State Space Model with Continuous Limit of HiPPO Matrix: Eigenvalue Analysis and Explicit Solution Formula

As a lightweight model for sequence processing, an LSSL that uses the HiPPO matrix has been proposed. In this paper, as the continuous limit of the HiPPO Matrix, we propose Continuized-HiPPO Operator. Furthermore, as examples of advantages obtained by using this operator, we show that one can analyze the asymptotic behavior of eigenvalues of a matrix related to the HiPPO matrix, and that the LSSL induced by the continuous limit Continuized-HiPPO Operator admits an explicit solution.

This paper studies the HiPPO matrix in the state space models, which are efficient sequence models to remember past information. We replace this matrix with a continuous operator, making its behavior easier to analyze. This viewpoint lets us prove an eigenvalue formula related to a previous conjecture and derive an explicit solution for the corresponding state space model.