Factor models for big data

Abstract

<p>This dissertation is motivated by clustering dendritic spines which have attracted interest in neuroscience because the morphology of spines are closely related to brain functionality. However, modeling and analyzing the morphological data is challenging because they involve both directional and non-directional features and there is very little work available on characterizing the dependence among these features in a practically useful manner. In fact, there are very few methods available for modeling the dependence among directional components. Thus, in this collection of works, we present novel methodologies, matrix-free algorithms and real-world applications for modeling and illustrating the variability of data on a high-dimensional sphere and clustered multivariate data associated with directional features.</p> <p>We develop a matrix-free computational algorithm for fitting high-dimensional Gaussian data using a factor model, which can explain the variability of a large set of variables using a small set of factors. Then, we describe a novel family of distributions on the unit sphere that is obtained by radially projecting a Gaussian random variable with factor covariance structure. For practical applications, we further establish a novel matrix-free computational framework for computing maximum likelihood estimates and demonstrate the broad scope of the latent factor model by analyzing data from social networks, resting state functional magnetic resonance imaging experiments, genetics and digital image databases. Finally, we extend the latent factor model to model and cluster the spine morphological data. Our approach produces three spine groups with distinct morphological features, reveals a relationship among the directional variables and their correlations, and characterizes the variability of all the directional and the non-directional features.</p>