# On Hypothesis Transfer Learning of Functional Linear Models

arXiv: https://arxiv.org/abs/2206.04277 · HTML: https://mlsanity.com/paper/2206.04277/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 54 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 13 of 13)

- [Few-Shot Learning via Learning the Representation, Provably](https://mlsanity.com/paper/2002.09434/) (2020)
- [On the Theory of Transfer Learning: The Importance of Task Diversity](https://mlsanity.com/paper/2006.11650/) (2020)
- [Transfer Learning for High-dimensional Linear Regression: Prediction, Estimation, and Minimax Optimality](https://mlsanity.com/paper/2006.10593/) (2020)
- [Transfer Learning under High-dimensional Generalized Linear Models](https://mlsanity.com/paper/2105.14328/) (2021)
- [Provable Guarantees for Gradient-Based Meta-Learning](https://mlsanity.com/paper/1902.10644/) (2019)
- [Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier](https://mlsanity.com/paper/1906.02903/) (2019)
- [Learning-to-Learn Stochastic Gradient Descent with Biased Regularization](https://mlsanity.com/paper/1903.10399/) (2019)
- [Hypothesis Transfer Learning via Transformation Functions](https://mlsanity.com/paper/1612.01020/) (2016)
- [Representation Learning Beyond Linear Prediction Functions](https://mlsanity.com/paper/2105.14989/) (2021)
- [Fast Rates by Transferring from Auxiliary Hypotheses](https://mlsanity.com/paper/1412.1619/) (2014)
- [Functional optimal transport: map estimation and domain adaptation for functional data](https://mlsanity.com/paper/2102.03895/) (2021)
- [Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework](https://mlsanity.com/paper/1902.03674/) (2019)
- [Optimal Estimation for the Functional Cox Model](https://mlsanity.com/paper/1601.07496/) (2016)

## Built on by (most cited first, 2 of 2)

- [Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer](https://mlsanity.com/paper/2501.10870/) (2025)
- [Coefficient Shape Transfer Learning for Functional Linear Regression](https://mlsanity.com/paper/2506.11367/) (2025)

## Abstract (arXiv)

We study the transfer learning (TL) for the functional linear regression (FLR) under the Reproducing Kernel Hilbert Space (RKHS) framework, observing that the TL techniques in existing high-dimensional linear regression are not compatible with the truncation-based FLR methods, as functional data are intrinsically infinite-dimensional and generated by smooth underlying processes. We measure the similarity across tasks using RKHS distance, allowing the type of information being transferred to be tied to the properties of the imposed RKHS. Building on the hypothesis offset transfer learning paradigm, two algorithms are proposed: one conducts the transfer when positive sources are known, while the other leverages aggregation techniques to achieve robust transfer without prior information about the sources. We establish asymptotic lower bounds for this learning problem and show that the proposed algorithms enjoy a matching upper bound. These analyses provide statistical insights into factors that contribute to the dynamics of the transfer. We also extend the results to functional generalized linear models. The effectiveness of the proposed algorithms is demonstrated via extensive synthetic data as well as real-world data applications.