Deep neural network classifier for multidimensional functional data

Abstract

We propose a new approach, called as functional deep neural network (FDNN), for classifying multidimensional functional data. Specifically, a deep neural network is trained based on the principal components of the training data which shall be used to predict the class label of a future data function. Unlike the popular functional discriminant analysis approaches which only work for one‐dimensional functional data, the proposed FDNN approach applies to general non‐Gaussian multidimensional functional data. Moreover, when the log density ratio possesses a locally connected functional modular structure, we show that FDNN achieves minimax optimality. The superiority of our approach is demonstrated through both simulated and real‐world datasets.

Review on functional data classification

“…Hall et al (2001), Bongiorno and Goia (2016), X. Dai et al (2017), and Zhang and Sakhanenko (2019) employed the kernel density estimation methods to obtain by considering independent FPCS, where , and are density function estimators for the ‐th FPCS in the ‐th group. To overcome the restrictive independence assumption, S. Wang, Cao, and Shang (2023) additionally took into account a broader scenario encompassing situations where intricate are present. Particularly, incorporate convoluted interactions among FPCS.…”

“…Figure 3 portrays a representative sample of the MNIST dataset, emphasizing the varying intensity levels. S. Wang, Cao, and Shang (2023) developed a deep neural network based classifier to conduct multi‐class classification for this dataset.…”

“…In case of 3D analysis, the number of observed voxels in each patient's brain sample amounts to . S. Wang, Cao, and Shang (2023) have proposed multi‐dimensional functional data classification methods for both 2D and 3D ADNI data.…”

“…Naturally, given any generic classifier , it has such that the excess risk automatically bounds the MEMR from above. A rich body of literature has discussed the rates in different scenarios, see Biau et al (2010), Baíllo and Cuesta‐Albertos (2011), Meister (2016), S. Wang, Cao, and Shang (2023), and S. Wang, Shang, et al (2023). Nonetheless, deriving the lower bound of convergence often proves challenging due to its exhaustive consideration of all potential classifiers.…”

“…Building upon the same framework, S. Wang, Cao, and Shang (2023) undertook a more in‐depth investigation into the realm of minimax optimality, extending beyond the scope of Gaussian functional data. In particular, they assume that the log‐likelihood ratio belongs to a complex function space defined by Hölder smoothness, which extends from the parametric context in S. Wang, Shang, et al (2023) to a non‐parametric problem.…”