This repository provides an implementation of the MGLasso (Multiscale Graphical Lasso) algorithm: an approach for estimating sparse Gaussian Graphical Models with the addition of a group-fused Lasso penalty.

MGLasso is described in the paper Inference of Multiscale Gaussian Graphical Model. MGLasso has these contributions:

To solve the MGLasso problem, we seek the regression vectors \(\beta^i\) that minimize

$J{\lambda_1, \lambda_2}(\boldsymbol{\beta}; \mathbf{X} ) = \frac{1}{2} \sum{i=1}p \left \lVert \mathbf{X}i - \mathbf{X}{\setminus i} \boldsymbol{\beta}i \right \rVert2 2 + \lambda_1 \sum{i = 1}p
\left \lVert \boldsymbol{\beta}i \right \rVert1 + \lambda_2 \sum{i < j} \left \lVert \boldsymbol{\beta}i - \tau_{ij}(\boldsymbol{\beta}j) \right \rVert_2.$

MGLasso package is based on the python implementation of the solver CONESTA available in pylearn-parsimony library.

Package requirements and installation

# install.packages('mglasso')
install_pylearn_parsimony(envname = "rmglasso", method = "conda")
reticulate::use_condaenv("rmglasso", required = TRUE)

The conesta_solver is delay loaded. See reticulate::import_from_path for details.

An example of use is given below.

Illustration on a simple model

Simulate a block diagonal model

We simulate a \(3\)-block diagonal model where each block contains \(3\) variables. The intra-block correlation level is set to \(0.85\) while the correlations outside the blocks are kept to \(0\).

n = 50
K = 3
p = 9
rho = 0.85
blocs <- list()

for (j in 1:K) {
  bloc <- matrix(rho, nrow = p/K, ncol = p/K)
  for(i in 1:(p/K)) { bloc[i,i] <- 1 }
  blocs[[j]] <- bloc

mat.correlation <- Matrix::bdiag(blocs)
corrplot::corrplot(as.matrix(mat.correlation), method = "color", tl.col="black")

Simulate gaussian data from the covariance matrix

X <- mvtnorm::rmvnorm(n, mean = rep(0,p), sigma = as.matrix(mat.correlation))
colnames(X) <- LETTERS[1:9]

Run mglasso()

We set the sparsity level \(\lambda_1\) to \(0.2\) and rescaled it with the size of the sample.

X <- scale(X)    
res <- mglasso(X, lambda1 = 0.2*n, lambda2_start = 0.1, fuse_thresh = 1e-3, verbose = FALSE)

To launch a unique run of the objective function call the conesta function.

temp <- mglasso::conesta(X, lam1 = 0.2*n, lam2 = 0.1)

Estimated clustering path

We plot the clustering path of mglasso method on the 2 principal components axis of \(X\). The path is drawn on the predicted \(X\)'s.

mglasso:::plot_clusterpath(as.matrix(X), res)

Estimated adjacency matrices along the clustering path

As the the fusion penalty increases from level9 to level1 we observe a progressive fusion of adjacent edges.



Edmond, Sanou; Christophe, Ambroise; Geneviève, Robin; (2022): Inference of Multiscale Gaussian Graphical Model. ArXiv. Preprint.