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Class Angular Distortion Index for Dimensionality Reduction

This paper introduces the Class Angular Distortion Index (CADI), a differentiable metric based on internal angles that evaluates the faithfulness of cluster organization in dimensionality reduction projections, addressing limitations of existing metrics that fail to capture relative cluster arrangements or assume spherical shapes.

Original authors: Kaviru Gunaratne, Stephen Kobourov, Jacob Miller

Published 2026-05-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Kaviru Gunaratne, Stephen Kobourov, Jacob Miller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: The "Flat Map" Dilemma

Imagine you have a giant, 3D sculpture made of different colored balls of clay. Some balls are nested inside others (like Russian dolls), some are shaped like long rings, and some are twisted into complex knots.

Now, imagine you want to take a photo of this sculpture to show it to someone who only understands 2D drawings (like a piece of paper). This is what Dimensionality Reduction (DR) does: it squashes complex, high-dimensional data down into a simple 2D or 3D picture so humans can see it.

The problem is that squashing a 3D object onto a 2D page inevitably distorts it.

The Old Way: For years, popular tools like t-SNE and UMAP have been the "artists" of this world. They are great at grouping similar things together. If you have a pile of red clay and a pile of blue clay, they will make two distinct, tight blobs on the paper.
The Catch: While they make the blobs look pretty, they often mess up the relationship between the blobs. They might put the red blob right next to the blue one, even if in the original 3D world, they were on opposite sides of the room. Or, they might crush a long, twisted ring into a tight circle, making it look like a solid ball.

The Old Rulers: Why They Failed

Scientists needed a way to grade these "photos" to see which one was the most honest. They used old rulers like the Silhouette Score and Davies-Bouldin Index.

Think of these old rulers as judges who only care about one thing: "Are the groups tight and separated?"

If the judge sees two tight, round, separated blobs, they give a high score.
If the judge sees a long, winding chain or a set of nested rings, they get confused. They might think the nested rings are just one big messy blob, or they might prefer a random, messy drawing because it accidentally looks like separated blobs.

The paper argues that these judges are blind to shape and structure. They don't care if a ring is actually a ring or if one ball is inside another; they just want to see clean, round, separated circles.

The New Solution: CADI (The "Angle" Judge)

The authors introduce a new metric called CADI (Class Angular Distortion Index).

The Analogy: The Tourist with a Camera
Imagine you are a tourist standing in a museum looking at a sculpture. To understand the shape of the room, you don't just look at the distance between objects; you look at the angles.

If you stand at point A and look at points B and C, what is the angle between your line of sight to B and your line of sight to C?
If you move to a different spot, does that angle change?

CADI works the same way. Instead of measuring how far apart clusters are, it measures the angles formed by triplets of points.

It picks a point in one group (Class A).
It picks two points in a different group (Class B).
It measures the angle formed at the first point.

If the 2D picture preserves the "shape" of the world, those angles will stay the same. If the picture squashes a ring into a ball, those angles will change wildly. CADI calculates the "error" in these angles. A lower score means the angles are preserved, and the map is more honest.

Why This Matters: The "Russian Doll" Test

The paper tests this with a specific example: Nested Spheres (like Russian dolls).

The Reality: You have a small sphere inside a medium one, inside a large one.
The Old Judges (Silhouette): They hate this. They want to see three separate, round balls. They might rate a random, messy scatter plot as "good" because the points happen to look separated, even though the nesting structure is completely lost.
The New Judge (CADI): It looks at the angles. It sees that the "inner" sphere is being viewed from the "outer" sphere at a specific angle. It knows that if the projection flattens this into a mess, the angles will be wrong. It correctly identifies the projection that keeps the "dolls" nested.

The New Tool: AngleEmbedding

Because CADI is based on math that can be calculated step-by-step (it's "differentiable"), the authors didn't just build a ruler; they built a new artist.

They created a new method called AngleEmbedding.

Think of this as a new type of camera that doesn't just try to group things together. Instead, it tries to minimize the angle errors.
It uses a neural network (a type of AI) to constantly adjust the picture until the angles between the groups match the original 3D world as closely as possible.
The result? Projections that might look a bit "messier" or less "perfectly separated" than t-SNE, but they tell the truth about how the groups relate to each other (e.g., showing that one group is actually inside another, or that they form a chain).

Summary of Findings

Old metrics are biased: They love round, separated blobs and hate complex shapes like rings, chains, or nested structures. They often give high scores to bad, random maps.
CADI sees the shape: By measuring angles, CADI can tell the difference between a true nested structure and a messy pile. It rewards maps that keep the "skeleton" of the data intact.
AngleEmbedding works: The new method they built using CADI successfully recreates complex structures (like the nested spheres and linked rings) that other popular tools fail to show correctly.

The Bottom Line: If you want to see how different groups of data are separated, the old tools (t-SNE/UMAP) are fine. But if you want to understand the true geometry and relationships between those groups (like who is inside whom, or how they are chained together), CADI is the better judge, and AngleEmbedding is the better artist.

1. Problem Statement

Dimensionality Reduction (DR) techniques (e.g., t-SNE, UMAP, PCA) are essential for visualizing high-dimensional data. However, a critical limitation in current DR evaluation is the misleading representation of cluster relationships.

The Core Issue: While many DR methods successfully separate clusters (making them distinct "blobs"), they often distort the global geometric relationships between these clusters. For instance, nested structures, curved manifolds, or intertwined clusters may be flattened or arbitrarily rearranged in the projection.
Limitations of Existing Metrics: Current quality metrics fail to capture these distortions because they rely on flawed assumptions:
- Separability Assumption: Metrics like Silhouette Score and Davies-Bouldin Index assume clusters are compact, globular, and well-separated. They reward projections that create tight, isolated "blobs," even if this destroys the true topological structure (e.g., turning nested spheres into separate circles).
- Centroid Bias: Metrics like Label Trustworthiness & Continuity (Label-T&C) rely on cluster centroids. If clusters overlap or are nested (centroids coincide), these metrics fail to distinguish between a faithful projection and a random one.
- Pointwise vs. Cluster-level: Most metrics evaluate point-to-point or pairwise distances, missing higher-order geometric relationships between groups of points.

2. Methodology: Class Angular Distortion Index (CADI)

The authors propose CADI, a new metric designed to evaluate the preservation of inter-cluster geometric relationships using internal angles.

Core Concept

Instead of measuring distances between centroids or individual points, CADI measures the internal angles formed by triplets of points where the reference point belongs to one class, and the other two points belong to a different class.

Intuition: By observing how a cluster $C_b$ "looks" from the perspective of a point in cluster $C_a$ (via the angle $\angle jik$ ), the metric captures the shape and relative orientation of clusters.
Differentiability: Because it relies on cosine similarity of angles, CADI is differentiable, allowing it to be used as a loss function for optimization.

Mathematical Definition

Given a dataset $X$ and a projection $Y$ , with a partition of classes $C$ :
$\text{CADI}(X, Y, C) = \frac{1}{T} \sum_{i \in C_a, j,k \in C_b, j \neq k} \left( \cos(\theta_X(j,i,k)) - \cos(\theta_Y(j,i,k)) \right)^2$
Where:

$i$ is a reference point in class $C_a$ .
$j, k$ are points in a different class $C_b$ .
$\theta$ represents the internal angle at $i$ formed by vectors to $j$ and $k$ .
$T$ is the total number of sampled triplets.
Goal: Minimize CADI (closer to 0 is better).

Practical Implementation

Sampling: Computing the full sum is $O(n^3)$ . The authors demonstrate that sampling $k = O(n)$ triplets (specifically $40n$ ) yields stable results with negligible variance, making the metric scalable.
Invariance: CADI is invariant to translation, rotation, and uniform scaling, as it relies solely on angles.
AngleEmbedding: The authors introduce a new DR algorithm, AngleEmbedding, which uses CADI as a loss function optimized via a parametric neural network (feedforward network) to generate projections that specifically preserve inter-class angular relationships.

3. Key Contributions

New Metric (CADI): A cluster-level quality metric that moves beyond the "globularity" and "separability" assumptions of existing metrics. It specifically targets the preservation of complex structures like nested, curved, or intertwined clusters.
Demonstration of Failure Modes: The paper provides extensive evidence that existing metrics (Silhouette, Label-T&C, etc.) often rank random projections or "blob" projections higher than projections that faithfully preserve complex topology.
AngleEmbedding Algorithm: A novel supervised DR technique that optimizes CADI, proving that the metric can be effectively used to drive the generation of high-fidelity projections.
Comprehensive Benchmarking: Evaluation across 20+ synthetic and real-world datasets (including nested spheres, rings, gene expression, and text data) comparing CADI against 8 other metrics and 8 DR algorithms.

4. Results

The evaluation highlights significant discrepancies between CADI and existing metrics:

Synthetic Data (Nested Spheres & Rings):
- Scenario: Data consists of 5 nested spheres or a chain of interlocked rings.
- Existing Metrics: Silhouette Score and Label-T&C often rank t-SNE or UMAP (which break the rings into separate blobs or scatter the spheres) as the best, or even rank Random Projections highly because centroids overlap.
- CADI: Correctly identifies AngleEmbedding and PCA (which preserve the linear or nested ordering) as the best, assigning the worst scores to methods that destroy the topology.
Real-World Data (Liver, USPS, TREC):
- Liver Data: Cancer cells (HCC) have higher variance than healthy cells. t-SNE creates equal-sized, separated blobs, hiding this variance. CADI rewards AngleEmbedding and PCA, which correctly show the HCC cluster as more spread out.
- USPS (Digits): Cluster sizes vary significantly by digit. t-SNE forces equal-sized clusters. CADI rewards projections where cluster sizes reflect the true data distribution (preserved by AngleEmbedding).
- Correlation: CADI shows low correlation with other metrics (e.g., Spearman correlation with Silhouette is ~0.24), confirming it measures a distinct aspect of projection quality.
Hyperparameter Tuning: CADI was successfully used to guide hyperparameter selection for UMAP, identifying configurations that better preserve global structure even when visual inspection is ambiguous.

5. Significance and Conclusion

Paradigm Shift: The paper argues that for tasks involving cluster relationship analysis (e.g., understanding hierarchy, nesting, or global topology), current "separability-focused" metrics are misleading. CADI offers a necessary alternative for evaluating structural fidelity.
Tool Selection: The authors clarify the appropriate use cases:
- Use t-SNE/UMAP if the goal is simply to identify the number of clusters or for initial exploration.
- Use CADI/AngleEmbedding if the goal is to understand how clusters relate to one another, their relative positions, and their geometric shapes.
Future Work: The authors suggest extending AngleEmbedding to non-Euclidean spaces (spheres, hyperboloids) and further theoretical analysis of the aggregation of angular errors.

In summary, CADI addresses a critical blind spot in dimensionality reduction evaluation by quantifying the distortion of inter-cluster geometry, providing a more reliable guide for analysts working with complex, non-globular data structures.