Gradient Flows in Filtering and Fisher-Rao Geometry
Gradient Flows in Filtering and Fisher-Rao Geometry
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过滤和 Fisher-Rao 几何中的梯度流
DOI：
10.23919/acc.2018.8431003
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
T. Georgiou
中科院分区：
文献类型：
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作者：
A. Halder;T. Georgiou
不确定性传播和过滤可以被解释为相对于概率密度函数的无限维流形中的合适度量的梯度流。这样的观点已经提出了在最近的文献中，和一个系统的方法来制定和解决同样的线性高斯系统已经出现在我们以前的工作中，梯度流是通过近端运营商实现相对于Wasserstein度量产生的最佳质量传输。本文导出了信息几何中关于Fisher-Rao度量的近似算子形式的演化方程。我们开发的线性高斯的情况下，详细地表明，模板两步优化过程中提出的作者仍然适用。我们的目标是提供新的几何解释的已知方程的过滤，并澄清不同的度量选择的含义。
Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and a systematic way to formulate and solve the same for linear Gaussian systems has appeared in our previous work where the gradient flows were realized via proximal operators with respect to Wasserstein metric arising in optimal mass transport. In this paper, we derive the evolution equations as proximal operators with respect to Fisher-Rao metric arising in information geometry. We develop the linear Gaussian case in detail and show that a template two step optimization procedure proposed earlier by the authors still applies. Our objective is to provide new geometric interpretations of known equations in filtering, and to clarify the implication of different choices of metric.