# Kernel-based estimation for partially functional linear model: Minimax rates and randomized sketches

arXiv: https://arxiv.org/abs/2110.09042 · HTML: https://mlsanity.com/paper/2110.09042/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 18 Oct 2021.

## Builds on (most cited first, 5 of 5)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015)
- [Randomized sketches for kernels: Fast and optimal non-parametric regression](https://mlsanity.com/paper/1501.06195/) (2015)
- [Beyond Least-Squares: Fast Rates for Regularized Empirical Risk Minimization through Self-Concordance](https://mlsanity.com/paper/1902.03046/) (2019)
- [The Nystr"om method for convex loss functions](https://mlsanity.com/paper/2006.10016/) (2020)

## Abstract (arXiv)

This paper considers the partially functional linear model (PFLM) where all predictive features consist of a functional covariate and a high dimensional scalar vector. Over an infinite dimensional reproducing kernel Hilbert space, the proposed estimation for PFLM is a least square approach with two mixed regularizations of a function-norm and an ℓ1-norm. Our main task in this paper is to establish the minimax rates for PFLM under high dimensional setting, and the optimal minimax rates of estimation is established by using various techniques in empirical process theory for analyzing kernel classes. In addition, we propose an efficient numerical algorithm based on randomized sketches of the kernel matrix. Several numerical experiments are implemented to support our method and optimization strategy.