Class Angular Distortion Index for Dimensionality Reduction

arXiv cs.LG / 5/4/2026

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Key Points

The paper highlights a key limitation of many dimensionality reduction (DR) methods: they may preserve local neighborhoods while producing misleading global cluster arrangements in 2D/3D projections.
It introduces the Class Angular Distortion Index (CADI), a new cluster faithfulness metric that evaluates how internal angles among point triples are distorted from the original space to the projection.
The authors argue that existing cluster quality metrics often measure only separability or implicitly assume spherical, globular clusters, which can lead to incorrect conclusions when cluster geometry is more complex.
Experiments on real and synthetic datasets show that CADI can succeed in situations where prior metrics fail, producing more interpretable assessments of cluster organization.
Because CADI is based on angle computations, it is differentiable and can be used to optimize DR directly, demonstrated via a CADI-based DR technique.

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.