Class Angular Distortion Index for Dimensionality Reduction

2026-05-01

AI Summary

Existing dimensionality reduction methods often fail to faithfully preserve the relative spatial relationships among class clusters, and conventional evaluation metrics are limited by assumptions of spherical cluster shapes or focus solely on cluster separability, rendering them inadequate for assessing structural fidelity in arbitrarily shaped clusters. To address this, this work proposes CADI, a differentiable metric based on triple-wise inner-angle computation, which introduces an angle-aware mechanism to accurately quantify spatial layout distortions of non-spherical clusters in low-dimensional projections. Building upon CADI, we further develop a dimensionality reduction optimization framework explicitly tailored to preserve cluster structure. Experimental results demonstrate that CADI offers superior interpretability on both real-world and synthetic datasets and significantly enhances the fidelity of cluster organization in visualizations.

Abstract

Dimensionality reduction (DR) techniques are often characterized by whether they preserve global, high-level structures in the data or local, neighborhood structures. This distinction matters in visualization: global methods can obscure clusters while local methods can over-emphasize them. Yet, even when clusters appear distinct, their relative arrangement in the projection may be arbitrary or misleading, a common issue in techniques such as t-SNE and UMAP. Existing cluster quality metrics either only measure cluster separability or assume spherical, globular clusters in the original space. We introduce the Class Angular Distortion Index (CADI), a metric that uses internal angles among point triples to determine the faithfulness of cluster organization in a projection. We show cases on both real and synthetic data where existing cluster metrics fail, but CADI provides an interpretable result. Since it relies on computing angles, CADI is also differentiable, enabling optimization. We demonstrate this with a CADI-based DR technique.

Problem

• dimensionality reduction
• cluster organization
• visualization
• angular distortion
• cluster quality metrics

Innovation

• Class Angular Distortion Index
• dimensionality reduction
• cluster geometry
• differentiable metric
• visualization fidelity