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 multi-modal single cell Hi-C data, we consider the tensor response partial least squares (PLS) model withY=B×1X+F⁠, where the correlated high-dimensional predictorsX∈Rn×d1and sparse and noisy high-dimensional responsesY∈Rn×∏mdmare observed, the low rank and sparse partial least squares coefficient tensorB∈R∏mdmis unknown, andF∈Rn×∏mdmis the noise tensor. In this work, we study the problem of estimating the partial least squares coefficientBand 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 simultaneously perform variable selection, dimension reduction and tensor response denoising. We establish asymptotic guarantees for SHOPS in the high-dimensional regime and validate these results through comprehensive simulation studies, demonstrating SHOPS‘s advantages over baseline approaches. Finally, application of SHOPS to the motivating multi-modal single-cell Hi-C data reveals novel biological insights into gene regulation by multiple regulatory elements.