🤖 AI Summary
This work addresses a critical limitation in existing dimensionality reduction quality metrics, which predominantly focus on pairwise relationships while overlooking structural distortions in empty regions—areas that often define key visual layouts in scatterplots. To bridge this gap, the paper introduces the Gap Index (GI), the first metric specifically designed to quantify geometric distortion in low-dimensional empty regions. GI leverages Delaunay triangulation to construct “empty triangles” in the embedding space and measures local spatial deformation by comparing their geometry with corresponding structures in the high-dimensional space. The index can be aggregated into a scalar quality score or used to visualize regional distortion patterns. Computationally efficient and highly sensitive to visually salient fine-scale structural changes, GI demonstrably outperforms conventional metrics in capturing such subtle yet perceptually important distortions.
📝 Abstract
Quality metrics play a crucial role in the proper use of dimensionality reduction projections for visual analysis of high-dimensional data. They quantify the degree of distortion of a projection compared to the high-dimensional data and provide a reliable indication of how confident users can be in the structures they see in the resulting layouts. However, most popular metrics focus on capturing direct relationships between points (e.g., distances or neighborhoods) while neglecting distortions in empty areas of the layout, even though these often compose visually relevant features of a 2D layout. In this paper, we introduce the Gap Index (GI), a quality metric for 2D projections that captures visual distortion by measuring spatial distortion in empty areas of a projection. It does so by decomposing the space into empty triangles, which are then compared to their high-dimensional counterparts to compute the deformation. This per-triangle deformation can be aggregated into a single scalar value or overlaid on a projection to visualize regional distortion patterns. Results show that, contrary to popular quality metrics, the GI is sensitive to small structural deformations that have high visual impact. It is also fast to compute and interpretable.