The Rates of Minimax and Randomized Sketches

Minimax Rates and Randomized Sketches for Kernel-based Estimation in Partially Functional Linear Models

Authors: Shaogao Lv, Xin He, Junhui Wang; Journal of Machine Learning Research, 24(55):1−38, 2023.

Abstract

This study focuses on the partially functional linear model (PFLM), where predictive features comprise a functional covariate and a high-dimensional scalar vector. In this paper, we propose a least square approach for estimating the PFLM over an infinite-dimensional reproducing kernel Hilbert space. The estimation involves two mixed regularizations, namely a function-norm and an $\ell_1$-norm. Our main objective is to establish the minimax rates for PFLM under the high-dimensional setting. We achieve this by utilizing various techniques in empirical process theory to analyze kernel classes and determine the optimal minimax rates of estimation. Additionally, we present an efficient numerical algorithm based on randomized sketches of the kernel matrix. Several numerical experiments are conducted to demonstrate the effectiveness of our method and optimization strategy.

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