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.

翻译：广义标签多伯努利（GLMB）密度在多目标系统应用中广泛出现，类似于高斯分布在单目标滤波中的作用。然而，计算GLMB滤波密度需要解决NP难题。为缓解这一计算瓶颈，我们开发了一种线性复杂度的吉布斯采样框架用于GLMB密度计算。具体而言，我们提出了一种调温吉布斯采样器，利用GLMB滤波密度的结构特性，实现了$\mathcal{O}(T(P+M))$的复杂度，其中$T$为算法迭代次数，$P$和$M$分别为假设的目标数和测量数。这一创新使GLMB滤波器的实现复杂度从$\mathcal{O}(TP^{2}M)$降至$\mathcal{O}(T(P+M+\log T)+PM)$。此外，所提出的框架提供了跟踪性能与计算负载之间权衡的灵活性。我们证明了所提吉布斯采样器的收敛性，并通过数值研究验证了所提GLMB滤波器实现的有效性。