Kernel Regularization in Frequency Domain: Encoding High-Frequency Decay Property

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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) memory and O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) flops, where n is the impulse response length usually satisfying N ≪ n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> in regularization frameworks.