Hyperbolic partial differential equations generalize LSSL with the HiPPO matrix

Abstract The LSSL with HiPPO matrix is a model for the modeling of time series data that can handle long-range dependencies. In a previous paper, we showed that the LSSL with HiPPO matrix can be expressed as a discretization of a partial differential equation. In this paper, we generalize this result. We show the LSSL with HiPPO matrix can be regarded as another discretization of the partial differential equation under another condition as well, and also propose a generalization of the HiPPO matrix by different discretizations of partial differential equations. Our result can extend existing state-space models to infinite-dimensional settings. In addition, we perform thorough numerical experiments to evaluate the proposed generalized model and confirm that the performance of the model can certainly be improved by this generalization.