In the context of state-space models, skeleton-based smoothing algorithms rely on a backward sampling step which by default has a $\mathcal O(N^2)$ complexity (where $N$ is the number of particles). Existing improvements in the literature are unsatisfactory: a popular rejection sampling -- based approach, as we shall show, might lead to badly behaved execution time; another rejection sampler with stopping lacks complexity analysis; yet another MCMC-inspired algorithm comes with no stability guarantee. We provide several results that close these gaps. In particular, we prove a novel non-asymptotic stability theorem, thus enabling smoothing with truly linear complexity and adequate theoretical justification. We propose a general framework which unites most skeleton-based smoothing algorithms in the literature and allows to simultaneously prove their convergence and stability, both in online and offline contexts. Furthermore, we derive, as a special case of that framework, a new coupling-based smoothing algorithm applicable to models with intractable transition densities. We elaborate practical recommendations and confirm those with numerical experiments.

翻译：在状态空间模型框架下，基于骨架的平滑算法依赖于一个默认计算复杂度为$\mathcal O(N^2)$（其中$N$为粒子数）的向后采样步骤。现有文献中的改进方案存在不足：一种流行的基于拒绝采样的方法（如我们将要展示的）会导致执行时间不稳定；另一种带停止条件的拒绝采样器缺乏复杂度分析；还有一种受MCMC启发的算法无法保证稳定性。我们提供了多项研究成果来填补这些空白。具体而言，我们证明了一个新颖的非渐近稳定性定理，从而实现了具有真正线性复杂度和充分理论论证的平滑算法。我们提出了一个统一框架，该框架涵盖了文献中大多数基于骨架的平滑算法，并能够同时证明这些算法在在线和离线场景下的收敛性与稳定性。此外，作为该框架的特例，我们推导出一种适用于转移密度不易处理模型的新型耦合平滑算法。我们制定了实用建议，并通过数值实验验证了这些建议。