An analysis of convex relaxations for MAP estimation
Mudigonda Pawan Kumar, Vladimir Kolmogorov and Philip Torr
In: NIPS 2007, 3-6 Dec 2007, Vancouver, Canada.
The problem of obtaining the maximum a posteriori estimate of a general discrete
random field (i.e. a random field defined using a finite and discrete set of
labels) is known to be NP-hard. However, due to its central importance in many
applications, several approximate 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 for a special case and independently by Chekuri et al.,
Koster et al., and Wainwright et al. for the general case; (ii) QP-RL: the
quadratic programming (QP) relaxation by Ravikumar and Lafferty; and (iii)
SOCP-MS: the second order cone programming (SOCP) relaxation first proposed
by Muramatsu and Suzuki for two label problems and later extended by Kumar et
al. 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 SOCP-MS. 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 strictly dominate the previous approaches.