Enabling robots to act in unstructured and unknown environments requires versatile state estimation techniques. While traditional state estimation methods require known models and make strong assumptions about the dynamics, such versatile techniques should be able to deal with high dimensional observations and non-linear, unknown system dynamics. The recent framework for nonparametric inference allows to perform inference on arbitrary probability distributions. High-dimensional embeddings of distributions into reproducing kernel Hilbert spaces are manipulated by kernelized inference rules, most prominently the kernel Bayes’ rule (KBR). However, the computational demands of the KBR do not scale with the number of samples. In this paper, we present two techniques to increase the computational efficiency of non-parametric inference. First, the kernel Kalman rule (KKR) is presented as an approximate alternative to the KBR that estimates the embedding of the state based on a recursive least squares objective. Based on the KKR we present the kernel Kalman filter (KKF) that updates an embedding of the belief state and learns the system and observation models from data. We further derive the kernel forward backward smoother (KFBS) based on a forward and backward KKF and a smoothing update in Hilbert space. Second, we present the subspace conditional embedding operator as a sparsification technique that still leverages from the full data set. We apply this sparsification to the KKR and derive the corresponding sparse KKF and KFBS algorithms. We show on nonlinear state estimation tasks that our approaches provide a significantly improved estimation accuracy while the computational demands are considerably decreased.

中文翻译：

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核卡尔曼规则

使机器人能够在非结构化和未知环境中行动需要通用的状态估计技术。虽然传统的状态估计方法需要已知模型并对动力学做出强有力的假设，但这种通用技术应该能够处理高维观测和非线性、未知的系统动力学。最近的非参数推理框架允许对任意概率分布进行推理。分布到再现内核希尔伯特空间的高维嵌入由内核化推理规则操纵，最突出的是内核贝叶斯规则 (KBR)。然而，KBR 的计算需求不随样本数量而变化。在本文中，我们提出了两种技术来提高非参数推理的计算效率。第一的，核卡尔曼规则 (KKR) 被呈现为 KBR 的近似替代方案，它基于递归最小二乘目标估计状态的嵌入。基于 KKR，我们提出了内核卡尔曼滤波器 (KKF)，它更新置信状态的嵌入并从数据中学习系统和观察模型。我们进一步推导出基于前向和后向 KKF 以及希尔伯特空间中的平滑更新的内核前向后向平滑器（KFBS）。其次，我们将子空间条件嵌入算子作为一种仍然利用完整数据集的稀疏化技术。我们将这种稀疏化应用于 KKR 并推导出相应的稀疏 KKF 和 KFBS 算法。