Functional principal component analysis (FPCA) is an important technique for dimension reduction in functional data analysis (FDA). Classical FPCA method is based on the Karhunen-Lo\`{e}ve expansion, which assumes a linear structure of the observed functional data. However, the assumption may not always be satisfied, and the FPCA method can become inefficient when the data deviates from the linear assumption. In this paper, we propose a novel FPCA method that is suitable for data with a nonlinear structure by neural network approach. We construct networks that can be applied to functional data and explore the corresponding universal approximation property. The main use of our proposed nonlinear FPCA method is curve reconstruction. We conduct a simulation study to evaluate the performance of our method. The proposed method is also applied to two real-world data sets to further demonstrate its superiority.

翻译：函数主成分分析（FPCA）是函数型数据分析（FDA）中重要的降维技术。经典FPCA方法基于Karhunen-Loève展开，假设观测的函数型数据具有线性结构。然而该假设并非总能成立，当数据偏离线性假设时，FPCA方法可能效率低下。本文提出一种适用于非线性结构数据的新型FPCA方法，该方法采用神经网络途径。我们构建了可应用于函数型数据的网络，并探讨了相应的通用逼近性质。所提出的非线性FPCA方法主要应用于曲线重构。我们通过模拟研究评估了所提方法的性能，并将其应用于两个真实数据集以进一步验证其优越性。