Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields
This addresses a fundamental challenge in nonconvex optimization for computer vision and machine learning, offering a tractable method to improve inference in continuous MRFs, though it appears incremental as it builds on existing discretization and optimal transport theory.
The paper tackles the duality gap issue in dual decomposition for nonconvex optimization, specifically in MAP-inference for continuous Markov random fields, by reformulating the problem in measure space and using piecewise polynomial discretization to reduce it to a finite semidefinite program, showing experimental and theoretical success in gap reduction and scalability in stereo matching.
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.