Functional Attentive Interpretable Regression
Abstract
In function-on-function regression, the coefficient surface β(s,t) may exhibit complex support structure---from localized patches to global patterns such as disconnected regions, bands, or rings---where effect similarity does not align with Euclidean proximity. Projection-based methods that rely on fixed basis expansions can obscure such structure, while direct smoothing approaches risk oversmoothing the surface and its boundaries. We propose Functional Attentive Interpretable Regression (FAIR), which represents β(s,t) directly through coordinate features and uses self-attention to learn effect-adaptive neighborhoods, enabling information sharing at both local and global scales. A scalar compression network maps these learned representations to the coefficient surface. Sparsity and smoothness penalties applied over these neighborhoods promote localized support with coherent boundaries. We establish a sieve equivalence to tensor-product spline spaces and derive convergence rates. Simulations and applications to oceanographic and hydrological data demonstrate that FAIR recovers support geometry more accurately than existing methods while achieving superior prediction, particularly under sparse sampling.
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BibTeX
@misc{wang2026functional,
title = {{Functional Attentive Interpretable Regression}},
author = {Haixu Wang and Tianyu Guan and Jiguo Cao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05846},
eprint = {2609.05846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05846},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Functional Attentive Interpretable Regression
AU - Haixu Wang
AU - Tianyu Guan
AU - Jiguo Cao
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05846
UR - https://arxiv.org/abs/2609.05846
AB - In function-on-function regression, the coefficient surface β(s,t) may exhibit complex support structure---from localized patches to global patterns such as disconnected regions, bands, or rings---where effect similarity does not align with Euclidean proximity. Projection-based methods that rely on fixed basis expansions can obscure such structure, while direct smoothing approaches risk oversmoothing the surface and its boundaries. We propose Functional Attentive Interpretable Regression (FAIR), which represents β(s,t) directly through coordinate features and uses self-attention to learn effect-adaptive neighborhoods, enabling information sharing at both local and global scales. A scalar compression network maps these learned representations to the coefficient surface. Sparsity and smoothness penalties applied over these neighborhoods promote localized support with coherent boundaries. We establish a sieve equivalence to tensor-product spline spaces and derive convergence rates. Simulations and applications to oceanographic and hydrological data demonstrate that FAIR recovers support geometry more accurately than existing methods while achieving superior prediction, particularly under sparse sampling.
ER -