Hello,

I am working on an implementation of the graphical Lasso (Glasso) (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3019769/pdf/kxm045.pdf) in pure Julia. In the algorithm, we have to use *a square root* of sample covariance matrix `W11`

. I am thinking of using the Cholesky factor as it will speed-up calculations for the lasso optimization. I am getting error that `W11`

is not positive definite. How would you address this?

```
using SCS
using Convex
using Distributions
using LinearAlgebra
using Statistics
function glasso(X, ρ; tol = 1e-2, Niter = 100, withθ = true)
Nx, Ne = size(X)
S = cov(X')
W = S + ρ*I
Wold = copy(W)
W11 = zeros(Nx-1, Nx-1)
b = zeros(Nx-1)
B = zeros(Nx-1, Nx-1)
w12 = zeros(Nx-1)
# Iterate until convergence
for i=1:Niter
for j=1:Nx
jminus = setdiff(1:Nx, j)
# Compute a Cholesky factorization of W11
W11 .= W[jminus, jminus]
W11sqrt .= LowerTriangular(cholesky(Symmetric(W11)).L)
# W11svd = svd(W11)
# W11sqrt = W11svd.U * Diagonal(sqrt.(W11svd.S)) * W11svd.Vt #W_11^(1/2)
b .= 0.5*(W11sqrt \ S[jminus,j]) # W_11^(-1/2) * s_12
# Solve the lasso problem
β_lasso = Variable(Nx-1)
lasso_problem = minimize(sumsquares(W11sqrt*β_lasso - b) + ρ*norm(β_lasso, 1))
# Solve the problem by calling solve!
solve!(lasso_problem, SCS.Optimizer; silent_solver = true)
# TODO check resolution status
# @assert lasso_problem.status
w12 .= 2.0*(W11*evaluate(β_lasso))
# Update W
W[j, jminus] .= copy(w12)
W[jminus, j] .= copy(w12)
end
# Stop criterion
if norm(W - Wold,1) < tol
break
end
Wold = copy(W)
end
return W
# if withθ == true
# # Build precision matrix
# θ = zeros(Nx, Nx)
# end
end
```

```
Nx = 40
μX = zeros(Nx)
ΣX = PDiagMat(ones(Nx))
πX = Distributions.MvNormal(μX, ΣX)
Ne = 100
X = rand(πX, Ne)
W = glasso(X, 0.01)
```