Accurate estimation of the states of a nonlinear dynamical system is crucial for their design, synthesis, and analysis. Particle filters are estimators constructed by simulating trajectories from a sampling distribution and averaging them based on their importance weight. For particle filters to be computationally tractable, it must be feasible to simulate the trajectories by drawing from the sampling distribution. Simultaneously, these trajectories need to reflect the reality of the nonlinear dynamical system so that the resulting estimators are accurate. Thus, the crux of particle filters lies in designing sampling distributions that are both easy to sample from and lead to accurate estimators. In this work, we propose to learn the sampling distributions. We put forward four methods for learning sampling distributions from observed measurements. Three of the methods are parametric methods in which we learn the mean and covariance matrix of a multivariate Gaussian distribution; each methods exploits a different aspect of the data (generic, time structure, graph structure). The fourth method is a nonparametric alternative in which we directly learn a transform of a uniform random variable. All four methods are trained in an unsupervised manner by maximizing the likelihood that the states may have produced the observed measurements. Our computational experiments demonstrate that learned sampling distributions exhibit better performance than designed, minimum-degeneracy sampling distributions.

翻译：非线性动态系统状态的精确估计对其设计、综合与分析至关重要。粒子滤波器是一种估计器，通过从采样分布中模拟轨迹并根据重要性权重进行加权平均来构建。为使粒子滤波器具有计算可行性，必须能够通过从采样分布中抽取样本来模拟轨迹。同时，这些轨迹需要反映非线性动态系统的实际状态，以确保所得到的估计器具有准确性。因此，粒子滤波器的核心在于设计既易于采样又能生成精确估计器的采样分布。本文提出学习采样分布的方法。我们提出了四种从观测数据中学习采样分布的方法。其中三种为参数化方法，通过学习多元高斯分布的均值和协方差矩阵实现；每种方法利用数据的不同特征（通用性、时间结构、图结构）。第四种方法是一种非参数替代方案，直接学习均匀随机变量的变换。所有四种方法均通过最大化状态生成观测数据的可能性进行无监督训练。我们的计算实验表明，学习得到的采样分布比人工设计的、最小退化采样分布表现出更优的性能。