Learning Exponential Family Graphical Models with Latent Variables using Regularized Conditional Likelihood
This work addresses the challenge of latent-variable graphical modeling for researchers and practitioners in statistics and machine learning, offering a more flexible approach compared to previous methods, though it appears incremental as it builds on existing convex relaxation techniques.
The paper tackles the problem of fitting graphical models when observed variables are influenced by latent variables, which can cause confounding dependencies, by proposing a new convex relaxation framework based on regularized conditional likelihood for exponential family graphical models. The result is a method applicable in broader settings, such as non-Gaussian data, without requiring knowledge of latent variable distributions, demonstrated through numerical experiments on synthetic and real data.
Fitting a graphical model to a collection of random variables given sample observations is a challenging task if the observed variables are influenced by latent variables, which can induce significant confounding statistical dependencies among the observed variables. We present a new convex relaxation framework based on regularized conditional likelihood for latent-variable graphical modeling in which the conditional distribution of the observed variables conditioned on the latent variables is given by an exponential family graphical model. In comparison to previously proposed tractable methods that proceed by characterizing the marginal distribution of the observed variables, our approach is applicable in a broader range of settings as it does not require knowledge about the specific form of distribution of the latent variables and it can be specialized to yield tractable approaches to problems in which the observed data are not well-modeled as Gaussian. We demonstrate the utility and flexibility of our framework via a series of numerical experiments on synthetic as well as real data.