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(Sep 09 1-3pm) Pre-meeting Overview: New Outer Bounds on the Marginal Polytope (Sontag, Jaakkola)

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Meeting will be in room 243.

This week we’ll be covering the following paper from NIPS 2007 (best student paper) – Sontag and Jaakola, “New Outer Bounds on the Marginal Polytope“.

Aside from the paper, we will also need to deal with two organizational issues, when to have our regular meeting for the rest of fall semester, and what topics we want to cover (at least decide upon the first topic to cover).  See the previous organizational post for details.

Paper Overview:

This paper presents a new variational algorithm for inference in discrete (binary & non-binary) MRF’s. Recall that the main difficulty in exact inference is in explicitly characterizing the marginal polytope and in exactly computing the conjugate dual. The main contribution of this paper is a new [tighter] outer-bound on the marginal polytope. For the conjugate dual, their algorithm utilizes existing approximations such as the log-determinant and tree-reweighted (TRW) approximations.

A broader goal of this paper is in highlighting an emerging connection between polyhedral combinatorics and probabilistic inference. To this end, their outer-bound for the marginal polytope is based on a cutting-plane algorithm (Algorithm 1 on page 3). A key aspect cutting-plane algorithms is in having an efficient mechanism for detecting violated constraints (Step 5 of Algorithm 1). One contribution in this paper is in using the cycle inequalities for which efficient separation algorithms are known (Section 2, page 5). A second main contribution is extending these inequalities and the separation mechanism to non-binary MRF’s (Section 4). Notice that the extension to non-pairwise MRF’s is trivial since any non-pairwise MRF can be easily converted to a pairwise one by introducing auxiliary variables (as described in Appendix E.3 of Wainwright & Jordan). The first two pages of the paper provide a nice & succinct summary of most of the points we covered in Wainwright & Jordan.

The experiments in this paper show the improvement in their inference procedure on computing marginals and MAP estimation for protein side-chain prediction.

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