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Abstract: Classical statistical techniques for classification have historically been tailored for scenarios wherein the number of observations significantly exceeds the number of features, a paradigm characteristic of low-dimensional datasets. However, recent advancements in technologies have ushered in a transformative era in data collection practices across diverse domains, resulting in the acquisition of extensive feature measurements. As a result of this shift, datasets have transitioned into a high-dimensional realm wherein the number of features significantly exceeds the number of observations, rendering classical statistical techniques such as least squares ill-suited. Analyzing such high-dimensional datasets presents immediate challenges owing to the intricacies of the dataset complexity and the wealth of information encapsulated within each data point. Fortunately, high-dimensional datasets often exhibit redundancy, where many individual features can be expressed through a combination of other features, thereby indicating an exploitable property suggesting that such high-dimensional datasets possess an intrinsic, underlying lower-dimensional structure. An analysis of dimensionality reduction techniques, specifically leveraging Geometric Multi-Resolution Analysis (GMRA) reveals an intrinsic low-dimensional structure across various types of datasets, including signals, images, and graphs and networks. Employing Geometric Multi-Resolution Analysis (GMRA) showcases the effective computation of low-dimensional representations within high-dimensional data spaces, thus offering concise yet informative depictions of the original datasets.

Thesis Committee: Professor Chin (Chair), Professor Vaze, Professor Raymond