Tightening Fractional Covering Upper Bounds on the Partition Function for High-Order Region Graphs
This work addresses computational challenges in probabilistic inference for researchers and practitioners dealing with large graphical models, though it appears incremental as it builds on existing fractional covering and entropy barrier methods.
The paper tackles the problem of computing tighter upper bounds on partition functions for high-order region graphs by developing a new approach based on fractional covering bounds on entropy, resulting in concave and convex optimization programs that can be solved efficiently for large-scale problems with thousands of regions.
In this paper we present a new approach for tightening upper bounds on the partition function. Our upper bounds are based on fractional covering bounds on the entropy function, and result in a concave program to compute these bounds and a convex program to tighten them. To solve these programs effectively for general region graphs we utilize the entropy barrier method, thus decomposing the original programs by their dual programs and solve them with dual block optimization scheme. The entropy barrier method provides an elegant framework to generalize the message-passing scheme to high-order region graph, as well as to solve the block dual steps in closed-form. This is a key for computational relevancy for large problems with thousands of regions.