Class Angular Distortion Index for Dimensionality Reduction

TL;DR

The paper introduces the Class Angular Distortion Index (CADI), a new metric for evaluating how well dimensionality reduction preserves cluster organization, addressing limitations of existing metrics and enabling optimization.

Contribution

It proposes CADI, a differentiable angular-based metric for assessing cluster organization in DR, and demonstrates its use in a new DR technique.

Findings

CADI effectively measures cluster organization fidelity in DR.

Existing metrics often fail to capture cluster arrangement accurately.

CADI-based DR technique can be optimized using gradient-based methods.

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.