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Abstract
With the rapid development of computing power, various technologies have drawn much attention for the great successes in many disciplines. Deep learning outperforms other machine learning methods in major computer vision competitions. Quantum search methods significantly expedite the intractable searching process for classical algorithms. We leverage those emerging technologies to multiple important areas in statistical learning. The first part of the dissertation focuses on deep neural networks for medical imaging data. We propose a deep learning paradigm to address several challenges in medical imaging analysis, especially for functional magnetic resonance imaging data. We develop a statistical sampling algorithm based on regions of interest to augment small data for deep learning modeling. We utilize the ComBat harmonization for multi-source batch effect correction. In addition, we implement the prediction uncertainty in deep learning through a dropout approximation of the deep Gaussian process. We assess the performance of our proposed method using various examples. The second part studies high dimensional multi-response linear models. We propose to estimate the coefficient matrix by selecting linearly independent columns without the low rank constraint. The computationally intensive column selection problem is tackled by a novel non-oracular quantum search algorithm which is substantially faster than classical search algorithms. We examine some theoretical properties of the proposed estimation approach. Intensive numerical studies also elucidate the advantages of the quantum-enhanced algorithm.