Dual Decomposition for Joint Discrete-Continuous Optimization

We analyse convex formulations for combined discrete-continuous MAP inference using the dual decomposition method. As a consquence we can provide a more intuitive derivation for the resulting convex relaxation than presented in the literature. Further, we show how to strengthen the relaxation by reparametrizing the potentials, hence convex relaxations for discrete-continuous inference does not share an important feature of LP relaxations for discrete labeling problems: incorporating unary potentials into higher order ones affects the quality of the relaxation. We argue that the convex model for discrete-continuous inference is very general and can be used as alternative for alternation-based methods often employed for such joint inference tasks.