An Analysis of Convex Relaxations for MAP Estimation of Discrete MRFs
Mudigonda Pawan Kumar, V. Kolmogorov and Philip Torr
Journal of Machine Learning Research
The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random
field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard.
However, due to its central importance in many applications, several approximation algorithms have
been proposed in the literature. In this paper, we present an analysis of three such algorithms based
on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger
(1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), andWainwright
et al. (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of
Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation
first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended
by Kumar et al. (2006) for a general label set.
We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove
that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP,
the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCPMS.
We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations
which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP
relaxations which define constraints using random variables that form cycles or cliques in the graphical
model representation of the random field. Using some examples we show that the new SOCP
relaxations strictly dominate the previous approaches.