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Gap Index

Gap Index quantifies how strongly empty triangular regions in a 2D projection are deformed relative to the corresponding regions in the original space. A score of 0 indicates no regional area distortion; the score is bounded by 1.

The metric was introduced by Jaume Ros, Alessio Arleo, and Fernando Paulovich in Measuring Distortion in the Empty Regions of Dimensionality Reduction Scatterplots with the Gap Index.

Standard ZADU interface

from zadu import ZADU

specs = [{"id": "gi", "params": {"metric": "euclidean"}}]
score = ZADU(specs, original).measure(projection)[0]["gap_index"]

The projection must have exactly two columns and contain at least three non-collinear points. Its Delaunay triangulation defines the regions.

Distance choices

metric may be:

  • "euclidean" for the optimized coordinate path;
  • the name of a SciPy distance function such as "cityblock";
  • a callable accepting two original-space rows; or
  • "precomputed" when original is a finite, symmetric, non-negative (n, n) distance matrix with a zero diagonal.

The projection-space triangle edges are always Euclidean, matching the published formulation.

from scipy.spatial.distance import pdist, squareform
from zadu.measures import gap_index

distances = squareform(pdist(original))
score = gap_index.gap_index(distances, projection, metric="precomputed")

Regional details

Use the direct compute() function to obtain the triangulation and its per-region values:

from zadu.measures import gap_index

result = gap_index.compute(original, projection)

print(result.score)
print(result.triangles)
print(result.deformations)
print(result.original_relative_areas)
print(result.embedded_relative_areas)

The standard scheduled API intentionally returns only the finite scalar score. The direct detailed result is useful for scientific analysis and custom visualization.

Provenance

ZADU adapted the authors' MIT-licensed reference implementation at revision 0a11e4887864fe5d41526d8487eea33685b8f0b4. The port adds ZADU's measure contract, validation, typed detailed results, bounded vectorization for Euclidean and precomputed areas, and regression tests pinned to an upstream golden result.

The original algorithm and implementation remain credited to Ros, Arleo, and Paulovich. See the repository's third-party notice and retained MIT license.