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Exactly sparse Gaussian variational inference with application to derivative-free batch nonlinear state estimation
The International Journal of Robotics Research ( IF 9.2 ) Pub Date : 2020-07-29 , DOI: 10.1177/0278364920937608
Timothy D Barfoot 1 , James R Forbes 2 , David J Yoon 1
Affiliation  

We present a Gaussian variational inference (GVI) technique that can be applied to large-scale nonlinear batch state estimation problems. The main contribution is to show how to fit both the mean and (inverse) covariance of a Gaussian to the posterior efficiently, by exploiting factorization of the joint likelihood of the state and data, as is common in practical problems. This is different than maximum a posteriori (MAP) estimation, which seeks the point estimate for the state that maximizes the posterior (i.e., the mode). The proposed exactly sparse Gaussian variational inference (ESGVI) technique stores the inverse covariance matrix, which is typically very sparse (e.g., block-tridiagonal for classic state estimation). We show that the only blocks of the (dense) covariance matrix that are required during the calculations correspond to the non-zero blocks of the inverse covariance matrix, and further show how to calculate these blocks efficiently in the general GVI problem. ESGVI operates iteratively, and while we can use analytical derivatives at each iteration, Gaussian cubature can be substituted, thereby producing an efficient derivative-free batch formulation. ESGVI simplifies to precisely the Rauch–Tung–Striebel (RTS) smoother in the batch linear estimation case, but goes beyond the ‘extended’ RTS smoother in the nonlinear case because it finds the best-fit Gaussian (mean and covariance), not the MAP point estimate. We demonstrate the technique on controlled simulation problems and a batch nonlinear simultaneous localization and mapping problem with an experimental dataset.

中文翻译:

应用于无导数批量非线性状态估计的精确稀疏高斯变分推理

我们提出了一种高斯变分推理 (GVI) 技术,该技术可应用于大规模非线性批量状态估计问题。主要贡献是展示如何通过利用状态和数据的联合似然的分解来有效地将高斯的均值和(逆)协方差拟合到后验,这在实际问题中很常见。这与最大后验 (MAP) 估计不同,后者寻求最大化后验(即模式)的状态的点估计。所提出的精确稀疏高斯变分推理 (ESGVI) 技术存储逆协方差矩阵,该矩阵通常非常稀疏(例如,用于经典状态估计的块三对角)。我们展示了计算过程中所需的(密集)协方差矩阵的唯一块对应于逆协方差矩阵的非零块,并进一步展示了如何在一般 GVI 问题中有效地计算这些块。ESGVI 迭代运行,虽然我们可以在每次迭代中使用解析导数,但可以替代高斯体积,从而产生高效的无导数批处理公式。ESGVI 在批量线性估计情况下精确地简化为 Rauch-Tung-Striebel (RTS) 平滑器,但在非线性情况下超越了“扩展”RTS 平滑器,因为它找到了最佳拟合的高斯(均值和协方差),而不是MAP 点估计。
更新日期:2020-07-29
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