Tensor network square root Kalman filter for online Gaussian process regression
This work addresses a critical stability problem in high-dimensional recursive estimation for applications like system identification, though it is incremental as it builds on existing tensor network methods.
The authors tackled the issue of filter divergence in tensor network Kalman filters due to loss of positive definiteness in covariance matrices by developing a tensor network square root Kalman filter for online Gaussian process regression. They demonstrated equivalence to conventional methods at full rank and applied it to a real-life system identification problem, outperforming the state-of-the-art tensor network Kalman filter in prediction accuracy and uncertainty quantification while estimating 4^14 parameters on a standard laptop.
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.