The United States health-care system is considered to be unsustainable due to its unbearably high cost. Many of the resources are spent on acute conditions rather than aiming at preventing them. Preventive medicine methods, therefore, are viewed as a potential remedy since they can help reduce the occurrence of acute health episodes. The work in this dissertation tackles two distinct problems related to the prevention of acute disease. Specifically, we consider: (1) early detection of incorrect or abnormal postures of the human body and (2) the prediction of hospitalization due to heart related diseases. The solution to the former problem could be used to prevent people from unexpected injuries or alert caregivers in the event of a fall. The latter study could possibly help improve health outcomes and save considerable costs due to preventable hospitalizations. For body posture detection, we place wireless sensor nodes on different parts of the human body and use the pairwise measurements of signal strength corresponding to all sensor transmitter/receiver pairs to estimate body posture. We develop a composite hypothesis testing approach which uses a Generalized Likelihood Test (GLT) as the decision rule. The GLT distinguishes between a set of probability density function (pdf) families constructed using a custom pdf interpolation technique. The GLT is compared with the simple Likelihood Test and Multiple Support Vector Machines. The measurements from the wireless sensor nodes are highly variable and these methods have different degrees of adaptability to this variability. Besides, these methods also handle multiple observations differently. Our analysis and experimental results suggest that GLT is more accurate and suitable for the problem. For hospitalization prediction, our objective is to explore the possibility of effectively predicting heart-related hospitalizations based on the available medical history of the patients. We extensively explored the ways of extracting information from patients' Electronic Health Records (EHRs) and organizing the information in a uniform way across all patients. We applied various machine learning algorithms including Support Vector Machines, AdaBoost with Trees, and Logistic Regression adapted to the problem at hand. We also developed a new classifier based on a variant of the likelihood ratio test. The new classifier has a classification performance competitive with those more complex alternatives, but has the additional advantage of producing results that are more interpretable. Following this direction of increasing interpretability, which is important in the medical setting, we designed a new method that discovers hidden clusters and, at the same time, makes decisions. This new method introduces an alternating clustering and classification approach with guaranteed convergence and explicit performance bounds. Experimental results with actual EHRs from the Boston Medical Center demonstrate prediction rate of 82% under 30% false alarm rate, which could lead to considerable savings when used in practice.

<p dir="ltr">In most theoretical studies on missing data analysis, data is typically assumed to be missing according to a specific probabilistic model. However, such assumption may not accurately reflect real-world situations, and sometimes missing is not purely random. In this thesis, our focus is on analyzing incomplete data matrices without relying on any probabilistic model assumptions for the missing schemes. To characterize a missing scheme deterministically, we employ a graph whose adjacency matrix is a binary matrix that indicates whether each matrix entry is observed or not. Leveraging its graph properties, we mathematically represent the missing pattern of an incomplete data matrix and conduct a theoretical analysis of how this non-random missing pattern affects the solvability of specific problems related to incomplete data. This dissertation primarily focuses on three types of incomplete data problems characterized by their low-rank nature: structured prediction, matrix completion, and sparse PCA.</p><p dir="ltr">First, we investigate a basic structured prediction problem, which involves recovering binary node labels on a fixed undirected graph, where noisy binary observations corresponding to edges are given. Essentially, this setting parallels a simple binary rank-1 symmetric matrix completion problem, where missing entries are determined by a fixed undirected graph. Our aim is to establish the fundamental limit bounds of this problem, revealing a close association between the limits and graph properties, such as connectivity.</p><p dir="ltr">Second, we move on to the general low-rank matrix completion problem. In this study, we establish provable guarantees for exact and approximate low-rank matrix completion problems that can be applied to any non-random missing pattern, by utilizing the observation graph corresponding to the missing scheme. We theoretically and experimentally show that the standard constrained nuclear norm minimization algorithm can successfully recover the true matrix when the observation graph is well-connected and has similar node degrees. We also verify that matrix completion is achievable with a near-optimal sample complexity rate when the observation graph has uniform node degrees and its adjacency matrix has a large spectral gap.</p><p dir="ltr">Finally, we address the sparse PCA problem, featuring an approximate low-rank attribute. Missing data is common in situations where sparse PCA is useful, such as single-cell RNA sequence data analysis. We propose a semidefinite relaxation of the non-convex $\ell_1$-regularized PCA problem to solve sparse PCA on incomplete data. We demonstrate that the method is particularly effective when the observation pattern has favorable properties. Our theory is substantiated through synthetic and real data analysis, showcasing the superior performance of our algorithm compared to other sparse PCA approaches, especially when the observed data pattern has specific characteristics.</p>