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 discretecontinuous inference is very general and can be used as alternative for alternation-based methods often employed for such joint inference tasks.