Cross-Fusion Distance: A Novel Metric for Measuring Fusion and Separability Between Data Groups in Representation Space

TL;DR

Cross-Fusion Distance (CFD) introduces a principled, scale-invariant metric to quantify fusion and separability between latent data groups under domain shift. Deriving CFD from a variance decomposition of the fused representation cloud, the authors define the Cross-Fusion Score $\mathrm{CFS}$ and distance $\mathrm{CFD} = -\log(\mathrm{CFS})$, achieving a kernel-free, closed-form, and linear-time computation $O(n \cdot d)$. CFD isolates fusion-altering geometry (inter-group displacement and within-group dispersion) from fusion-preserving variations (scaling, topology), and shows monotonicity and robustness in synthetic tests, while aligning with real-world cross-domain degradation in biomedical batch-effect settings. Empirical validation includes ground-truth anchored RDR calibration on histopathology data and strong correlation with cross-domain performance degradation, surpassing Wasserstein, MMD, Hausdorff, and Chamfer as a deployment-oriented batch-effect metric. Overall, CFD offers a theoretically grounded, interpretable, and scalable tool for representation-space analysis under domain shift with direct practical relevance to deployment robustness.

Abstract

Quantifying degrees of fusion and separability between data groups in representation space is a fundamental problem in representation learning, particularly under domain shift. A meaningful metric should capture fusion-altering factors like geometric displacement between representation groups, whose variations change the extent of fusion, while remaining invariant to fusion-preserving factors such as global scaling and sampling-induced layout changes, whose variations do not. Existing distributional distance metrics conflate these factors, leading to measures that are not informative of the true extent of fusion between data groups. We introduce Cross-Fusion Distance (CFD), a principled measure that isolates fusion-altering geometry while remaining robust to fusion-preserving variations, with linear computational complexity. We characterize the invariance and sensitivity properties of CFD theoretically and validate them in controlled synthetic experiments. For practical utility on real-world datasets with domain shift, CFD aligns more closely with downstream generalization degradation than commonly used alternatives. Overall, CFD provides a theoretically grounded and interpretable distance measure for representation learning.

Figures (6)

• Figure 1: Wasserstein distance decomposition: $WD_{ts}$ - the translational cost arising from geometric displacement; $WD_{df}$ - the deformation cost due to structural differences.
• Figure 2: Empirical runtime comparison of distance measures as a function of sample size.
• Figure 3: Synthetic evaluation of distance measures. CFD primarily tracks geometric displacement and dispersion variation while remaining stable to global scaling, topological deformation, and outliers.
• Figure B.1: Synthetic evaluation of distance measures for latent dimensionalities $d =32$ and sample sizes $n=300$.
• Figure B.2: Synthetic evaluation of distance measures for latent dimensionalities $d =32$ and sample sizes $n=1000$.

Theorems & Definitions (10)

• Proposition 1.1: Variance Decomposition
• proof
• Proposition 1.2: Bounds of Cross-Fusion Score
• proof
• Proposition 1.3: Non-negativity and Zero Case
• proof
• Proposition 1.4: Monotonicity in Geometric Displacement
• proof
• Proposition 1.5: Limit Cases
• proof