# Sparse Inverse Covariance Estimation

## Introduction

Assume we are given i.i.d. observations $$x_i \sim N(0,\Sigma)$$ for $$i = 1,\ldots,m$$, and the covariance matrix $$\Sigma \in {\mathbf S}_+^n$$, the set of symmetric positive semidefinite matrices, has a sparse inverse $$S = \Sigma^{-1}$$. Let $$Q = \frac{1}{m-1}\sum_{i=1}^m (x_i - \bar x)(x_i - \bar x)^T$$ be our sample covariance. One way to estimate $$\Sigma$$ is to maximize the log-likelihood with the prior knowledge that $$S$$ is sparse (Friedman, Hastie, and Tibshirani 2008), which amounts to the optimization problem:

$\begin{array}{ll} \underset{S}{\mbox{maximize}} & \log\det(S) - \mbox{tr}(SQ) \\ \mbox{subject to} & S \in {\mathbf S}_+^n, \quad \sum_{i=1}^n \sum_{j=1}^n |S_{ij}| \leq \alpha. \end{array}$

The parameter $$\alpha \geq 0$$ controls the degree of sparsity. The problem is convex, so we can solve it using CVXR.

## Example

We’ll create a sparse positive semi-definite matrix $$S$$ using synthetic data

set.seed(1)
n <- 10      ## Dimension of matrix
m <- 1000    ## Number of samples

## Create sparse, symmetric PSD matrix S
A <- rsparsematrix(n, n, 0.15, rand.x = stats::rnorm)
Strue <- A %*% t(A) + 0.05 * diag(rep(1, n))    ## Force matrix to be strictly positive definite

We can now create the covariance matrix $$R$$ as the inverse of $$S$$.

R <- base::solve(Strue)

As test data, we sample from a multivariate normal with the fact that if $$Y \sim N(0, I)$$, then $$R^{1/2}Y \sim N(0, R)$$ since $$R$$ is symmetric.

x_sample <- matrix(stats::rnorm(n * m), nrow = m, ncol = n) %*% t(expm::sqrtm(R))
Q <- cov(x_sample)    ## Sample covariance matrix

Finally, we solve our convex program for a set of $$\alpha$$ values.

Version 1.0 Note: Positive semi-definite variables are now designated using PSD = TRUE rather than the Semidef function!

alphas <- c(10, 8, 6, 4, 1)
if (packageVersion("CVXR") > "0.99-7") {
S  <- Variable(n, n, PSD = TRUE)
} else {
S <- Semidef(n)    ## Variable constrained to positive semidefinite cone
}

obj <- Maximize(log_det(S) - matrix_trace(S %*% Q))

S.est <- lapply(alphas,
function(alpha) {
constraints <- list(sum(abs(S)) <= alpha)
## Form and solve optimization problem
prob <- Problem(obj, constraints)
result <- solve(prob)

## Create covariance matrix
R_hat <- base::solve(result$getValue(S)) Sres <- result$getValue(S)
Sres[abs(Sres) <= 1e-4] <- 0
Sres
})

In the code above, the Semidef constructor restricts S to the positive semidefinite cone. In our objective, we use CVXR functions for the log-determinant and trace. The expression matrix_trace(S %*% Q) is equivalent to sum(diag(S %*% Q))}, but the former is preferred because it is more efficient than making nested function calls.

However, a standalone atom does not exist for the determinant, so we cannot replace log_det(S) with log(det(S)) since det is undefined for a Semidef object.

## Results

The figures below depict the solutions for the above dataset with $$m = 1000, n = 10$$, and $$S$$ containing 26% non-zero entries, represented by the dark squares in the images below. The sparsity of our inverse covariance estimate decreases for higher $$\alpha$$, so that when $$\alpha = 1$$, most of the off-diagonal entries are zero, while if $$\alpha = 10$$, over half the matrix is dense. At $$\alpha = 4$$, we achieve the true percentage of non-zeros.

do.call(multiplot, args = c(list(plotSpMat(Strue)),
mapply(plotSpMat, S.est, alphas, SIMPLIFY = FALSE),
list(layout = matrix(1:6, nrow = 2, byrow = TRUE)))) ## Session Info

sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-apple-darwin19.5.0 (64-bit)
## Running under: macOS Catalina 10.15.7
##
## Matrix products: default
## BLAS/LAPACK: /usr/local/Cellar/openblas/0.3.10_1/lib/libopenblasp-r0.3.10.dylib
##
## locale:
##  en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
##  grid      stats     graphics  grDevices datasets  utils     methods
##  base
##
## other attached packages:
##  expm_0.999-5  Matrix_1.2-18 ggplot2_3.3.2 CVXR_1.0-9
##
## loaded via a namespace (and not attached):
##   tidyselect_1.1.0   xfun_0.15          slam_0.1-47        purrr_0.3.4
##   lattice_0.20-41    Rmosek_9.2.3       colorspace_1.4-1   vctrs_0.3.2
##   generics_0.0.2     htmltools_0.5.0    yaml_2.2.1         gmp_0.6-0
##  rlang_0.4.7        pillar_1.4.6       glue_1.4.1         Rmpfr_0.8-1
##  withr_2.2.0        Rcplex_0.3-3       bit64_0.9-7        RColorBrewer_1.1-2
##  scs_1.3-3          lifecycle_0.2.0    stringr_1.4.0      munsell_0.5.0
##  blogdown_0.19      gtable_0.3.0       gurobi_9.0.3.1     codetools_0.2-16
##  evaluate_0.14      labeling_0.3       knitr_1.28         cccp_0.2-4
##  Rcpp_1.0.5         scales_1.1.1       farver_2.0.3       bit_1.1-15.2
##  digest_0.6.25      stringi_1.4.6      bookdown_0.19      dplyr_1.0.0
##  Rglpk_0.6-4        tools_4.0.2        magrittr_1.5       tibble_3.0.3
##  crayon_1.3.4       pkgconfig_2.0.3    ellipsis_0.3.1     rcbc_0.1.0.9001
##  assertthat_0.2.1   rmarkdown_2.3      R6_2.4.1           compiler_4.0.2`

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