Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields

Abstract

Dual decomposition approaches in nonconvex optimization may suffer from a duality gap. This poses a challenge when applying them directly to nonconvex problems such as MAP-inference in a Markov random field (MRF) with continuous state spaces. To eliminate such gaps, this paper considers a reformulation of the original nonconvex task in the space of measures. This infinite-dimensional reformulation is then approximated by a semi-infinite one, which is obtained via a piecewise polynomial discretization in the dual. We provide a geometric intuition behind the primal problem induced by the dual discretization and draw connections to optimization over moment spaces. In contrast to existing discretizations which suffer from a grid bias, we show that a piecewise polynomial discretization better preserves the continuous nature of our problem. Invoking results from optimal transport theory and convex algebraic geometry we reduce the semi-infinite program to a finite one and provide a practical implementation based on semidefinite programming. We show, experimentally and in theory, that the approach successfully reduces the duality gap. To showcase the scalability of our approach, we apply it to the stereo matching problem between two images.

A Cutting-Plane Method for Sublabel-Accurate Relaxation of Problems with Product Label Spaces

“…A well-known approach to the global optimization of ( 3) is a lifting or stochastic relaxation procedure, which has been considered in diverse fields such as polynomial optimization (Lasserre, J-B. 2000), continuous Markov random fields (Fix and Agarwal 2014;Peng et al 2011;Bauermeister et al 2021), variational methods (Pock et al 2008), and black-box optimization (de Boer et al 2005;Ollivier et al 2017;Schaul 2011). The idea is to relax the search space in (3) from γ ∈ to probability distributionsu ∈ P( ) and solve 1 min u∈P( )…”