Interior point solution of fractional Bethe permanent

Many combinatorial problems in fields such as object tracking involve reasoning over correspondence, e.g, calculating the probability that a measurement belongs to a particular track. Recent studies have shown that loopy belief propagation (LBP) provides a highly desirable option in the trade-off between accuracy and computational complexity in this task. LBP can be understood as a particular method for optimising the Bethe free energy (BFE). In this paper, we directly optimise the BFE using an interior point Newton method. Exploiting the structure of the constraints, we arrive at an algorithm offers improvements in computation in cases in which LBP converges very slowly. The method also solves the recently-proposed fractional free energy (FFE); we use this to demonstrate that FFE can offer marginal estimates with improved accuracy.