The conditional censored graphical lasso estimator

Abstract

The Gaussian graphical model is one of the most used tools for inferring genetic networks. Nowadays, the data are often collected from different sources or under different biological conditions, resulting in heterogeneous datasets that exhibit a dependency structure that varies across groups. The complex structure of these data is typically recovered using regularized inferential procedures that use two penalties, one that encourages sparsity within each graph and the other that encourages common structures among the different groups. To this date, these approaches have not been developed for handling the case of censored data. However, these data are often generated by gene expression technologies such as RT-qPCR experiments. In this paper, we fill this gap and propose an extension of joint Gaussian graphical modelling to account for censored, or more generally missing, data.

Sparse inference of the human hematopoietic system from heterogeneous and partially observed genomic data

“…The choice of this parametrization, which is different to the one proposed in earlier related work (Augugliaro et al, 2020b;Sohn and Kim, 2012), comes from the specific scientific questions of the current analysis. Indeed, the density function of Z given above can be factorized as follows:…”

“…Finally, considering the sub-problem (5.12) and using again (5.13), we have: (5.15) which represents an extension of the conditional glasso problem studied by various authors (Rothman et al, 2010;Yin and Li, 2011;Augugliaro et al, 2020b) to the case of multiple conditions. Since for any fixed { ψ}, the function Q y|x ({ ψ}, {B}, {Θ}) is a bi-convex function of {B} and {Θ}, the maximization problem (5.15) can be carried out by repeating the two sub-steps of estimation of {B} and estimation of {Θ}, respectively, until a convergence criterion is met.…”

“…Going then into the detail of problem (5.16), while it could be solved, in principle, by the multi-lasso algorithm of Augugliaro et al (2020b), this algorithm tends to be unstable when the response vectors are highly correlated, as it is based on the idea of optimizing one column of each regression coefficient matrix at a time until a convergence criterion is met. This shortcoming motivated us to develop a dedicated ADMM.…”

“…With regards to the first problem, using Ibrahim et al (2008) and Augugliaro et al (2020b), we replace the exact log-likelihood function with the Q-function used in the M-Step of the main algorithm, i.e. we use the following approximation:…”

“…On the other hand, in a high-dimensional setting, one can use two values small enough to avoid the instability of the model. With respect to the largest values of the two tuning parameters, combining the results of Theorems 1 and 2 of Danaher et al (2014) and of Theorem 3 of Augugliaro et al (2020b) the formulae for computing λ max and ρ max are given, respectively, by…”