标题： 高维中自助粒子滤波器的尖锐失败率 显示英文标题

标题： Sharp failure rates for the bootstrap particle filter in high dimensions

摘要： 我们证明，在高维高斯粒子滤波器中，样本重要性权重的最大值除非集合大小随系统维度呈指数增长，否则将收敛到1。 我们的工作受到Bengtsson、Bickel和Li（2007）的推导启发，并与之平行；然而，我们削弱了他们对先验分布协方差矩阵特征值的假设，并严格建立了他们关于权重崩溃发生时机的强猜想。 具体而言，我们去除了非零特征值远离零的假设，尽管相关向量的维度趋于无穷大，这一假设本质上允许有效系统维度保持有界。 此外，在最大特征值增长速率的一些限制下，我们放宽了他们的假设，即特征值被从上方限制，从而允许系统由单一模式主导。 显示英文摘要

摘要： We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.