nrow = n)
  Y <- LL %*% t(FF) + E
  YYt <- (1/p)*tcrossprod(Y)
  return(list(Y = Y, YYt = YYt, LL = LL, FF = FF, K = ncol(LL)))
}
sim_args = list(pop_sizes = rep(40, 4), n_genes = 1000, branch_sds = rep(2,7), indiv_sd = 1, seed = 1)
sim_data <- sim_4pops(sim_args)

This is a heatmap of the scaled Gram matrix:

plot_heatmap(sim_data$YYt, colors_range = c('blue','gray96','red'), brks = seq(-max(abs(sim_data$YYt)), max(abs(sim_data$YYt)), length.out = 50))

This is a scatter plot of the true loadings matrix:

pop_vec <- c(rep('A', 40), rep('B', 40), rep('C', 40), rep('D', 40))
plot_loadings(sim_data$LL, pop_vec)

This is a plot of the eigenvalues of the Gram matrix:

S_eigen <- eigen(sim_data$YYt)
plot(S_eigen$values) + abline(a = 0, b = 0, col = 'red')

integer(0)

This is the minimum eigenvalue:

min(S_eigen$values)
[1] 0.3724341

Split point-Laplace Fit

I will load in previously saved results for the point-Laplace fit:

symebcovmf_laplace_fit <- readRDS('data/symebcovmf_laplace_tree_backfit_20000.rds')

Now, we split the estimate into its positive and negative parts.

LL_split_init <- cbind(pmax(symebcovmf_laplace_fit$L_pm, 0), pmax(-symebcovmf_laplace_fit$L_pm, 0))

This is a plot of the estimate with its positive and negative parts split into different factors, \(\hat{L}_{split}\).

plot_loadings(LL_split_init, pop_vec)

This is a heatmap of the estimate with its positive and negative parts split into different factors, \(\hat{L}_{split}\).

plot_heatmap(LL_split_init)

Backfit with Split Initialization

Now, we backfit from the split initialization (this is what GBCD does). The backfit is run with the point-exponential prior.

First, we initialize a symEBcovMF object using \(\hat{L}_{split}\).

symebcovmf_laplace_split_init_obj <- sym_ebcovmf_init(sim_data$YYt)

split_L_norms <- apply(LL_split_init, 2, function(x){sqrt(sum(x^2))})
idx.keep <- which(split_L_norms > 0)
split_L_normalized <- apply(LL_split_init[,idx.keep], 2, function(x){x/sqrt(sum(x^2))})
symebcovmf_laplace_split_init_obj$L_pm <- split_L_normalized

symebcovmf_laplace_split_init_obj$lambda <- rep(symebcovmf_laplace_fit$lambda, times = 2)[idx.keep]

symebcovmf_laplace_split_init_obj$resid_s2 <- estimate_resid_s2(S = sim_data$YYt, 
  L = symebcovmf_laplace_split_init_obj$L_pm, 
  lambda = symebcovmf_laplace_split_init_obj$lambda, 
  n = nrow(sim_data$Y), 
  K = length(symebcovmf_laplace_split_init_obj$lambda))

symebcovmf_laplace_split_init_obj$elbo <- compute_elbo(S = sim_data$YYt, 
  L = symebcovmf_laplace_split_init_obj$L_pm, 
  lambda = symebcovmf_laplace_split_init_obj$lambda, 
  resid_s2 = symebcovmf_laplace_split_init_obj$resid_s2, 
  n = nrow(sim_data$Y), 
  K = length(symebcovmf_laplace_split_init_obj$lambda), 
  KL = rep(0, length(symebcovmf_laplace_split_init_obj$lambda)))

I refit the lambda values keeping the factors fixed.

# need to check this
symebcovmf_laplace_split_obj <- refit_lambda(S = sim_data$YYt, symebcovmf_laplace_split_init_obj, maxiter = 500)

This is a plot of the initial loadings estimate:

plot_loadings(symebcovmf_laplace_split_init_obj$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_obj$lambda)), pop_vec)

This is a plot of the initial estimate of the scaled Gram matrix:

plot_heatmap(tcrossprod(symebcovmf_laplace_split_init_obj$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_obj$lambda))), brks = seq(0, max(abs(tcrossprod(symebcovmf_laplace_split_init_obj$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_obj$lambda))))), length.out = 50))

Now, we run the backfit with point-exponential prior.

symebcovmf_laplace_split_init_backfit <- sym_ebcovmf_backfit(sim_data$YYt, symebcovmf_laplace_split_init_obj, ebnm_fn = ebnm::ebnm_point_exponential, backfit_maxiter = 500)

This is a plot of the estimate from the backfit initialized with the split estimate, \(\hat{L}_{split-backfit}\).

plot_loadings(symebcovmf_laplace_split_init_backfit$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_backfit$lambda)), pop_vec)

This is a heatmap of the estimate from the backfit initialized with the split estimate, \(\hat{L}_{split-backfit}\).

plot_heatmap(symebcovmf_laplace_split_init_backfit$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_backfit$lambda)))

This is the crossproduct similarity of the loadings estimate from the backfit initialized with the split estimate, \(\hat{L}_{split-backfit}\).

compute_crossprod_similarity(symebcovmf_laplace_split_init_backfit$L_pm, sim_data$LL)
[1] 0.999089

This is a heatmap of the estimated Gram matrix, \(\hat{L}_{split-backfit}\hat{\Lambda}\hat{L}_{split-backfit}'\).

gram_est <- tcrossprod(symebcovmf_laplace_split_init_backfit$L_pm %*% diag(sqrt(symebcovmf_laplace_split_init_backfit$lambda)))
plot_heatmap(gram_est, brks = seq(0, max(abs(gram_est)), length.out = 50))

This is the L2 norm of the difference, excluding diagonal entries.

compute_L2_fit <- function(est, dat){
  score <- sum((dat - est)^2) - sum((diag(dat) - diag(est))^2)
  return(score)
}

compute_L2_fit(gram_est, sim_data$YYt)
[1] 30.4981

This is a plot of fitted vs. observed values:

ggplot(data = NULL, aes(x = as.vector(sim_data$YYt), y = as.vector(gram_est))) + geom_point() + geom_abline(slope = 1, intercept = 0, color = 'red') + xlab('Observed Values') + ylab('Fitted Values')

This is a plot of the progression of the ELBO:

plot(symebcovmf_laplace_split_init_backfit$backfit_iter_elbo_vec, ylab = 'ELBO')

Observations

The estimate from the backfit still captures the tree components and yields a very high crossproduct similarity value. One observation is some of the factors are not fully sparse. For example, in factor 3, which primarily captures the the population effect for population 3, we see that members of populations 1 and 2 now have small non-zero loading. Based on my intuition, I would expect the fully sparse representation to have a higher objective function value, so I’m not exactly sure why these factors have these small loading values in other populations. I imagine these small loading values could be shrunk to zero if you used a binary (or a generalized binary prior that leans toward being binary) to improve the fit.

GBCD

For comparison, I also try applying GBCD with point-exponential prior.

library(gbcd)
gbcd_fit <- fit_gbcd(sim_data$Y, Kmax = 7, ebnm::ebnm_point_exponential)
[1] "Form cell by cell covariance matrix..."
   user  system elapsed 
  0.008   0.000   0.008 
[1] "Initialize GEP membership matrix L..."
Adding factor 1 to flash object...
Wrapping up...
Done.
Adding factor 2 to flash object...
Adding factor 3 to flash object...
Adding factor 4 to flash object...
Adding factor 5 to flash object...
Factor doesn't significantly increase objective and won't be added.
Wrapping up...
Done.
Backfitting 4 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e+04...
  Difference between iterations is within 1.0e+03...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  --Maximum number of iterations reached!
Wrapping up...
Done.
Backfitting 4 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e+03...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  Difference between iterations is within 1.0e-03...
Wrapping up...
Done.
Backfitting 4 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e+00...
Wrapping up...
Done.
   user  system elapsed 
  3.393   0.071   3.465 
[1] "Estimate GEP membership matrix L..."
Backfitting 7 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e+04...
  Difference between iterations is within 1.0e+03...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  --Maximum number of iterations reached!
Wrapping up...
Done.
Backfitting 7 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e+02...
  Difference between iterations is within 1.0e+01...
  Difference between iterations is within 1.0e+00...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  --Maximum number of iterations reached!
Wrapping up...
Done.
Backfitting 7 factors (tolerance: 3.81e-04)...
  Difference between iterations is within 1.0e-01...
  Difference between iterations is within 1.0e-02...
  --Maximum number of iterations reached!
Wrapping up...
Done.
   user  system elapsed 
  6.711   0.085   6.797 
[1] "Estimate GEP signature matrix F..."
Backfitting 7 factors (tolerance: 2.38e-03)...
  --Estimate of factor 7 is numerically zero!
  Difference between iterations is within 1.0e+00...
  --Maximum number of iterations reached!
Wrapping up...
Done.
   user  system elapsed 
 17.773   0.266  18.047

This is a plot of the loadings estimate from GBCD, \(\hat{L}_{gbcd}\).

plot_loadings(gbcd_fit$L, pop_vec)

This is a heatmap of the loadings estimate from GBCD, \(\hat{L}_{gbcd}\).

plot_heatmap(gbcd_fit$L)

This is the crossproduct similarity of the loadings estimate from GBCD, \(\hat{L}_{gbcd}\).

compute_crossprod_similarity(gbcd_fit$L, sim_data$LL)
[1] 0.9975294

Observations

The estimate from GBCD also captures all the tree components and yields a high crossproduct similarity value. The crossproduct similarity is just slightly lower than that of symEBcovMF backfit initialized with the point-Laplace fit plus splitting. We also find that some of the GBCD factors have small non-zero loadings. For example, factor 5, which primarily captures population 2, has small non-zero loadings for members of populations 3 and 4. Given that the GBCD estimate also has this characteristic, I am inclined to think non-zero loadings are a result of model misspecification.


sessionInfo()
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS 15.4.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

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/New_York
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] gbcd_0.1-20     ggplot2_3.5.1   pheatmap_1.0.12 ebnm_1.1-34    
[5] workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.1     viridisLite_0.4.2    dplyr_1.1.4         
 [4] farver_2.1.2         fastmap_1.2.0        lazyeval_0.2.2      
 [7] promises_1.3.0       digest_0.6.37        lifecycle_1.0.4     
[10] processx_3.8.4       invgamma_1.1         magrittr_2.0.3      
[13] compiler_4.3.2       progress_1.2.3       rlang_1.1.4         
[16] sass_0.4.9           tools_4.3.2          utf8_1.2.4          
[19] yaml_2.3.10          data.table_1.16.0    knitr_1.48          
[22] prettyunits_1.2.0    labeling_0.4.3       htmlwidgets_1.6.4   
[25] scatterplot3d_0.3-44 RColorBrewer_1.1-3   Rtsne_0.17          
[28] purrr_1.0.2          withr_3.0.1          flashier_1.0.53     
[31] grid_4.3.2           fansi_1.0.6          git2r_0.33.0        
[34] fastTopics_0.6-192   colorspace_2.1-1     scales_1.3.0        
[37] gtools_3.9.5         cli_3.6.3            crayon_1.5.3        
[40] rmarkdown_2.28       generics_0.1.3       RcppParallel_5.1.9  
[43] rstudioapi_0.16.0    httr_1.4.7           pbapply_1.7-2       
[46] cachem_1.1.0         stringr_1.5.1        splines_4.3.2       
[49] parallel_4.3.2       softImpute_1.4-1     vctrs_0.6.5         
[52] Matrix_1.6-5         jsonlite_1.8.9       callr_3.7.6         
[55] hms_1.1.3            mixsqp_0.3-54        ggrepel_0.9.6       
[58] irlba_2.3.5.1        horseshoe_0.2.0      trust_0.1-8         
[61] plotly_4.10.4        tidyr_1.3.1          jquerylib_0.1.4     
[64] glue_1.8.0           ps_1.7.7             uwot_0.1.16         
[67] cowplot_1.1.3        stringi_1.8.4        Polychrome_1.5.1    
[70] gtable_0.3.5         later_1.3.2          quadprog_1.5-8      
[73] munsell_0.5.1        tibble_3.2.1         pillar_1.9.0        
[76] htmltools_0.5.8.1    truncnorm_1.0-9      R6_2.5.1            
[79] rprojroot_2.0.4      evaluate_1.0.0       lpSolve_5.6.20      
[82] lattice_0.22-6       highr_0.11           RhpcBLASctl_0.23-42 
[85] SQUAREM_2021.1       ashr_2.2-66          httpuv_1.6.15       
[88] bslib_0.8.0          Rcpp_1.0.13          deconvolveR_1.2-1   
[91] whisker_0.4.1        xfun_0.48            fs_1.6.4            
[94] getPass_0.2-4        pkgconfig_2.0.3