Convergence analysis of regularised Nyström method for functional linear regression

Abstract The functional linear regression model has been widely studied and utilized for dealing with functional predictors. In this paper, we study the Nyström 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 Nyström 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 Nyström subsampling for functional linear regression model, effectively reducing the computational complexity from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> represents the size of the observed empirical dataset and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> </mml:math> 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 numerical examples of Nyström subsampling in the reproducing kernel Hilbert space framework for the functional linear regression model.