Highly Adaptive Principal Component Regression

2026-02-11

AI Summary

This work addresses the high computational complexity of Highly Adaptive Lasso (HAL) in high-dimensional nonparametric regression by proposing a dimensionality reduction approach based on principal component analysis (PCA), yielding the PCHAL and PCHAR estimators. The method substantially reduces computational cost without relying on the outcome variable, while maintaining empirical performance comparable to that of the original HAL and HAR estimators. Theoretical analysis reveals a spectral connection between the principal components of the HAL/HAR Gram operator and the discrete sine basis, uncovering an intrinsic Fourier-type structure. This insight provides both a novel perspective and practical tools for efficient nonparametric estimation in high-dimensional settings.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The Highly Adaptive Lasso (HAL) is a nonparametric regression method that achieves almost dimension-free convergence rates under minimal smoothness assumptions, but its implementation can be computationally prohibitive in high dimensions due to the large basis matrix it requires. The Highly Adaptive Ridge (HAR) has been proposed as a scalable alternative. Building on both procedures, we introduce the Principal Component based Highly Adaptive Lasso (PCHAL) and Principal Component based Highly Adaptive Ridge (PCHAR). These estimators constitute an outcome-blind dimension reduction which offer substantial gains in computational efficiency and match the empirical performances of HAL and HAR. We also uncover a striking spectral link between the leading principal components of the HAL/HAR Gram operator and a discrete sinusoidal basis, revealing an explicit Fourier-type structure underlying the PC truncation.

Problem

• Highly Adaptive Lasso
• high-dimensional regression
• computational efficiency
• dimension reduction
• nonparametric regression

Innovation

• Highly Adaptive Lasso
• Principal Component Regression
• Dimension Reduction
• Gram Operator
• Fourier Structure