Sparse inverse covariance estimation (graphical lasso)

approxsol.jl

# Fattahi, Salar, and Somayeh Sojoudi. 2019. “Graphical Lasso and Thresholding:
# Equivalence and Closed-Form Solutions.” Journal of Machine Learning Research:
# JMLR. https://www.jmlr.org/papers/volume20/17-501/17-501.pdf.
function approxsol(Σ, λ)
    # Residue of Σ relative to λ (see def. 11).
    Σres = zeros(size(Σ))
    for j in axes(Σ, 2), i in axes(Σ, 1)
        if i ≠ j && abs(Σ[i,j]) > λ
            Σres[i,j] = Σ[i,j] - λ*sign(Σ[i,j])
        end
    end
    # Approximated solution (see eq. 13).
    v = zeros(size(Σ, 1))
    for j in axes(Σ, 2), i in 1:j-1
        if !iszero(Σres[i,j])
            v[i] += Σres[i,j]^2/(Σ[i,i]*Σ[j,j] - Σres[i,j]^2)
        end
    end
    A = zeros(size(Σ))
    for j in axes(Σ, 2), i in axes(Σ, 1)
        if i == j
            A[i,j] = (1 + v[i])/Σ[i,i]
        elseif !iszero(Σres[i,j])
            A[i,j] = -Σres[i,j]/(Σ[i,i]*Σ[j,j] - Σres[i,j]^2)
        end
    end
    return A
end