The data sets arising in modern science and engineering are often extremely large, befitting the era of big data. But these data sets are not only large in the number of samples they have, they may also have a large number of features, placing each data point in a high-dimensional space. However, unique problems arise when the dimension of the data has the same or even greater order than the sample size. This scenario in statistics is known as the High Dimension, Low Sample Size problem (HDLSS). In this paradigm, many standard statistical estimators are shown to perform sub-optimally and in some cases can not be computed at all. To overcome the barriers found in HDLSS scenarios, one must make additional assumptions on the data, either with explicit formulations or with implicit beliefs about the behavior of the data. The first type of research leads to structural assumptions placed on the probability model that generates the data, which allow for alterations to classical methods to yield theoretically optimal estimators for the chosen well-defined tasks. The second type of research, in contrast, makes general assumptions usually based on the the causal nature of chosen real-world data application, where the data is assumed to have dependencies between the parameters. This dissertation develops two novel algorithms that successfully operate in the paradigm of HDLSS. We first propose the Generalized Eigenvalue (GEV) estimator, a unified sparse projection regression framework for estimating generalized eigenvector problems. Unlike existing work, we reformulate a sequence of computationally intractable non-convex generalized Rayleigh quotient optimization problems into a computationally efficient simultaneous linear regression problem, padded with a sparse penalty to deal with high-dimensional predictors. We showcase the applications of our method by considering three iconic problems in statistics: the sliced inverse regression (SIR), linear discriminant analysis (LDA), and canonical correlation analysis (CCA). We show the reformulated linear regression problem is able to recover the same projection space obtained by the original generalized eigenvalue problem. Statistically, we establish the nonasymptotic error bounds for the proposed estimator in the applications of SIR and LDA, and prove these rates are minimax optimal. We present how the GEV is applied to the CCA problem, and adapt the method for a robust Huber-loss based formulation for noisy data. We test our framework on both synthetic and real datasets and demonstrate its superior performance compared with other state-of-the-art methods in high dimensional statistics. The second algorithm is the scJEGNN, a graphical neural network (GNN) tailored to the task of data integration for HDLSS single-cell sequencing data. We show that with its unique model, the GNN is able to leverage structural information of the biological data relations in order to perform a joint embedding of multiple modalities of single-cell gene expression data. The model is applied to data from the NeurIPS 2021 competition for Open Problems in Single-Cell Analysis, and we demonstrate that our model is able to outperform top teams from the joint embedding task.
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