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

TL;DR

This paper introduces a novel measure-based reformulation and discretization method for continuous Markov random fields, effectively reducing duality gaps in nonconvex MAP-inference problems using semidefinite programming.

Contribution

It proposes a semi-infinite reformulation with piecewise polynomial discretization that better preserves the continuous problem structure and reduces duality gaps in nonconvex optimization.

Findings

Successfully reduces duality gap in continuous MRFs

Demonstrates scalability on stereo matching tasks

Provides a practical semidefinite programming implementation

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.