On Learning Continuous Pairwise Markov Random Fields

Abstract

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) [39] to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) [39] can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression à la Lasso, which may be of interest in its own right.

Learning Continuous Exponential Families Beyond Gaussian

“…Unlike pseudo-likelihood, the Interaction Screening estimator does not include a normalization factor, and can be generalized to the case of noisy or corrupted data (Goel et al, 2019). In a recent work (Shah et al, 2020), the Interaction Screening method has been adapted to the setting of continuous graphical models, however this analysis was restricted to distributions with pairwise interactions, and more importantly to distributions with bounded support. Indeed, as we explain later in this paper, the vanilla Interaction Screening estimator introduced in (Vuffray et al, 2020) does not apply to continuous distributions with unbounded support.…”