Convergence Analysis of regularised Nystr\"om method for Functional Linear Regression

The functional linear regression model has been widely studied and utilized for dealing with functional predictors. In this paper, we study the Nystr\"om subsampling method, a strategy used to tackle the computational complexities inherent in big data analytics, especially within the domain of functional linear regression model in the framework of reproducing kernel Hilbert space. By adopting a Nystr\"om subsampling strategy, our aim is to mitigate the computational overhead associated with kernel methods, which often struggle to scale gracefully with dataset size. Specifically, we investigate a regularization-based approach combined with Nystr\"om subsampling for functional linear regression model, effectively reducing the computational complexity from $O(n^3)$ to $O(m^2 n)$, where $n$ represents the size of the observed empirical dataset and $m$ is the size of subsampled dataset. Notably, we establish that these methodologies will achieve optimal convergence rates, provided that the subsampling level is appropriately selected. We have also demonstrated a numerical example of Nystr\"om subsampling in the RKHS framework for the functional linear regression model.