Graphical Learning Optimization and Dimensionality Reduction with Geometric Multi-Resolution Analysis

This paper employs Geometric Multi-Resolution Analysis (GMRA) as a technique for dimensionality reduction and explores its impact on high-dimensional graphical learning tasks. The burgeoning surge in data collection practices, driven by technological advancements across diverse domains, has resulted in an influx of datasets wherein the number of features significantly exceeds the number of observations-a paradigm characteristic of high-dimensional datasets. Analyzing such high-dimensional datasets presents immediate challenges owing to the intricacies of dataset complexity as well as the wealth of information encapsulated within each data point. GMRA exploits redundant representations in such high-dimensional datasets, embedding the high-dimensional data into an intrinsic, underlying lower-dimensional structure. This process aims to preserve essential features while reducing dimensionality and facilitate analysis by mitigating the computational complexities associated with analyzing high-dimensional datasets. This paper proposes a novel application of Geometric Multi-Resolution Analysis to dimensionality reduction in graph embeddings. Empirically, its efficacy is validated by its performance in computing the intrinsic, underlying lower-dimensional structure for a comprehensive set of graph learning tasks, including node classification, edge classification, link prediction, anomaly detection, and graph clustering.