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Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable

Zhanfeng Wang, Xinyu Li, Hao Ding, Jian Qing Shi

TL;DR

This paper addresses regression when responses lie on Riemannian manifolds, where standard extrinsic GP methods can distort geometry. It introduces intrinsic Gaussian process regression (iGPR) by building an intrinsic covariance function through parallel transport in tangent spaces and by consolidating information in a common tangent space at a base point, using a coregionalization-based kernel. The authors prove information consistency and posterior consistency, and show that predictions are invariant to the choice of tangent frame or basepoint curve $ obreakline oldsymbol{ u}(t)$, provided appropriate transport is applied. Through simulations on SPD manifolds and real-data examples involving flight trajectories on $ ext{S}^2$ and diffusion tensors on SPD manifolds, iGPR delivers superior predictive accuracy and uncertainty quantification, highlighting its practical value for manifold-valued data analysis.

Abstract

Extrinsic Gaussian process regression methods, such as wrapped Gaussian process, have been developed to analyze manifold data. However, there is a lack of intrinsic Gaussian process methods for studying complex data with manifold-valued response variables. In this paper, we first apply the parallel transport operator on Riemannian manifold to propose an intrinsic covariance structure that addresses a critical aspect of constructing a well-defined Gaussian process regression model. We then propose a novel intrinsic Gaussian process regression model for manifold-valued data, which can be applied to data situated not only on Euclidean submanifolds but also on manifolds without a natural ambient space. We establish the asymptotic properties of the proposed models, including information consistency and posterior consistency, and we also show that the posterior distribution of the regression function is invariant to the choice of orthonormal frames for the coordinate representations of the covariance function. Numerical studies, including simulation and real examples, indicate that the proposed methods work well.

Intrinsic Gaussian Process Regression Modeling for Manifold-valued Response Variable

TL;DR

This paper addresses regression when responses lie on Riemannian manifolds, where standard extrinsic GP methods can distort geometry. It introduces intrinsic Gaussian process regression (iGPR) by building an intrinsic covariance function through parallel transport in tangent spaces and by consolidating information in a common tangent space at a base point, using a coregionalization-based kernel. The authors prove information consistency and posterior consistency, and show that predictions are invariant to the choice of tangent frame or basepoint curve , provided appropriate transport is applied. Through simulations on SPD manifolds and real-data examples involving flight trajectories on and diffusion tensors on SPD manifolds, iGPR delivers superior predictive accuracy and uncertainty quantification, highlighting its practical value for manifold-valued data analysis.

Abstract

Extrinsic Gaussian process regression methods, such as wrapped Gaussian process, have been developed to analyze manifold data. However, there is a lack of intrinsic Gaussian process methods for studying complex data with manifold-valued response variables. In this paper, we first apply the parallel transport operator on Riemannian manifold to propose an intrinsic covariance structure that addresses a critical aspect of constructing a well-defined Gaussian process regression model. We then propose a novel intrinsic Gaussian process regression model for manifold-valued data, which can be applied to data situated not only on Euclidean submanifolds but also on manifolds without a natural ambient space. We establish the asymptotic properties of the proposed models, including information consistency and posterior consistency, and we also show that the posterior distribution of the regression function is invariant to the choice of orthonormal frames for the coordinate representations of the covariance function. Numerical studies, including simulation and real examples, indicate that the proposed methods work well.

Paper Structure

This paper contains 11 sections, 6 theorems, 8 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Background and notations
  4. Intrinsic covariance function of tangent space
  5. Intrinsic Gaussian process regression
  6. iGPR with observation noise
  7. Simulation studies
  8. Real data analysis
  9. Flight trajectory data
  10. Diffusion tensor imaging data
  11. Conclusions

Key Result

Proposition 2.2

Assuming that $A$ and $B$ are any two points on a measurable curve $g$ of $\mathcal{M}$, and $V_{A,1}, V_{A,2} \in T_{A} \mathcal{M}$ are any two different elements in the tangent space of $A$. For any $a,b \in \mathbb{R}$, we have $\Gamma_{A \rightarrow B}(aV_{A,1}+bV_{A,2}) = a\Gamma_{A \rightarro

Figures (3)

  • Figure 1: Ambient movement and parallel transport movement of the tangent vectors.
  • Figure 2: Logarithm map of the manifold-valued data and schematic diagram of equivalent version of iGPR, where the blue solid line represents BPF.
  • Figure 3: The three-dimensional ellipsoid visualization of the sampled DTI dataset.

Theorems & Definitions (11)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 1
  • Theorem 2.8
  • Theorem 2.9
  • ...and 1 more