# Distributed Learning with Discretely Observed Functional Data

arXiv: https://arxiv.org/abs/2410.02376 · HTML: https://mlsanity.com/paper/2410.02376/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 41 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 8 of 8)

- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Optimal Rates For Regularization Of Statistical Inverse Learning Problems](https://mlsanity.com/paper/1604.04054/) (2016)
- [Divide and Conquer Kernel Ridge Regression: A Distributed Algorithm with Minimax Optimal Rates](https://mlsanity.com/paper/1305.5029/) (2013)
- [Distributed learning with regularized least squares](https://mlsanity.com/paper/1608.03339/) (2016)
- [Minimax Optimal Rates of Estimation in High Dimensional Additive Models: Universal Phase Transition](https://mlsanity.com/paper/1503.02817/) (2015)
- [The Falling Factorial Basis and Its Statistical Applications](https://mlsanity.com/paper/1405.0558/) (2014)
- [Functional Linear Regression with Mixed Predictors](https://mlsanity.com/paper/2012.00460/) (2020)
- [Statistical Optimality of Divide and Conquer Kernel-based Functional Linear Regression](https://mlsanity.com/paper/2211.10968/) (2022)

## Built on by (most cited first, 1 of 1)

- [Generalization Analysis of Distributed Kernel-based Robust Gradient Descent Algorithms](https://mlsanity.com/paper/2609.11712/) (2026)

## Abstract (arXiv)

By selecting different filter functions, spectral algorithms can generate various regularization methods to solve statistical inverse problems within the learning-from-samples framework. This paper combines distributed spectral algorithms with Sobolev kernels to tackle the functional linear regression problem. The design and mathematical analysis of the algorithms require only that the functional covariates are observed at discrete sample points. Furthermore, the hypothesis function spaces of the algorithms are the Sobolev spaces generated by the Sobolev kernels, optimizing both approximation capability and flexibility. Through the establishment of regularity conditions for the target function and functional covariate, we derive matching upper and lower bounds for the convergence of the distributed spectral algorithms in the Sobolev norm. This demonstrates that the proposed regularity conditions are reasonable and that the convergence analysis under these conditions is tight, capturing the essential characteristics of functional linear regression. The analytical techniques and estimates developed in this paper also enhance existing results in the previous literature.