# Functional Attentive Interpretable Regression

arXiv: https://arxiv.org/abs/2609.05846 · HTML: https://mlsanity.com/paper/2609.05846/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 25 references that are in the map.
- First posted on arXiv 5 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Functional Linear Regression with Mixed Predictors](https://mlsanity.com/paper/2012.00460/) (2020)
- [Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework](https://mlsanity.com/paper/1902.03674/) (2019)
- [Robust Function-on-Function Regression](https://mlsanity.com/paper/1908.11601/) (2019)
- [A partial least squares approach for function-on-function interaction regression](https://mlsanity.com/paper/2012.05313/) (2020)
- [Prediction in functional regression with discretely observed and noisy covariates](https://mlsanity.com/paper/2112.06486/) (2021)

## Abstract (arXiv)

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.