Title: High-Dimensional Dependency Structure Learning for Physical Processes,
Abstract: In this paper, we consider the use of structure learning methods for
probabilistic graphical models to identify statistical dependencies in
high-dimensional physical processes. Such processes are often synthetically
characterized using PDEs (partial differential equations) and are observed in a
variety of natural phenomena, including geoscience data capturing atmospheric
and hydrological phenomena. Classical structure learning approaches such as the
PC algorithm and variants are challenging to apply due to their high
computational and sample requirements. Modern approaches, often based on sparse
regression and variants, do come with finite sample guarantees, but are usually
highly sensitive to the choice of hyper-parameters, e.g., parameter $\lambda$
for sparsity inducing constraint or regularization. In this paper, we present
ACLIME-ADMM, an efficient two-step algorithm for adaptive structure learning,
which estimates an edge specific parameter $\lambda_{ij}$ in the first step,
and uses these parameters to learn the structure in the second step. Both steps
of our algorithm use (inexact) ADMM to solve suitable linear programs, and all
iterations can be done in closed form in an efficient block parallel manner. We
compare ACLIME-ADMM with baselines on both synthetic data simulated by partial
differential equations (PDEs) that model advection-diffusion processes, and
real data (50 years) of daily global geopotential heights to study information
flow in the atmosphere. ACLIME-ADMM is shown to be efficient, stable, and
competitive, usually better than the baselines especially on difficult
problems. On real data, ACLIME-ADMM recovers the underlying structure of global
atmospheric circulation, including switches in wind directions at the equator
and tropics entirely from the data.