Moment Matching by Kernel-Based Learning

We introduce a kernel-based moment matching theory which relies upon a novel data-driven model reduction method that employs the estimation of moments within a Reproducing Kernel Hilbert Space. We demonstrate that moment estimation can be enhanced by appropriately tuning the regularization term, regardless of the kernel choice. Additionally, we present conditions to ensure that the Reproducing Kernel Hilbert Space contains only functions which are bona fide moments. While exact moment matching with finite data is impractical in this scenario, we introduce the concepts of weak moment matching and moment matching almost everywhere onto the $\mathcal {L}_{2}$ -space. Additionally, we address scenarios in which the dataset contains noisy measurements of outputs that are not yet in a steady-state, which typically biases the estimation due to the effect of the output transients. We further prove that estimating over a Reproducing Kernel Hilbert Space can ensure weak moment matching asymptotically and, with additional assumptions, also moment matching almost everywhere despite these transients. Finally, we provide a probabilistic bound that guarantees weak moment matching for an arbitrarily finite amount of data.