An Optimal Transport Formulation of Bayes’ Law for Nonlinear Filtering Algorithms
An Optimal Transport Formulation of Bayes’ Law for Nonlinear Filtering Algorithms
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DOI：
10.1109/cdc51059.2022.9992776
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发表时间：
2022-03
期刊：
影响因子：
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通讯作者：
A. Taghvaei;Bamdad Hosseini
中科院分区：
文献类型：
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作者：
A. Taghvaei;Bamdad Hosseini
本文利用最优运输理论给出了贝叶斯定律的变分表示。变分表示是在（状态，观察）的联合分布和它们的独立耦合之间的最佳传输。通过对传输映射施加一定的结构，变分问题的解决方案被用来构造一个Brenier型映射，该映射将观测信号的任何值的先验分布传输到后验分布。新的配方是用来推导出最佳的运输形式的EnjunctiveKalman滤波器（EnKF）的离散时间滤波问题，并提出了一种新的扩展EnKF的非高斯设置，利用输入凸神经网络。最后，所提出的方法是用来推导出最佳的传输形式的反馈粒子填料（FPF）在连续的时间限制，这构成了它的第一个变分结构，而不明确使用非线性滤波方程或贝叶斯定律。
This paper presents a variational representation of the Bayes’ law using optimal transportation theory. The variational representation is in terms of the optimal transportation between the joint distribution of the (state, observation) and their independent coupling. By imposing certain structure on the transport map, the solution to the variational problem is used to construct a Brenier-type map that transports the prior distribution to the posterior distribution for any value of the observation signal. The new formulation is used to derive the optimal transport form of the Ensemble Kalman filter (EnKF) for the discrete-time filtering problem and propose a novel extension of EnKF to the non-Gaussian setting utilizing input convex neural networks. Finally, the proposed methodology is used to derive the optimal transport form of the feedback particle filler (FPF) in the continuous-time limit, which constitutes its first variational construction without explicitly using the nonlinear filtering equation or Bayes’ law.