# 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

suppressWarnings(suppressMessages(library(CVXR)))
suppressWarnings(suppressMessages(library(Matrix)))
suppressWarnings(suppressMessages(library(expm)))

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.

suppressMessages(suppressWarnings(library(CVXR)))

alphas <- c(10, 8, 6, 4, 1)
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 3.4.2 (2017-09-28)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.1
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] grid      methods   stats     graphics  grDevices datasets  utils
## [8] base
##
## other attached packages:
## [1] expm_0.999-2  Matrix_1.2-11 CVXR_0.94-4   ggplot2_2.2.1
##
## loaded via a namespace (and not attached):
##  [1] gmp_0.5-13.1       Rcpp_0.12.13       RColorBrewer_1.1-2
##  [4] compiler_3.4.2     plyr_1.8.4         R.methodsS3_1.7.1
##  [7] R.utils_2.6.0      tools_3.4.2        digest_0.6.12
## [10] bit_1.1-12         evaluate_0.10.1    tibble_1.3.4
## [13] gtable_0.2.0       lattice_0.20-35    rlang_0.1.2
## [16] yaml_2.1.14        blogdown_0.1.7     Rmpfr_0.6-1
## [19] ECOSolveR_0.3-2    stringr_1.2.0      knitr_1.17
## [22] rprojroot_1.2      bit64_0.9-7        R6_2.2.2
## [25] rmarkdown_1.6      bookdown_0.5       magrittr_1.5
## [28] backports_1.1.1    scales_0.5.0       htmltools_0.3.6
## [31] scs_1.1-1          colorspace_1.3-2   labeling_0.3
## [34] stringi_1.1.5      lazyeval_0.2.1     munsell_0.4.3
## [37] R.oo_1.21.0

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## References

Friedman, J., T. Hastie, and R. Tibshirani. 2008. “Sparse Inverse Covariance Estimation with the Graphical Lasso.” Biostatistics 9 (3): 432–41.