Abstract
This letter discusses the kernel regularization in the frequency domain. In particular, this letter proposes a new kernel which encodes prior knowledge on the rate of high frequency decay. The proposed kernel has a similar structure to the one of the first order spline kernel. By exploiting the known properties of such kernel, the determinant and the inverse of the Gram matrix of the proposed kernel are given in closed form. One of the important advantages of the proposed kernel is the computational burden reduction. In fact, it turns out that the complexity is linear in the dataset size N, while standard methods require O(n 2 ) memory and O(n 3 ) flops, where n is the impulse response length usually satisfying N ≪ n 2 in regularization frameworks.
Highlights
B LANCING model complexity and data fit is one of the key issues in system identification field (e.g., [1, Ch. 16])
Many works on kernel regularization have been reported; e.g., kernel design [6], [7], kernel properties [8]–[10], hyperparameter tuning [11]–[13], input design [14]–[16], and so on
This letter proposes a new kernel regularization method that exploits a prior knowledge in the frequency domain, i.e., the high frequency decay property
Summary
B LANCING model complexity and data fit is one of the key issues in system identification field (e.g., [1, Ch. 16]). In kernel-based identification for linear systems, the unknown impulse response is estimated via regularized least squares. The advantage of such approach w.r.t. classic parametric methods is that the trade-off between data fit and model complexity is ruled by a real parameter instead of a discrete value, allowing for more flexibility. This letter employs the high frequency decay rate to design the regularization term, and reformulates the regularized least squares problem in the frequency domain This reformulation drastically reduces the computational burden. The main contributions of this letter are the following: It proposes a quadratic regularization based on a prior knowledge in the frequency domain, i.e., the rate of high frequency decay.
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