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Bayesian Sphere-on-Sphere Regression with Optimal Transport Maps

Tin Lok James Ng, Kwok-Kun Kwong, Jiakun Liu, Andrew Zammit-Mangion

TL;DR

This work introduces a Bayesian framework for sphere-on-sphere regression by factorizing the regression map into an optimal transport map $S_{\nu}$ and a global rotation $R$, enabling uncertainty quantification of the inferred map. The model, FMSOS, leverages a semi-discrete OT construction with a discrete target measure $\nu$ to induce a flexible, interpretable regression that naturally clusters the covariate space via Laguerre cells. The authors establish posterior contraction rates for two prior specifications and prove a quantitative stability result for OT maps on the sphere, complemented by a practical MCMC inference scheme using dual formulation. Empirical validation includes simulations and two real-data applications (vector-cardography and cyclone trajectories) demonstrating improved fit over a pure rotation model and useful geometric interpretability. The approach offers a principled, generative, uncertainty-aware tool for spherical regression with potential extensions to broader manifolds and per-cell refinements.

Abstract

Spherical regression, where both covariate and response variables are defined on the sphere, is a required form of data analysis in several scientific disciplines, and has been the subject of substantial methodological development in recent years. Yet, it remains a challenging problem due to the complexities involved in constructing valid and expressive regression models between spherical domains, and the difficulties involved in quantifying uncertainty of estimated regression maps. To address these challenges, we propose casting spherical regression as a problem of optimal transport within a Bayesian framework. Through this approach, we obviate the need for directly parameterizing a spherical regression map, and are able to quantify uncertainty on the inferred map. We derive posterior contraction rates for the proposed model under two different prior specifications and, in doing so, obtain a result on the quantitative stability of optimal transport maps on the sphere, one that may be useful in other contexts. The utility of our approach is demonstrated empirically through a simulation study and through its application to real data.

Bayesian Sphere-on-Sphere Regression with Optimal Transport Maps

TL;DR

This work introduces a Bayesian framework for sphere-on-sphere regression by factorizing the regression map into an optimal transport map and a global rotation , enabling uncertainty quantification of the inferred map. The model, FMSOS, leverages a semi-discrete OT construction with a discrete target measure to induce a flexible, interpretable regression that naturally clusters the covariate space via Laguerre cells. The authors establish posterior contraction rates for two prior specifications and prove a quantitative stability result for OT maps on the sphere, complemented by a practical MCMC inference scheme using dual formulation. Empirical validation includes simulations and two real-data applications (vector-cardography and cyclone trajectories) demonstrating improved fit over a pure rotation model and useful geometric interpretability. The approach offers a principled, generative, uncertainty-aware tool for spherical regression with potential extensions to broader manifolds and per-cell refinements.

Abstract

Spherical regression, where both covariate and response variables are defined on the sphere, is a required form of data analysis in several scientific disciplines, and has been the subject of substantial methodological development in recent years. Yet, it remains a challenging problem due to the complexities involved in constructing valid and expressive regression models between spherical domains, and the difficulties involved in quantifying uncertainty of estimated regression maps. To address these challenges, we propose casting spherical regression as a problem of optimal transport within a Bayesian framework. Through this approach, we obviate the need for directly parameterizing a spherical regression map, and are able to quantify uncertainty on the inferred map. We derive posterior contraction rates for the proposed model under two different prior specifications and, in doing so, obtain a result on the quantitative stability of optimal transport maps on the sphere, one that may be useful in other contexts. The utility of our approach is demonstrated empirically through a simulation study and through its application to real data.
Paper Structure (32 sections, 20 theorems, 271 equations, 4 figures, 2 tables)

This paper contains 32 sections, 20 theorems, 271 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Optimal Transport
  4. Semi-discrete Optimal Transport
  5. Related Work on Normalizing Flows
  6. Related Work on Contraction Rates of Posterior Distributions
  7. Methodology
  8. Model
  9. Posterior Contraction Rates: General Regression Map
  10. Posterior Contraction: FMSOS
  11. Direct Prior Specification on Discrete Measures
  12. Prior Specification under Dual Formulation
  13. Bayesian Inference
  14. Simulation Study
  15. Data Applications
  16. ...and 17 more sections

Key Result

Theorem 1

Assume there exists sequences $(\epsilon_n)_{n=1}^{\infty}$, $(\underline{\epsilon}_n)_{n=1}^{\infty}$ such that $0 \le \underline{\epsilon}_n \le \epsilon_n \rightarrow 0$ and $\min(n \epsilon_n^2, n \underline{\epsilon}_n^2) \rightarrow \infty$ as $n \rightarrow \infty$. Let $({\cal F}_n)_{n=1}^{\ Then, as $n \rightarrow \infty$.

Figures (4)

  • Figure 1: Illustration depicting Voronoi (top-left) and Laguerre (bottom-left) tessellations for five atoms on the Euclidean domain $[0,2] \times [0,1]$, with $\psi(z) = 1$ for the Voronoi tessellation (top-right) and $\psi(z) = (z_1^3 + z_2^3)^\frac{1}{5}$ for the Laguerre tessellation (bottom-right).
  • Figure 2: The plots show a sample of randomly selected 20 covariate-response pairs. Left: Blue points show covariates, and green points represent responses, with dotted lines connecting each covariate to its corresponding response. Middle and Right: Blue points show covariates, while red points display the predicted mean responses, with arrows linking covariates to their predicted mean responses. The two panels correspond to two MCMC iterations.
  • Figure 3: The plots show a sample of randomly selected 20 covariate-response pairs. Left: Blue points indicate covariates, and green points represent responses, with arrows connecting each covariate to its corresponding response. Middle and Right: Blue points show covariates, while red points display the mean predicted responses, with arrows linking covariates to their predicted mean responses. The two panels correspond to two MCMC iterations.
  • Figure E1: Left: Blue points indicate covariates, and red points represent responses, with arrows connecting each covariate to its corresponding response. Middle and Right: Blue points show covariates, while red points display the predicted mean responses, with arrows linking covariates to their predicted mean responses. Top row: sample size = 100. Middle row: sample size = 500. Bottom row: sample size = 1000.

Theorems & Definitions (46)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Theorem 2
  • Remark 5
  • Remark 6
  • Lemma 1
  • proof
  • ...and 36 more