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:

We simultaneously infer a network and estimate a clustering structure by combining the neighborhood selection approach (Meinshausen and Bühlman, 2006) and convex clustering (Hocking et al. 2011).

We use a continuation with Nesterov smoothing in a shrinkage-thresholding algorithm (

`CONESTA`

, Hadj-Selem et al. 2018) to solve the optimization problem.

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 \rVert*2 ^{2} +
\lambda_1
\sum*{i = 1}

\left \lVert \boldsymbol{\beta}

`MGLasso`

package is based on the python implementation of the solver `CONESTA`

available in pylearn-parsimony library.

- Install the
`reticulate`

package and Miniconda if no conda distribution available on the OS.

```
install.packages('reticulate')
reticulate::install_miniconda()
```

- Install
`MGLasso`

, its python dependencies and configure the conda environment`rmglasso`

.

```
# install.packages('mglasso')
remotes::install_github("desanou/mglasso")
```

```
library(mglasso)
install_pylearn_parsimony(envname = "rmglasso", method = "conda")
reticulate::use_condaenv("rmglasso", required = TRUE)
reticulate::py_config()
```

The `conesta_solver`

is delay loaded. See `reticulate::import_from_path`

for details.

An example of use is given below.

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 \(0\).

```
library(Matrix)
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")
```

```
set.seed(11)
X <- mvtnorm::rmvnorm(n, mean = rep(0,p), sigma = as.matrix(mat.correlation))
colnames(X) <- LETTERS[1:9]
```

`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)
```

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.

```
library(ggplot2)
library(ggrepel)
mglasso:::plot_clusterpath(as.matrix(X), res)
```

As the the fusion penalty increases from `level9`

to `level1`

we observe a progressive fusion of adjacent edges.

```
plot_mglasso(res)
```

Edmond, Sanou; Christophe, Ambroise; Geneviève, Robin; (2022): Inference of Multiscale Gaussian Graphical Model. ArXiv. Preprint. https://doi.org/10.48550/arXiv.2202.05775