# Tensor network square root Kalman filter for online Gaussian process regression

arXiv: https://arxiv.org/abs/2409.03276 · HTML: https://mlsanity.com/paper/2409.03276/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 5 Sep 2024.

## Builds on (most cited first, 11 of 11)

- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Hilbert Space Methods for Reduced-Rank Gaussian Process Regression](https://mlsanity.com/paper/1401.5508/) (2014)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [Scalable Gaussian Processes with Billions of Inducing Inputs via Tensor Train Decomposition](https://mlsanity.com/paper/1710.07324/) (2017)
- [Kernel Interpolation for Scalable Online Gaussian Processes](https://mlsanity.com/paper/2103.01454/) (2021)
- [Large-Scale Learning with Fourier Features and Tensor Decompositions](https://mlsanity.com/paper/2109.01545/) (2021)
- [Recursive Estimation for Sparse Gaussian Process Regression](https://mlsanity.com/paper/1905.11711/) (2019)
- [Nonlinear system identification with regularized Tensor Network B-splines](https://mlsanity.com/paper/2003.07594/) (2020)
- [The Rank-Reduced Kalman Filter: Approximate Dynamical-Low-Rank Filtering In High Dimensions](https://mlsanity.com/paper/2306.07774/) (2023)
- [Blitzkriging: Kronecker-structured Stochastic Gaussian Processes](https://mlsanity.com/paper/1510.07965/) (2015)
- [A Tensor Network Kalman filter with an application in recursive MIMO Volterra system identification](https://mlsanity.com/paper/1610.05434/) (2016)

## Abstract (arXiv)

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 414 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.