Linear Complexity Gibbs Sampling for Generalized Labeled Multi-Bernoulli Filtering

Generalized Labeled Multi-Bernoulli (GLMB) densities arise in a host of multi-object system applications analogous to Gaussians in single-object filtering. However, computing the GLMB filtering density requires solving NP-hard problems. To alleviate this computational bottleneck, we develop a linear complexity Gibbs sampling framework for GLMB density computation. Specifically, we propose a tempered Gibbs sampler that exploits the structure of the GLMB filtering density to achieve an
$\mathcal {O}(T(P+M))$
complexity, where
$T$
is the number of iterations of the algorithm,
$P$
and
$M$
are the number hypothesized objects and measurements. This innovation enables the GLMB filter implementation to be reduced from an
$\mathcal {O}(TP^{2}M)$
complexity to
$\mathcal {O}(T(P+M+\log T)+PM)$
. Moreover, the proposed framework provides the flexibility for trade-offs between tracking performance and computational load. Convergence of the proposed Gibbs sampler is established, and numerical studies are presented to validate the proposed GLMB filter implementation.