Do we assume graphical LASSO explanatory variables to be normally distributed? And what if this assumption fails? I am working on a graphical LASSO (GLASSO) shrinkage of the variance-covariance matrix of financial log-returns data for 10 years. I tested for normality and the Jarque-Bera test (but also other tests) reject the null hypothesis of normal distributed assets return. If returns are not normally distribute, can I anyway apply the GLASSO method for reducing conditional dependence between covariates?
 A: Let us take a look at the objective of the graphical LASSO. Let us say your data consists of $X_i \in \mathbb R^p$, for $i=1,\dotsc,n$. For simplicity we assume the data are centered ($E[X_i]=0$) and finally we let $\mathbf{X}_n$ be the $n \times p$ design matrix with rows $X_i^\intercal$. Now let $S_n$ be the sample covariance of the observations, i.e., $S_n = \mathbf{X}_n^\intercal \mathbf{X}_n/n$. Fixing a penalty parameter $\lambda >0$, the Graphical LASSO seeks to maximize over covariance matrices $\Sigma \succ 0$, the following objective
$$\ell(\Sigma) =  \underbrace{-\frac{n}{2}\text{trace}(S_n \Sigma^{-1}) + \frac{n}{2} \log(det|\Sigma^{-1}|)}_{\text{Gaussian log-likelihood}} - \underbrace{\lambda \sum_{1 \leq j \neq k \leq p} |(\Sigma^{-1})_{jk}|}_{\text{Regularization term}}$$
Let us look at the two parts in turn:
The first part is indeed motivated by multivariate Gaussian measurements $X_i \sim \mathcal{N}(0,\Sigma)$, however it also makes sense for any multivariate distribution. Indeed, the maximizer of the first part (if we ignore regularization), is just $S_n$ itself, i.e., the sample covariance, which is a reasonable estimate of $\Sigma$ for any multivariate distribution (at least in the regime where $p \ll n$).
The second part, may also be interpreted generically. You want to regularize $S_n$ towards a $\Sigma$ that has a sparse inverse (precision matrix) with many entries $(\Sigma^{-1})_{jk}$ equal to $0$. For Gaussian measurements this has a particularly nice interpretation, since $(\Sigma^{-1})_{jk}=0$ means that the $X_{i,j}$ and $X_{i,k}$, i..e, the $j$-th, resp. $k$-th coordinates of $X_i$ are independent conditionally on the other $p-2$ coordinates. However, this penalty also makes sense for any multivariate distribution, for example $(\Sigma^{-1})_{jk}=0$ means that the partial correlation of the $j$-th and $k$-th variable are equal to $0$.
Let me mention some caveats though. First, if you have some more knowledge about your $X_i$'s, you could get better performance by using another objective (that keeps $\Sigma$ "close'' to $S_n$) or another regularizer of your choice. Presumably such choices could help more under non-Gaussianity. A second difficulty outside of Gaussianity could be inference, but I think even with Gaussianity, the Graphical LASSO is typically used in a more exploratory way or just to get point estimates of the covariance matrix or the partial correlation graph. So that would still be fine.
As a final remark: the situation is very similar e.g., to the regular LASSO. The LASSO penalty is the sum of the log-likelihood of homoskedastic Gaussian measurements and the $L_1$ regularizer. But the objective (negative of squared euclidean norm of residuals) makes sense also for other noise models and we use it all the time!
