Concatenated Distance Covariance (CCdCov)

• Concatenated Distance Covariance (CCdCov) is a multivariate dependence measure that applies classical distance covariance to concatenated variable blocks, nullifying only when full independence holds.
• It decomposes the joint dependency between model predictions and grouped protected attributes, effectively capturing both marginal and intersectional relationships.
• Its computational framework, which involves double-centering distance matrices and leveraging unbiased estimators, extends to high-dimensional and non-Euclidean data settings.

Concatenated Distance Covariance (CCdCov) is a multivariate dependence measure constructed by applying the classical distance covariance to random vectors formed through the concatenation of multiple variable groups or feature blocks. By leveraging the characteristic property of distance covariance—vanishing if and only if independence holds—CCdCov quantifies the joint (potentially nonlinear and high-order) association between a target vector (such as model predictions) and a possibly heterogeneous set of attributes, such as protected features in fairness-regularized machine learning. CCdCov has been recently formalized and applied in fairness-gerrymandering contexts, providing a rigorous energy-based framework for detecting intersectional dependencies and enforcing independence constraints among complex, multi-type data sources (Lee et al., 9 Sep 2025).

1. Mathematical Formulation and Independence Characterization

At its core, CCdCov generalizes the definition of Brownian distance covariance. Let $X = (X_1, \ldots, X_k)$ and $Y = (Y_1, \ldots, Y_\ell)$ be collections of random vectors (blocks), potentially of varying type and distribution. The concatenated vectors are

$$X_\mathrm{concat} = (X_1, X_2, \ldots, X_k), \qquad Y_\mathrm{concat} = (Y_1, Y_2, \ldots, Y_\ell).$$

CCdCov is defined as

$$CCdCov(X_1, \ldots, X_k ; Y_1, \ldots, Y_\ell) := \mathcal{V}^2(X_\mathrm{concat}, Y_\mathrm{concat}),$$

where $\mathcal{V}^2$ denotes classic distance covariance between vectors in arbitrary dimensions (Székely et al., 2010).

The population version uses characteristic functions via

$$\mathcal{V}^2(X,Y) = \frac{1}{c_p c_q} \int_{\mathbb{R}^{p+q}} \frac{|f_{X,Y}(t,s)-f_X(t)f_Y(s)|^2}{|t|^{p+1}|s|^{q+1}}\,dt\,ds,$$

where $f_{X,Y}$, $f_X$, and $f_Y$ are the joint and marginal characteristic functions. This construction ensures

$$CCdCov(X_1, \ldots, X_k ; Y_1, \ldots, Y_\ell) = 0 \iff (X_1, \ldots, X_k) \perp (Y_1, \ldots, Y_\ell).$$

Consequently, CCdCov preserves the strict independence-detection property at the concatenated level.

2. Decomposition and Theoretical Properties

A key result in applied fairness contexts is the decomposition: $$CCdCov(\hat{\mathbf{y}}, \mathbf{s}_1, \ldots, \mathbf{s}_d) = \sum_{k=1}^d \widetilde{dCov}^2(\hat{\mathbf{y}}, \mathbf{s}_k) + \eta(\hat{\mathbf{y}}, \mathbf{s}_1, \ldots, \mathbf{s}_d),$$ where $\hat{\mathbf{y}}$ is typically a vector of model predictions, $\mathbf{s}_k$ are protected attribute blocks, $\widetilde{dCov}^2$ denotes unbiased distance covariance to an individual block, and $\eta$ is a correction capturing higher-order (intersectional) dependency (Lee et al., 9 Sep 2025). Unlike measures such as JdCov, which aggregate marginal, pairwise, and higher-order terms among both the target and protected blocks (thus penalizing intrinsic correlations within protected attributes), CCdCov restricts attention to the dependency between the target and the entire concatenated attribute vector, making it more stable in presence of natural protected attribute dependence.

Under appropriate moment and regularity conditions, all properties of classical distance covariance (consistency, affine invariance with normalization, and extension to general metric spaces of strong negative type) extend to the concatenated construction (Lyons, 2011, Janson, 2019).

3. Algorithmic Implementation and Computational Considerations

Empirical CCdCov is computed analogously to classical distance covariance, but using concatenated sample vectors:

1. Stack or concatenate the relevant blocks for each sample to form joint vectors.
2. Compute the pairwise distance matrices for both the concatenated predictions and concatenated protected features.
3. Double-center each distance matrix; then apply the unbiased U-statistic or (optionally) the easier-to-compute V-statistic estimator (Monroy-Castillo et al., 3 May 2024, Raymaekers et al., 18 Jun 2024).
4. Combine using the empirical formula:

$$\widehat{CCdCov} = \frac{1}{n(n-3)} \sum_{i \neq j} \tilde{A}_{ij} \tilde{B}_{ij},$$

where $\tilde{A}_{ij}$, $\tilde{B}_{ij}$ are the double-centered distance matrices.

Computational complexity in the worst case is $O(n^2)$, but for univariate or single-block settings fast algorithms are available with $O(n\log n)$ complexity (Chaudhuri et al., 2018), and optimization strategies can be adopted for large-scale applications. When generalizing to high-dimensions or metric spaces, additional normalization (such as standardization or working in reproducing kernel Hilbert spaces) is essential (Székely et al., 2010, Böttcher et al., 2017).

4. Applications in High-dimensional Inference and Fairness

CCdCov is particularly effective in fairness-sensitive machine learning, especially for addressing fairness gerrymandering. By concatenating all protected attributes (e.g., gender, ethnicity, age, region) into a single joint vector and computing the distance covariance to model predictions, CCdCov captures both marginal and intersectional disparities that would elude attribute-wise approaches (Lee et al., 9 Sep 2025). The loss function typically takes the form: $$\min_\Theta \left\{ \frac{1}{n} \sum_{i=1}^n \mathcal{L}(\hat{y}_{\Theta,i}, y_i) + \lambda \, CCdCov(\hat{\mathbf{y}}, (\mathbf{s}_1,\ldots,\mathbf{s}_d)) \right\},$$ where $\mathcal{L}$ is a prediction loss and $\lambda$ a regularization strength.

Empirical evidence in classification (e.g., the COMPAS dataset) and regression (motor insurance claims) demonstrates that CCdCov regularization enforces parity of prediction distributions across both individual and intersectional subgroups, rendering subgroup-dependent disparities nearly zero in appropriately regularized models. Importantly, CCdCov does not penalize inherent dependence among protected attributes, which prevents undesirable numerical instability (Lee et al., 9 Sep 2025).

Beyond fairness, the conceptual framework of concatenating several groups or blocks appears in mutual independence testing, ICA, functional dimension reduction, and joint dependence estimation for stochastic processes and time series (Matteson et al., 2013, Matsui et al., 2017, Yang et al., 2022).

5. Extensions and Limitations

Several theoretical and practical extensions have been proposed:

• General Metric and Functional Spaces: CCdCov applies under general metric (including Hilbert) spaces if the underlying metrics have strong negative type (Lyons, 2011, Janson, 2019). This includes structured data such as networks, text, or functional data using appropriate embeddings or kernels.
• Flexible Weighting and Negative Definite Functions: The use of continuous negative definite functions beyond Euclidean metrics (e.g., general Lévy measures) enables adaptation to heterogeneous concatenated data (Böttcher et al., 2017).
• Spectral Analysis: Recent advances in random matrix theory enable the analysis of high-dimensional CCdCov-based matrices via their spectral properties, where eigenvalue phase transitions can reveal subtle cross-block dependencies (Li et al., 2021).

There are several considerations:

• Normalization: Disparity in scale or heterogeneity across concatenated blocks can distort dependence measurement, requiring careful standardization or weighting.
• Interpretability: Additive decompositions (ADC formula) provide interpretable views of the latent feature correlations that drive CCdCov, though alignment with domain features remains an ongoing challenge (Wang et al., 2023).
• Hypothesis Testing: Asymptotic and bootstrap approximations for critical values become complex under high-dimensional concatenation and require adaptation for valid inference (Dehling et al., 2018).

6. Related Methods and Alternatives

CCdCov is contrasted with other multivariate dependence measures:

• JdCov (Joint Distance Covariance): Sums all inferable marginal, pairwise, and higher-order dependencies, but penalizes inherent structure within protected attribute blocks; less stable when these are naturally dependent (Lee et al., 9 Sep 2025).
• Distance Multivariance: Generalizes mutual independence testing for multiple multivariate random vectors, seen as an extended concatenation (Böttcher et al., 2017).
• Energy-based and RKHS-based Dependence: CCdCov is related (via kernel choices) to the Hilbert-Schmidt Independence Criterion (HSIC) (Wang et al., 2023).

Method	Captures Intersectional Dependency	Penalizes Intrinsic Block Dependence	Vanishes iff Full Independence
CCdCov	Yes	No	Yes
JdCov	Yes	Yes	No (if blocks are dependent)
Attribute-wise	No	No	No

7. Summary

CCdCov provides a principled, robust, and computationally tractable extension of distance covariance for joint dependence assessment across concatenated variable sets. Its genesis in Brownian distance covariance ensures rigorous independence detection, and its joint measurement of marginal and intersectional dependencies makes it practically essential in contemporary fairness-aware machine learning, high-dimensional inference, and complex dependency testing. Ongoing developments focus on optimal normalization, interpretability of latent structure, scalable computation, and adaptation to general metric, functional, and structured data spaces (Székely et al., 2010, Lee et al., 9 Sep 2025).