Classifying infinite-dimensional data with unified basis functions: An effective machine learning approach

Many real-world applications involve functional data, where each sample is a continuous function rather than a finite-dimensional vector or matrix. This infinite-dimensional nature poses significant challenges for classification tasks. Despite the progress in functional data analysis, existing methods and machine learning algorithms often struggle with the complexities of infinite-dimensional spaces. This paper introduces a novel approach that aims to simplify and improve the accuracy of infinite-dimensional classification tasks within functional data analysis. By addressing the algorithmic and computational complexities of existing high-dimensional classification methods, our approach provides an effective solution for the non-linear classification of infinite-dimensional functional data. The primary innovation lies in identifying a set of unified basis functions to accurately represent all observations originally defined in infinite-dimensional space. Subsequently, these observed functions in an infinite-dimensional space can be represented by their parameter vectors in a finite-dimensional space, and the original functional classification problem is converted to the conventional classification problem in a vector space (parameters) to solve, which leads to the simplicity of the proposed approach. We present the detailed design and implementation of this approach, including an effective framework for identifying functional data representations and B-spline knot placements. This approach not only facilitates the transformation between infinite-dimensional function spaces and finite-dimensional representative parameter spaces but also demonstrates significant accuracy improvements across various datasets in this field, as evidenced by empirical findings and comparative experimental results.