  Multimodal data are now prevailing in scientific research. A central question
in multimodal integrative analysis is to understand how two data modalities
associate and interact with each other given another modality or demographic
variables. The problem can be formulated as studying the associations among
three sets of random variables, a question that has received relatively less
attention in the literature. In this article, we propose a novel generalized
liquid association analysis method, which offers a new and unique angle to this
important class of problems of studying three-way associations. We extend the
notion of liquid association of \citet{li2002LA} from the univariate setting to
the sparse, multivariate, and high-dimensional setting. We establish a
population dimension reduction model, transform the problem to sparse Tucker
decomposition of a three-way tensor, and develop a higher-order orthogonal
iteration algorithm for parameter estimation. We derive the non-asymptotic
error bound and asymptotic consistency of the proposed estimator, while
allowing the variable dimensions to be larger than and diverge with the sample
size. We demonstrate the efficacy of the method through both simulations and a
multimodal neuroimaging application for Alzheimer's disease research.
