Abstract:
The scale of modern datasets, with more and more variables measured on more and more observations, presents many statistical challenges, but also opportunities to discover and exploit the rich structure that is often present in the data. In neuroimaging studies, multiple kinds of brain imaging are conducted on the same participant, with each modality of imaging having its own further structure, and many associated phenotypic measurements taken on the participants. Understanding the complicated and noisy underlying relationships between all of these measurements holds promise for scientific and treatment breakthroughs in the long term, and requires sophisticated methods designed to uncover this structure. This thesis presents three projects on learning structure in high-dimensional datasets motivated by applications in neuroimaging.

The first project considers the setting where many networks are observed on a common node set: each observation comprises edge weights, covariates observed at each node, and a response. In our neuroimaging application, the edge weights correspond to functional connectivity between brain regions, node covariates encode task activations at each brain region, and performance on a behavioral task is the response. The goal is to use the edge weights and node covariates to predict the response and to identify a parsimonious and interpretable set of predictive features. We propose an approach that uses feature groups defined according to a community structure believed to exist in the network (naturally occurring in neuroimaging applications). We propose two schemes for forming feature groups where each group incorporates both edge weights and node covariates, and derive optimization algorithms for both using an overlapping group LASSO penalty. Empirical results on synthetic data show that our method, relative to competing approaches, has similar or improved prediction error along with superior support recovery, enabling a more interpretable and potentially a more accurate understanding of the underlying process. We also apply the method to neuroimaging data.

The second project focuses on inference for structure learned using Canonical Correlation Analysis (CCA). CCA is a method for analyzing a sample of pairs of random vectors; it learns a sequence of paired linear transformations of the original variables that are maximally correlated within pairs while uncorrelated across pairs. CCA outputs both canonical correlations as well as the canonical directions which define the transformations. While inference for canonical correlations is well developed, conducting inference for canonical directions is more challenging and not well-studied, but is key to interpretability. We propose a computational bootstrap method for inference on CCA direction (kombootcca). We conduct thorough simulation studies that range from simple and well-controlled to complex but realistic and validate the statistical properties of kombootcca while comparing it to several competitors. We also apply the kombootcca method to a brain imaging dataset and discover linked patterns in brain connectivity and behavioral scores.

The third project proposes a new method for matrix CCA (matcca), which works with pairs of random matrices rather than pairs of random vectors, motivated by a neuroimaging application where the brain imaging data takes the form of a high-dimensional covariance matrix. Our matcca method uses a nuclear norm penalty that encourages the canonical directions associated with the matrix-variate data to have low rank structure when arranged into a matrix. Results from both synthetic and neuroimaging data show that matcca is very effective at recovering low rank signals even in noisy cases with few observations.