Nonlinear Functional Sufficient Dimension Reduction via Principal Fitted Components

In this paper, we propose a novel functional nonlinear sufficient dimension reduction method based on the principal fitted component model. Our approach extends the concept of principal fitted components to functional data, covering the case where both the predictors and responses are functions. We consider a general framework in which the predictor and response can each be viewed as elements of potentially infinite dimensional Hilbert spaces. This includes the important scalar on function and function on function cases as special instances. We generalize a nonlinear principal fitted component model within the framework of reproducing kernel Hilbert space, leveraging the nested Hilbert spaces theory to characterize nonlinear structures in functional data. The first space accommodates functions of random curves and the second space captures their nonlinear relationships. To establish the theoretical validity of our approach, we develop an asymptotic theory that characterizes the convergence behavior of the proposed estimator under mild regularity conditions. Extensive simulation studies demonstrate that our method outperforms existing functional sufficient dimension reduction methods, particularly in scenarios with complex nonlinear dependencies. The effectiveness of the proposed method is further validated through real data analysis.