提出新型张量网络平方根卡尔曼滤波，解决高维在线高斯过程回归中的发散问题。
Tensor network square root Kalman filter for online Gaussian process regression
- 采用平方根形式保持协方差矩阵正定性，避免传统方法的数值发散
- 在标准笔记本上成功估计4^14个参数，精度与不确定性量化优于现有方法
- 适用于需要实时更新且高维参数的系统辨识任务
当前最先进的张量网络卡尔曼滤波器虽能缓解高维递归估计的维度灾难，但因需进行截断近似，常导致协方差矩阵失去正定性，引发滤波发散。本文首次提出张量网络平方根卡尔曼滤波器，将其应用于高维在线高斯过程回归。实验表明，当使用满秩张量网络时，该方法等价于经典卡尔曼滤波。在真实系统辨识任务中，我们在普通笔记本上成功估计了$4^{14}$个参数，所获模型在预测精度和不确定性量化方面均优于当前最优的张量网络卡尔曼滤波器。
原文摘要 · Abstract (English)
The state-of-the-art tensor network Kalman filter lifts the curse of dimensionality for high-dimensional recursive estimation problems. However, the required rounding operation can cause filter divergence due to the loss of positive definiteness of covariance matrices. We solve this issue by developing, for the first time, a tensor network square root Kalman filter, and apply it to high-dimensional online Gaussian process regression. In our experiments, we demonstrate that our method is equivalent to the conventional Kalman filter when choosing a full-rank tensor network. Furthermore, we apply our method to a real-life system identification problem where we estimate $4^{14}$ parameters on a standard laptop. The estimated model outperforms the state-of-the-art tensor network Kalman filter in terms of prediction accuracy and uncertainty quantification.
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