Tensor network square root Kalman filter for online Gaussian process regression

TL;DR

This paper introduces a tensor network square root Kalman filter that maintains positive definiteness of covariance matrices, enabling high-dimensional online Gaussian process regression with improved accuracy and stability.

Contribution

It develops the first tensor network square root Kalman filter, addressing divergence issues and demonstrating superior performance in high-dimensional regression tasks.

Findings

Equivalent to conventional Kalman filter with full-rank tensor network

Successfully estimates 4^{14} parameters on a standard laptop

Outperforms existing tensor network Kalman filter in accuracy and uncertainty quantification

Abstract

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.