The Fréchet correlation coefficient for heterogeneous random objects

1. Problem Statement

Modern data analysis increasingly involves heterogeneous random objects where predictors ($X$) and responses ($Y$) reside in different, potentially non-Euclidean metric spaces (e.g., distributions in Wasserstein space, symmetric positive definite (SPD) matrices, or circular data).

• Limitation of Classical Methods: Traditional measures of explanatory power, such as the Euclidean coefficient of determination ($R^2$) and Pearson correlation, rely on linear projections and Euclidean geometry. They are not directly applicable to metric spaces where linear projections do not exist.
• Limitation of Existing Dependence Measures: While recent methods (e.g., distance covariance, ball covariance) can detect dependence in metric spaces, they are typically symmetric ($X$ vs. $Y$) and do not provide a directional measure of how much variation in $Y$ is explained by $X$. They also often lack a clear interpretation as an "explained variance" ratio.
• Core Question: How can one define a directional, scale-free, and interpretable coefficient that quantifies the proportion of Fréchet variation in a response $Y$ explained by a predictor $X$, allowing for the comparison of heterogeneous predictors on a common scale?

2. Methodology

2.1 The Fréchet Correlation Coefficient (FCC)

The authors define the FCC ($\rho$) as the relative reduction in the Fréchet variance of $Y$ after conditioning on $X$.

• Definitions:
  • Fréchet Mean ($\mu_F$): The minimizer of expected squared distance: $\mu_F = \arg\min_{\mu} E[d^2_Y(Y, \mu)]$.
  • Fréchet Variance ($V_F$): $E[d^2_Y(Y, \mu_F)]$.
  • Conditional Fréchet Mean ($\mu_{F,x}$): The minimizer of expected squared distance given $X=x$.
  • Residual Variance: $E[V(X)] = E[d^2_Y(Y, \mu_{F,X})]$.
• Formula:
  $$\rho = 1 - \frac{E[V(X)]}{V_F}$$
  This represents the proportion of total Fréchet variance explained by the predictor.
• Properties:
  • Directional: $\rho(X, Y) \neq \rho(Y, X)$ generally.
  • Interpretability: $\rho \in [0, 1]$.
    • $\rho = 1$ if and only if $Y$ is almost surely a measurable function of $X$ (perfect prediction).
    • $\rho = 0$ if and only if the conditional Fréchet mean equals the global Fréchet mean almost surely (no Fréchet mean dependence). Note that independence implies $\rho=0$, but $\rho=0$ does not necessarily imply independence (it only implies no dependence in the mean).

2.2 Partition-Based Estimator

Direct nonparametric estimation of the conditional Fréchet mean function $\mu_{F,x}$ is computationally difficult and statistically unstable in general metric spaces. The authors propose a partition-based estimator to avoid explicit function estimation.

• Procedure:
  1. Partition the predictor space $\mathcal{X}$ into $M_n$ cells $\{\Omega_m\}_{m=1}^{M_n}$. The partition should depend only on the geometry of $X$ (e.g., via Voronoi cells around random prototypes or clustering), not on $Y$.
  2. Compute the sample Fréchet mean and variance within each cell $m$.
  3. Estimate the residual variance as a weighted average of cell-wise variances.
  4. Estimate $\rho$ as:
     $$\hat{\rho}_{M_n} = 1 - \frac{\sum_{m=1}^{M_n} \frac{n_m}{n} \hat{V}_m}{\hat{V}_F}$$
     where $n_m$ is the sample size in cell $m$, $\hat{V}_m$ is the cell-wise sample variance, and $\hat{V}_F$ is the global sample variance.
• Advantage: This approach avoids the "curse of dimensionality" associated with estimating a full regression surface and is computationally efficient.

3. Key Contributions

1. Theoretical Formulation: Introduced the FCC as a unified, directional measure of explained variation for random objects in general metric spaces. Established that it generalizes the concept of $R^2$ to non-Euclidean settings.
2. Estimation Strategy: Proposed a novel, efficient partition-based estimator that bypasses the need for explicit nonparametric estimation of conditional Fréchet means.
3. Asymptotic Theory:
   • Proved consistency of the estimator under mild regularity conditions (boundedness, uniqueness of means).
   • Derived null asymptotic distributions for the hypothesis of no Fréchet mean dependence ($H_0: \rho = 0$) under two regimes:
     • Fixed-partition regime: The limit follows a weighted chi-square distribution.
     • Growing-partition regime: Depending on the cumulative variance structure, the limit is either asymptotically normal (Central Limit Theorem) or a second-order Gaussian chaos (weighted sum of chi-squares).
   • These results cover both finite-dimensional Riemannian manifolds and infinite-dimensional Wasserstein spaces (via $L^2$ quantile embedding).
4. Empirical Validation: Demonstrated through simulations that FCC is competitive with or superior to existing dependence measures (Ball covariance, Energy distance, Chatterjee's correlation) in detecting nonlinear and non-Euclidean dependencies.

4. Key Results

• Boundary Cases: The paper rigorously proves that $\rho=1$ corresponds to perfect functional dependence and $\rho=0$ corresponds to the absence of Fréchet mean dependence.
• Noise Sensitivity: In representative models (Wasserstein distributions and SPD matrices), FCC decreases monotonically as the noise level increases, validating its interpretation as a signal-to-noise ratio for explained variation.
• Simulation Performance:
  • In Euclidean settings, FCC performs comparably to state-of-the-art rank-based and distance-based methods.
  • In Non-Euclidean settings (e.g., spherical data, Wasserstein distributions, SPD matrices), FCC demonstrates high power, often outperforming competitors in complex, nonlinear, or non-injective dependence structures.
  • The method successfully handles heterogeneous predictors (e.g., comparing a scalar predictor vs. a matrix predictor in explaining a distributional response).

5. Significance

• Unification: FCC provides a common scale for ranking heterogeneous predictors (e.g., comparing the explanatory power of a brain connectome matrix vs. a sleep cycle vector on a cognitive distribution response).
• Model-Free: It requires no specification of a parametric regression model, making it robust to complex, unknown relationships.
• Directionality: Unlike omnibus independence tests, FCC specifically answers the scientific question: "How much of the variation in $Y$ is explained by $X$?" This is crucial for feature selection and variable importance in multimodal studies.
• Practicality: The partition-based estimator is computationally feasible and easy to implement, making it a practical tool for modern data science applications involving complex data types (images, graphs, distributions, matrices).

In summary, the paper fills a critical gap in statistical methodology by extending the concept of the coefficient of determination ($R^2$) to the realm of heterogeneous, non-Euclidean random objects, providing both a theoretical foundation and a practical tool for modern multimodal data analysis.