Reduced-Rank Regularized Multivariate Model for High-Dimensional Data

Abstract

This article proposes a modeling framework for high-dimensional experimental data, such as brain images or microarrays, that discovers statistically significant structures most relevant to the experimental covariates. To deal with the curse of dimensionality, three regularization schemes are used: a reduced-rank model, penalization of the covariance matrix, and regularization of the basis-expanded predictor set. The latter allows us to flexibly model associations while controlling for overfitting. The modeling framework is derived from a reduced-rank multiresponse linear model, which offers a familiar interface for researchers. The novel regularizations of both sides of the model make it applicable in high-dimensional settings, without a need for prior dimension reduction, and can model nonlinear relationships. An efficient, dual-space algorithm is proposed to estimate its components in low-dimensional space. It permits the use of the bootstrap, to provide pointwise standard error bands on association graphs, and other resampling techniques to optimize hyperparameters. We evaluate the model on a small neuroimaging dataset, and in a simulation study using simple images corrupted by additive Gaussian iid and random field noise components with signal-to-noise ratios below 0.1. Our model compares well with a general linear model (GLM) even when the nonlinear associations are specified explicitly in GLM.