A Multi-Scan Labeled Random Finite Set Model for Multi-object State Estimation

State space models in which the system state is a finite set--called the multi-object state--have generated considerable interest in recent years. Smoothing for state space models provides better estimation performance than filtering by using the full posterior rather than the filtering density. In multi-object state estimation, the Bayes multi-object filtering recursion admits an analytic solution known as the Generalized Labeled Multi-Bernoulli (GLMB) filter. In this work, we extend the analytic GLMB recursion to propagate the multi-object posterior. We also propose an implementation of this so-called multi-scan GLMB posterior recursion using a similar approach to the GLMB filter implementation

Graph matching: relax or not?

We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly-stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving 2n equality and n^2 inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic