Adaptive nonlinear learning framework with dynamic representations for scalar-on-function regression.

Abstract

Continuous functional data, such as biomedical signals and sensor measurements, are increasingly prevalent in modern scientific and industrial applications. Effectively modelling scalar-on-function regression, where scalar responses depend on infinite-dimensional predictors, remains a significant challenge in statistics and machine learning. A universal framework for this task is still lacking. This gap arises primarily from the difficulty of mapping infinite-dimensional functional predictors into finite-dimensional representations that are statistically estimable, computationally tractable, and compatible with nonlinear machine learning models, while still preserving the intrinsic temporal, spatial, or smooth structural information of the original functions. In this paper, we propose an adaptive nonlinear learning framework with dynamic representations for scalar-on-function regression that draws on the principles of functional data analysis while preserving the intrinsic structure of functional predictors. The framework adapts across multiple families of basis functions and sequentially optimises both the number and the locations of the basis components. As a result, it produces flexible functional representations that capture complex and localised structures without being tied to a single basis family. The resulting representations preserve the key characteristics of the original functional predictors while supporting accurate scalar-on-function prediction. Evaluation on seven real-world benchmark datasets from chemometrics, neuroscience, and meteorology confirms consistent and substantial improvements over existing methods. For example, the rRMSE is reduced from 0.100 to 0.003 on the Gasoline dataset and from 1.000 to 0.220 on the DTI dataset, while achieving best performance across all remaining benchmarks. These findings indicate that the proposed framework is both practical and scalable for scalar-on-function regression.