Geometric Evaluation of Multidimensional Embeddings via Convex Hulls and Ramsey-3,3 Triples

Accurately evaluating high-dimensional embeddings is crucial in machine learning and data analysis. This study introduces a geometric framework for assessing embeddings by integrating Banach and hyperbolic space mappings with convex hull construction and Ramsey-3,3 triple analysis. Extreme points of the data are identified, convex hulls are constructed, and all points are analyzed with respect to their proximity to nearest triples of anchor points. The resulting geometric band serves as an intrinsic metric for embedding fidelity, outlier detection, and model evaluation. The methodology is demonstrated on synthetic datasets, highlighting its ability to capture core structures and identify anomalies in both Euclidean and non-Euclidean embeddings.