Analysis of Information Loss in Dimensionality Projection: An Integrated, Hypersphere Volume-Based Approach

This paper presents a novel geometric approach to quantifying information loss in dimensionality reduction through a hypersphere volume-based metric. Unlike traditional data-centric methods that focus on preserving specific dataset properties like variance (PCA) or structural relationships (t-SNE, UMAP), this work introduces a universal measure based on the fundamental geometric properties of Euclidean space itself. The proposed metric calculates information loss as the relative decrease in "geometric entropy" - defined as the natural logarithm of n-dimensional unit hypersphere volumes - when projecting from higher to lower dimensions. Key findings demonstrate increasing geometric information loss as target dimensions decrease (10.27% for 4D→3D, 20.08% for 3D→2D, 39.45% for 2D→1D), directly reflecting the counterintuitive behavior of hypersphere volumes that peak around 5-6 dimensions before rapidly approaching zero. Rather than competing with existing methods, this metric serves as a complementary diagnostic baseline that, when compared against data-specific loss measures, reveals insights about data sparsity and intrinsic manifold structure. The work ultimately advocates for an integrated evaluation framework that combines universal geometric constraints with empirical data properties, proposing future extensions toward "effective volume" measures that incorporate actual data distributions while maintaining the theoretical elegance of pure geometric analysis.