Structural Compactness as a Complementary Criterion for Explanation Quality

Abstract

In the evaluation of attribution quality, the quantitative assessment of explanation legibility is particularly difficult, as it is influenced by varying shapes and internal organization of attributions not captured by simple statistics. To address this issue, we introduce Minimum Spanning Tree Compactness (MST-C), a graph-based structural metric that captures higher-order geometric properties of attributions, such as spread and cohesion. These components are combined into a single score that evaluates compactness, favoring attributions with salient points spread across a small area and spatially organized into few but cohesive clusters. We show that MST-C reliably distinguishes between explanation methods, exposes fundamental structural differences between models, and provides a robust, self-contained diagnostic for explanation compactness that complements existing notions of attribution complexity.

Figures (10)

• Figure 1: Overview of workflow. Given a sample, a prediction (a) and different attributions (b) are obtained. (c) scores are derived from each attribution, by (1) converting it into a graph, (2) obtaining its preserving spatial structure (3) computing the length, (4) capturing attribution spread and cohesion, and (5) aggregating these properties into a final score.
• Figure 2: Attribution heatmaps (upper rows) across methods and models for a single sample, with corresponding graph visualizations (bottom row) highlighting structural differences. Refer to Figure \ref{['fig:qual_comparison2']} in the appendix for additional examples.
• Figure 3: Correlation of with Sparseness, Complexity, and Effective Complexity across 250 samples, spanning different attribution methods and models. As shown, exhibits a strong positive correlation with Sparseness (up to 0.85 for LRP-$\varepsilon$-$z^+$-$\text{flat}$, on the SimCLR architecture) and negative correlations with Complexity and Effective Complexity.
• Figure 4: Comparison of samples with similar complexity metric scores but divergent values on the SimCLR architecture. While the samples receive comparable evaluations under Complexity, Effective Complexity, and Sparsity, they differ substantially in . The dashed lines highlight pairs of samples that obtain similar scores under each complexity metric but exhibit noticeably different spatial structures. This indicates that can reveal nuanced structural (and visually meaningful) differences in attribution maps that are not captured by traditional complexity metrics.
• Figure 5: Correlation of with Arras_2022 (left) and rong2022evaluating (right) across 250 samples, encompassing multiple attribution methods and models. As expected, no consistent correlation is observed between and metrics evaluating aspects of attribution correctness, such as localization () and faithfulness . In combination with results in Figure \ref{['fig:correlation_complexity']}, this underscores how captures aspects of attribution legibility, independent of attribution correctness.