Graphical Lasso

The graphical lasso learns a GMRF from data by estimating a sparse precision matrix. Given a data matrix $X$, the algorithm solves

$$\underset{\mathrm{\Omega }\succ 0}{max}\log det\mathrm{\Omega }-\mathrm{tr}(S\mathrm{\Omega })-\lambda \Vert \mathrm{\Omega }{\Vert }_1$$

where $S$ is the sample covariance and $\lambda$ is a sparsity-inducing penalty.

The implementation follows the approach of [4], which combines soft-thresholding of the sample covariance with a maximum-determinant positive-definite matrix completion via chordal graph techniques (using CliqueTrees.jl).

Restricted Graphical Lasso

Instead of a scalar threshold $\lambda$, you can pass a sparse matrix whose sparsity pattern defines which entries to penalize, and whose values define per-entry thresholds. This is useful when the conditional independence structure is partially known.

API Reference

GaussianMarkovRandomFields.graphical_lasso Function

graphical_lasso(X::AbstractMatrix, threshold; blocksize::Int=256, shift=zero(T), alg=nothing)

Learn a GMRF from data by solving the graphical lasso problem:

maximize  log det Ω - tr(ΣΩ) - λ ‖Ω‖₁
such that  Ω is positive definite

where Σ is the sample covariance and Ω is the GMRF's precision matrix.

threshold can be a scalar λ for uniform penalization, or a SparseMatrixCSC for per-entry penalties within a given sparsity pattern (restricted graphical lasso).

shift adds a constant to the diagonal of the sample covariance as regularization to improve convergence.

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