High-dimensional Hilbert–Schmidt linear regression with Hilbert manifold variables
The Annals of Statistics · 2025 · Volume 53 · Issue 6
Abstract
In this paper, we propose a novel high-dimensional linear regression technique for variables taking values in Hilbert manifolds. Our approach offers a flexible framework where the response and covariates originate from distinct spaces and are interconnected by Hilbert–Schmidt operators. Our methodology is designed for situations where the number of the covariates grows exponentially fast as the sample size increases and some of the covariates together with the response take values in infinite dimensional spaces. It is formulated under a general penalization scheme that includes various nonconvex penalty functions. Leveraging modern statistical theory for data residing on Hilbert manifolds, we establish the oracle property and derive error bounds for the proposed estimators. We also provide an efficient computational algorithm to solve the associated constrained optimization problem. The practical performance of the proposed method is demonstrated via numerical simulation and real data applications.
Keywords
62J07, 62R30, high-dimensional regression, Hilbert spaces, Hilbert–Schmidt operators, non-Euclidean data, Penalization, Riemannian manifolds, spectral decomposition, Wasserstein spaces
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