<p>Although deep learning models continue to gain momentum, their robustness and interpretability have always been a big concern because of the complexity of such models. In this dissertation, we studied several topics on the robustness and interpretability of convolutional neural networks (CNNs) and graph neural networks (GNNs). We first identified the structural problem of deep convolutional neural networks that leads to the adversarial examples and defined DNN uncertainty regions. We also argued that the generalization error, the large sample theoretical guarantee established for DNN, cannot adequately capture the phenomenon of adversarial examples. Secondly, we studied the dropout in GNNs, which is an effective regularization approach to prevent overfitting. Contrary to CNN, GNN usually has a shallow structure because a deep GNN normally sees performance degradation. We studied different dropout schemes and established a connection between dropout and over-smoothing in GNNs. Therefore we developed layer-wise compensation dropout, which allows GNN to go deeper without suffering performance degradation. We also developed a heteroscedastic dropout which effectively deals with a large number of missing node features due to heavy experimental noise or privacy issues. Lastly, we studied the interpretability of graph neural networks. We developed a self-interpretable GNN structure that denoises useless edges or features, leading to a more efficient message-passing process. The GNN prediction and explanation accuracy were boosted compared with baseline models. </p>

Despite considerable advances in missing data imputation methods over the last three decades, the problem of missing data remains largely unsolved. Many techniques have emerged in the literature as candidate solutions. These techniques can be categorised into two classes: statistical methods of data imputation and computational intelligence methods of data imputation. Due to the longstanding use of statistical methods in handling missing data problems, it takes quite some time for computational intelligence methods to gain profound attention even though these methods have analogous accuracy, in comparison to other approaches. The merits of both these classes have been discussed at length in the literature, but only limited studies make significant comparison to these classes. This thesis contributes to knowledge by firstly, conducting a comprehensive comparison of standard statistical methods of data imputation, namely, mean substitution (MS), regression imputation (RI), expectation maximization (EM), tree imputation (TI) and multiple imputation (MI) on missing completely at random (MCAR) data sets. Secondly, this study also compares the efficacy of these methods with a computational intelligence method of data imputation, ii namely, a neural network (NN) on missing not at random (MNAR) data sets. The significance difference in performance of the methods is presented. Thirdly, a novel procedure for handling missing data is presented. A hybrid combination of each of these statistical methods with a NN, known here as the post-processing procedure, was adopted to approximate MNAR data sets. Simulation studies for each of these imputation approaches have been conducted to assess the impact of missing values on partial least squares structural equation modelling (PLS-SEM) based on the estimated accuracy of both structural and measurement parameters. The best method to deal with particular missing data mechanisms is highly recognized. Several significant insights were deduced from the simulation results. It was figured that for the problem of MCAR by using statistical methods of data imputation, MI performs better than the other methods for all percentages of missing data. Another unique contribution is found when comparing the results before and after the NN post-processing procedure. This improvement in accuracy may be resulted from the neural network¿s ability to derive meaning from the imputed data set found by the statistical methods. Based on these results, the NN post-processing procedure is capable to assist MS in producing significant improvement in accuracy of the approximated values. This is a promising result, as MS is the weakest method in this study. This evidence is also informative as MS is often used as the default method available to users of PLS-SEM software. / Minister of Higher Education Malaysia and University Utara Malaysia

The Association of Research Libraries (ARL) provides an option for librarians to administer a combination of LibQUAL+® and LibQUAL+® Lite to measure users' perceptions of library service quality. LibQUAL+® Lite is a shorter version of LibQUAL+® that uses planned missing data in its design. The present study investigates the loss of information in commonly administered proportions of LibQUAL+® and LibQUAL+® Lite when compared to administering LibQUAL+® alone. Data from previous administrations of LibQUAL+® protocol (2005, N = 525; 2007, N = 3,261; and 2009, N = 2,103) were used to create simulated datasets representing various proportions of LibQUAL+® versus LibQUAL+® Lite administration (0.2:0.8, 0.4:0.6. 0.5:0.5, 0.6:0.4, and 0.8:0.2). Statistics (i.e., means, adequacy and superiority gaps, standard deviations, Pearson product-moment correlation coefficients, and polychoric correlation coefficients) from simulated and real data were compared. Confidence intervals captured the original values. Root mean square errors and absolute and relative biases of correlations showed that accuracy in the estimates decreased with increase in percentage of planned missing data. The recommendation is to avoid using combinations with more than 20% planned missing data.

<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>

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