Tensor Kalman Filter and Its Applications - Chang, Shih Yu; Wu, Hsiao-Chun (IEEE transactions on knowledge and data engineering, 2023)
Kalman filter is one of the most important estimation algorithms, which estimates certain unknown variables given the measurements observed over time subject to a dynamic system, for many applications in science and engineering including environmental science, ecometrics, robotics, financial analysis, data mining, etc. With the recent emergence of the Big Data era, it is often necessary to characterize multiple relationships among various kinds of signals/data in tensor form. The conventional Kalman filter paradigm is based on the low-dimensional state-space representation, which is restricted by the state-transition, observation-model, process-noise covariance, and observation-noise covariance matrices. However, we often need to express some or all of them in terms of tensors in practice. Very lately, the aforementioned Kalman filter in tensor form was tackled using tensor decomposition but the exact estimator has never been established so far. In this work, we propose a new generalized Kalman filter framework consisting of state, state-transition model, observation-model, process-noise covariance, and observation-noise covariance tensors of arbitrary orders by applying the Sherman-Morrison-Woodbury identity and block tensor inverse, which we call "Tensor Kalman Filter" (TKF). Our proposed new approach can produce the exact Kalman filter estimator without any need of tensor decomposition (approximation). The pertinent computational- and memory-complexity studies are also provided in this paper. Finally, numerical experiments are conducted to evaluate the prediction performance of the proposed new TKF over biomedical signal data in comparison with other existing high-dimensional prediction methods.