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Kantorovich Regression Analysis of Random Distributions with Mixed Predictors

Kaheon Kim, Changbo Zhu

Abstract

We study regression problems with distribution-valued responses and mixed distributional and Euclidean predictors. In quadratic cost, the negative gradient of the Kantorovich potential represents, at each source location, the displacement to its matched location under the optimal transport map. By constructing potentials from the Wasserstein barycenter to individual distributions, the proposed Kantorovich regression model approximates the response displacement field as a sum of predictor displacement fields, each adjusted by a functional parameter. Owing to the linear structure, Euclidean predictors can enter as scaling coefficients of $c$-concave parameter potentials. We characterize functional parameter classes ensuring intrinsicness of the model, establish asymptotic theory through uniform convergence of the empirical Wasserstein loss, and derive Gâteaux derivatives leading to first-order optimization algorithms. Real data applications include a mixed-predictor analysis of housing price distributions and an analysis of two-dimensional temperature distributions, demonstrating the flexibility and interpretability of the proposed framework.

Kantorovich Regression Analysis of Random Distributions with Mixed Predictors

Abstract

We study regression problems with distribution-valued responses and mixed distributional and Euclidean predictors. In quadratic cost, the negative gradient of the Kantorovich potential represents, at each source location, the displacement to its matched location under the optimal transport map. By constructing potentials from the Wasserstein barycenter to individual distributions, the proposed Kantorovich regression model approximates the response displacement field as a sum of predictor displacement fields, each adjusted by a functional parameter. Owing to the linear structure, Euclidean predictors can enter as scaling coefficients of -concave parameter potentials. We characterize functional parameter classes ensuring intrinsicness of the model, establish asymptotic theory through uniform convergence of the empirical Wasserstein loss, and derive Gâteaux derivatives leading to first-order optimization algorithms. Real data applications include a mixed-predictor analysis of housing price distributions and an analysis of two-dimensional temperature distributions, demonstrating the flexibility and interpretability of the proposed framework.
Paper Structure (15 sections, 11 theorems, 56 equations, 11 figures)

This paper contains 15 sections, 11 theorems, 56 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Optimal transport
  4. Wasserstein barycenter
  5. Kantorovich Regression
  6. Single distributional predictor
  7. Multiple distributional predictors
  8. Mixed predictor
  9. Asymptotic Theory
  10. Computational Algorithms
  11. Numerical Studies
  12. 1D synthetic data
  13. 1D real data
  14. 2D real data
  15. Conclusion

Key Result

Lemma 3.1

Given a $c$-concave function $\phi \in \mathcal{C}^2(\Omega)$ such that $\nabla \phi(x) \neq 0$ for any $x \in \Omega$ and $g\in \mathcal{C}^1(\mathbb{R})$, we let $R(x) := x - g (x) \nabla \phi (x)$. If $R$ is cyclically monotone, then $g$ is constant on each connected component of each level set $

Figures (11)

  • Figure 1: Illustration of the adjustment on $\phi$ by $f$. Column 1 (bottom row) shows the Kantorovich potential $\phi:[0, 1]^2 \rightarrow \mathbb{R}$ as a heat map, together with the gradient field $\nabla \phi$ drawn as arrows. From column 2 onward, the top row displays the derivatives $f'$ of the functional parameter $f$, while the bottom row shows $f\circledcirc\phi$ as heat maps with the corresponding gradient field $\nabla (f\circledcirc\phi)$ (arrows).
  • Figure 2: 1D illustration of KR model with non-decreasing concave functional parameter.
  • Figure 3: 1D illustration of KR with non-increasing convex functional parameter.
  • Figure 4: 2D illustration of KR with non-decreasing concave functional parameter.
  • Figure 5: 2D illustration of KR with non-increasing convex functional parameter.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Lemma 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Remark 3.4
  • Remark 3.5
  • Corollary 3.6
  • Theorem 3.7
  • Corollary 3.8
  • Theorem 4.3
  • Corollary 4.4
  • ...and 3 more