AdaPtive Noisy Data Augmentation (PANDA) for Simultaneous Construction Multiple Graph Models
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We extend the data augmentation technique (PANDA) by Li et al. (2018) for regularizing single graph model estimations to jointly learning the structures of multiple graphs. Our proposed approach provides an unified framework to effectively jointly train multiple graphical models, regardless of the types of nodes. We design and introduce two types of noises to augment the observed data. The first type of noises is to regularize the estimation of each graph while the second type of noises promotes either the structural similarities, referred as the joint group lasso (JGL) regularization, or numerical similarities, referred as the joint fused ridge (JFR) regularization, among the edges in the same position across multiple graphs. The computation in PANDA is straightforward and only involves obtaining maximum likelihood estimator in generalized linear models (GLMs) in an iterative manner. We also extend the JGL and JFR regularization beyond the graphical model settings to variable selection and estimation in GLMs. The multiple graph version of PANDA enjoys the theoretical properties established for single graphs including the almost sure (a.s) convergence of the noise-augmented loss function to its expectation and the a.s convergence of the minimizer of the former to the minimizer of the latter. The simulation studies suggest PANDA is non-inferior to existing joint estimation approaches in constructing multiple Gaussian graphical models (GGMs), and significantly improves over the differencing approach over separately estimated graphs in multiple Poisson graphical models (PGMs). We also applied PANDA to a real-life lung cancer microarray data to simultaneously construct four protein networks.
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