Last updated: 2021-06-16

Checks: 7 0

Knit directory: ebpmf_data_analysis/

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    Modified:   topicView-app/app.R

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Rmd 1ede7c9 zihao12 2021-06-16 multinom sampling

Introduction

  • I want to see how good is the estimator for word-word covariance matrix (proposed in here, probably easier to understand here)
  • Model is: \(M = AW\) where \(dim(M)= (p, n)\) and each column is probability vector. Then sampling process has \(x_i \sim \text{Multinom}(N, m_i), \forall i \in [n]\).
  • They propose an estimator \(Q\) for \(\frac{1}{n} M M^T\) and claim when \(n > \frac{50 logp}{N \eps^2}\), with high probability all entries of \(|Q - \frac{1}{n} M M^T|\) is at most \(\epsilon\). The proof looks right. (Though the wording is a bit confusing,on the spliting dictionary part).
  • However, the problem I find is, entries in \(\frac{1}{n} M M^T\) is very very small with large \(p\). Therefore, to get reasonable good estimate we need to set \(\epsilon\) to be even smaller!! If we instead want to bound the relative difference between \(Q\) and \(\frac{1}{n} M M^T\), we need a lot more samples than merely \(log(p)\). Thus I think the true dependency of \(n\) on \(p\) is hidden inside \(\epsilon\).
  • Below are my simulations
rm(list = ls())
set.seed(123)
## n: # documents
## p: # words in dictionary
## N: document length
## k: number of topics 

simulate_data <- function(n,p,N,k = 4,seed =123){
  set.seed(seed)
  W = matrix( runif(n = k * n), nrow = k)
  A = matrix( runif(n = k * p), ncol = k)
  A <- A %*% diag(1/colSums(A))
  W <- W %*% diag(1/colSums(W))
  
  M = A %*% W
  X = matrix(,nrow = p, ncol = n)
  for(j in 1:n){
    X[,j] = rmultinom(n = 1, size = N, prob = M[,j])
  }
  cov_word <- (1/n)* M %*% t(M)
  return(list(X = X,cov_word = cov_word))
}

est_cov_word <- function(X){
  n = ncol(X)
  p = nrow(X)
  N = mean(colSums(X))
  upper <- 1:round(p/2)
  lower <- (round(p/2) + 1) : p
  Y1 = X; Y2 = X
  for(i in 1:n){
    upper <- sample(1:p, round(p/2))
    Y1[-upper,i]<- 0
    Y2[upper, i]<-0
  }
  Q <- Y1 %*% t(Y2) * (4/(n*N^2))
  return(Q)
}

Setting 1

The covariance matrix entries are all very small. Even though the absolute difference is small, it’s still bad estimate.

n= 300
p= 1000
N = 100
k = 4

sim <- simulate_data(n = n, p = p, N = N, k = k)
covhat <- est_cov_word(sim$X)
idx <-sample(p, 100)
plot(sim$cov_word[idx,idx], covhat[idx,idx],xlab ="truth", ylab = "est")

Setting 2

I increase \(N\) so even direct estimate \(M\) is not too bad. The estimate of \(MM^T\) is much much better

n= 300
p= 1000
N = 10000
k = 4

sim <- simulate_data(n = n, p = p, N = N, k = k)
covhat <- est_cov_word(sim$X)
idx <-sample(p, 100)
plot(sim$cov_word[idx,idx], covhat[idx,idx],xlab ="truth", ylab = "est")

b how many samples do we need

I think a good estimate here should at least need \(\epsilon \leq 10^{-8}\). The dominating term is \(\epsilon^2\), which is huge…


sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.15.7

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

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

other attached packages:
[1] workflowr_1.6.2

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5      rprojroot_1.3-2 digest_0.6.25   later_1.1.0.1  
 [5] R6_2.4.1        backports_1.1.7 git2r_0.26.1    magrittr_1.5   
 [9] evaluate_0.14   stringi_1.6.2   rlang_0.4.5     fs_1.3.1       
[13] promises_1.1.1  whisker_0.3-2   rmarkdown_2.1   tools_3.5.1    
[17] stringr_1.4.0   glue_1.4.1      httpuv_1.5.4    xfun_0.8       
[21] yaml_2.2.0      compiler_3.5.1  htmltools_0.5.0 knitr_1.28