Geometric Dimensionality Reduction

Abstract

The emergence of AI models developed through computationally intensive training has resulted in a surge of research into dimensionality reduction techniques that spans across numerous mathematical disciplines. In this thesis we establish Geometric Dimensionality Reduction, a non-linear data compression technique that utilizes low dimensional manifolds embedded in dimensional spaces to form composite contraction-and-projection maps. Geometric Dimensionality Reduction is predominantly demonstrated through a novel algorithm entitled LGE (Livschitz-Gu-Eyunni) that utilizes Multicomplex rotation groups and polyspherical coordinates to define a single tuneable logarithmic map from ℝ 2푛 to ℝ 푛+1 with deterministic time complexity, geometric tunability, and semi-reversibility. Significant breakthroughs in the dissertation include deriving a closed form solution to the contraction map from ℝ 4 onto a constrained 푆 3 ; as well as reducing the generalized problem, projecting points embedded in ℝ 2푛 onto a parametrized 푆 2푛−1 , to solving 푛 non-linear equations. Additionally, a set of preconditioning measures have been defined that optimize clustering fidelity through invertible coordinate rotations. Contemporary methods of dimensionality reduction such as UMAP (Uniform Manifold Approximation & Projection) and t-SNE (t-distributed Stochastic Neighbor Embedding) suffer from irreversibility, hyperparameter tuning sensitivity, and computationally costly internal processes. Even LLE (Local Linear Embedding), which is designed to reduce the embedding dimension of intrinsically lower dimensional data, is sensitive to noise & variance due to the reliance on derived weights from the original local neighborhood. With a focus on topologically reducing the dimension that data is embedded in, rather than statistically determining the intrinsic dimension of data itself, Geometric Dimensionality Reduction presents a compelling alternate framework to classical compression algorithms.