  In this paper, we propose a new estimation procedure for discovering the
structure of Gaussian Markov random fields (MRFs) with false discovery rate
(FDR) control, making use of the sorted l1-norm (SL1) regularization. A
Gaussian MRF is an acyclic graph representing a multivariate Gaussian
distribution, where nodes are random variables and edges represent the
conditional dependence between the connected nodes. Since it is possible to
learn the edge structure of Gaussian MRFs directly from data, Gaussian MRFs
provide an excellent way to understand complex data by revealing the dependence
structure among many inputs features, such as genes, sensors, users, documents,
etc. In learning the graphical structure of Gaussian MRFs, it is desired to
discover the actual edges of the underlying but unknown probabilistic graphical
model-it becomes more complicated when the number of random variables
(features) p increases, compared to the number of data points n. In particular,
when p >> n, it is statistically unavoidable for any estimation procedure to
include false edges. Therefore, there have been many trials to reduce the false
detection of edges, in particular, using different types of regularization on
the learning parameters. Our method makes use of the SL1 regularization,
introduced recently for model selection in linear regression. We focus on the
benefit of SL1 regularization that it can be used to control the FDR of
detecting important random variables. Adapting SL1 for probabilistic graphical
models, we show that SL1 can be used for the structure learning of Gaussian
MRFs using our suggested procedure nsSLOPE (neighborhood selection Sorted L-One
Penalized Estimation), controlling the FDR of detecting edges.
