Nyström subsampling for functional linear regression

Kernel methods have proven to be highly effective for functional data analysis, demonstrating significant theoretical and practical success over the past two decades. However, their computational complexity and storage requirements hinder their direct application to large-scale functional data learning problems. In this paper, we address this limitation by investigating the theoretical properties of the Nyström subsampling method within the framework of the functional linear regression model and reproducing kernel Hilbert space. Our proposed algorithm not only overcomes the computational challenges but also achieves the minimax optimal rate of convergence for the excess prediction risk, provided an appropriate subsampling size is chosen. Our error analysis relies on the approximation of integral operators induced by the reproducing kernel and covariance function.