Multiscale Gaussian Graphical Models and Algorithms for Large-Scale Inference

Abstract

We propose a class of multiscale graphical models and algorithms to estimate means and approximate error variances of large-scale Gaussian processes efficiently. Based on emerging techniques for inference on Gaussian graphical models with cycles, we extend traditional multiscale tree models to pyramidal graphs, which incorporate both inter-and intra-scale interactions. In the spirit of multipole algorithms, we develop efficient inference methods in which variables far-apart communicate through coarser resolutions and nearby variables interact at finer resolutions. In addition, we propose methods to update the estimates rapidly when measurements are added or new knowledge of a local region is provided.

Low-Rank Variance Approximation in GMRF Models: Single and Multiscale Approaches

“…The main drawback of tree-structured models is that certain neighbors in the fine-scale model may become quite distant in the tree-structured model, which leads to blocky artifacts in the estimates. To avoid these artifacts, multiscale models that allow loops also received attention, e.g., [13] and [14]. We consider a class of multiscale models on pyramidal graphs with loops described in [13] and [15].…”

“…To avoid these artifacts, multiscale models that allow loops also received attention, e.g., [13] and [14]. We consider a class of multiscale models on pyramidal graphs with loops described in [13] and [15]. The different scales in this model constitute a coherent statistical model with nondeterministic interscale interactions.…”

“…Conditioned on the coarse-scale variables, the conditional correlations among the fine-scale variables are more local. We leave the question of learning such pyramidal models to [13] and describe an extension of our low-rank variance approximation to find variances when such a model is specified in Section IV-C.…”

“…Now let us repeat the analysis for 2-D lattices with a slower, power-law rate of decay, i.e., , where . Then the sum in (12) changes to (13) If , then the sum converges (and is equal to (2 1), the Riemann zeta function), and the errors can be made arbitrarily small by increasing . However, if , then for any , the sum diverges.…”

“…Suppose a continuous-time function is -Lipschitz 12 (this is related to how many times is continuously differentiable). Also suppose that the wavelet family has vanishing moments, 13 with . Then the wavelet coefficients satisfy .…”