Sparse higher-order partial least squares for simultaneous variable selection, dimension reduction and tensor denoising

Abstract Motivated by the challenge of estimating effects of DNA methylation on 3D genomic contacts captured by multimodal single-cell Hi-C data, we consider the tensor-response partial least squares model with $ \mathcal{Y}=\mathcal{B}\times_{1}X+\mathcal{F} $, where the correlated high-dimensional predictors $ X\in\mathbb{R}^{n\times d_{1}} $ and the sparse and noisy high-dimensional responses $ \mathcal{Y}\in\mathbb{R}^{n\times\prod_{m}d_{m}} $ are observed, the low-rank and sparse partial least squares coefficient tensor $ \mathcal{B}\in\mathbb{R}^{\prod_{m}d_{m}} $ is unknown, and $ \mathcal{F}\in\mathbb{R}^{n\times\prod_{m}d_{m}} $ is the noise tensor. In this work, we study the problem of estimating the partial least squares coefficient $ \mathcal{B} $ and identifying its active entries in the tensor partial least squares framework. We show that the consistency of the existing tensor partial least squares estimator (Zhao et al., 2012) cannot be guaranteed under a high-dimensional regime in which both the number of predictors and the response tensor dimensions grow faster than the sample size. To address this high dimensionality, we propose the sparse higher-order partial least squares estimator and an accompanying algorithm that can simultaneously perform variable selection, dimension reduction and tensor-response denoising. We establish asymptotic guarantees for the proposed estimator in the high-dimensional regime and validate these results through comprehensive simulation studies, demonstrating our method’s advantages over baseline approaches. Finally, application of the proposed estimator to the motivating multimodal single-cell Hi-C data provides novel biological insights into gene regulation by multiple regulatory elements.