Efficient Factorisation-based Gaussian Process Approaches for Online Tracking

Abstract

Target tracking often relies on complex models with non-stationary parameters. Gaussian process (GP) is a modelfree method that can achieve accurate performance. However, the inverse of the covariance matrix poses scalability challenges. Since the covariance matrix is typically dense, direct inversion and determinant evaluation methods suffer from cubic complexity to data size. This bottleneck limits the GP for long-term tracking or high-speed tracking. We present an efficient factorisationbased GP approach without any additional hyperparameters. The proposed approach reduces the computational complexity of the Cholesky decomposition by hierarchically factorising the covariance matrix into off-diagonal low-rank parts. Meanwhile, the resulting low-rank approximated Cholesky factor can also reduce the computation complexity of the inverse and the determinant operations. Numerical results based on offline and online tracking problems demonstrate the effectiveness of the proposed approach.

Gaussian Process Upper Confidence Bounds in Distributed Point Target Tracking Over Wireless Sensor Networks

“…The reduction in computational complexity of DGP can also be justified by looking into (22), where both the computations of determinant and inversion are only based on a much smaller matrix Σ (i) . In addition, as compared to (21), the factorized marginal likelihood can potentially be maximized in a decentralized manner like federated learning [57] since it is a summation over local marginal likelihood functions.…”

“…To cope with the clutter and for learning the GP hyperparameters, a method is designed to assign weights to different measurements based on the marginal likelihood (21). A weighted summation over measurements collected by one sensor is then calculated as the training data for DGP.…”

“…In addition, in Figure 2, the impact of the number of local GPs on the running time for computing the log marginal likelihood and its gradient is also presented based on the data set with a fixed size of 10000. We can find that having more local GPs can help reduce the running time since according to (21), if the smaller size of covariance matrices are generated, solving the matrix determinant and inversion can be much faster on these smaller matrices.…”

“…Since this paper focuses on the model-free approaches, the standard GP-based centralized tracking approach is simulated as the benchmark. This scheme relies on solving the MLE (21) to learn the hyperparameters, and the learning process requires the measurements to be transmitted in the WSN. To make fair comparisons, the standard GP-based centralized tracking approach is trained with the same sensor measurements in the sliding window, which means this approach uses the same amount of data for model training and hyperparameter learning as well as the DGPT approach.…”

“…Ignoring this uncertainty can have disastrous consequences, especially when the output of such models is then fed into higher-level decision-making procedures. Recently, GP regression has been applied to solve both point target tracking [15]- [21] and extended target tracking [22]- [24] problems.…”