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High-order Accurate Inference on Manifolds

Chengzhu Huang, Anru R. Zhang

TL;DR

The paper develops a framework for high-order statistical inference on Riemannian manifolds by integrating bootstrap methods with Riemannian Newton iterations and curvature-aware coordinate schemes. It uses fixed normal charts, second-order retracts, and double exponential mappings to achieve accurate hypothesis tests and confidence regions for manifold-valued parameters, with explicit distributional guarantees via Edgeworth-type analyses. The methodology is demonstrated across sphere, Stiefel, fixed-rank matrices, and rank-one tensor manifolds, including Gaussian location and barycenter applications, supported by simulation studies. This work enables reliable, high-precision uncertainty quantification in non-Euclidean parameter spaces common in modern data analysis, while highlighting the role of curvature and chart choice in inference.

Abstract

We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under a fixed normal chart centered at the true parameter, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings-including spheres, the Stiefel manifold, fixed-rank matrices manifolds, and rank-one tensor manifolds-and, for Euclidean submanifolds, introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.

High-order Accurate Inference on Manifolds

TL;DR

The paper develops a framework for high-order statistical inference on Riemannian manifolds by integrating bootstrap methods with Riemannian Newton iterations and curvature-aware coordinate schemes. It uses fixed normal charts, second-order retracts, and double exponential mappings to achieve accurate hypothesis tests and confidence regions for manifold-valued parameters, with explicit distributional guarantees via Edgeworth-type analyses. The methodology is demonstrated across sphere, Stiefel, fixed-rank matrices, and rank-one tensor manifolds, including Gaussian location and barycenter applications, supported by simulation studies. This work enables reliable, high-precision uncertainty quantification in non-Euclidean parameter spaces common in modern data analysis, while highlighting the role of curvature and chart choice in inference.

Abstract

We present a new framework for statistical inference on Riemannian manifolds that achieves high-order accuracy, addressing the challenges posed by non-Euclidean parameter spaces frequently encountered in modern data science. Our approach leverages a novel and computationally efficient procedure to reach higher-order asymptotic precision. In particular, we develop a bootstrap algorithm on Riemannian manifolds that is both computationally efficient and accurate for hypothesis testing and confidence region construction. Although locational hypothesis testing can be reformulated as a standard Euclidean problem, constructing high-order accurate confidence regions necessitates careful treatment of manifold geometry. To this end, we establish high-order asymptotics under a fixed normal chart centered at the true parameter, thereby enabling precise expansions that incorporate curvature effects. We demonstrate the versatility of this framework across various manifold settings-including spheres, the Stiefel manifold, fixed-rank matrices manifolds, and rank-one tensor manifolds-and, for Euclidean submanifolds, introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.
Paper Structure (70 sections, 19 theorems, 205 equations, 6 figures, 2 tables, 5 algorithms)

This paper contains 70 sections, 19 theorems, 205 equations, 6 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Challenges in Manifold Settings.
  3. Extremum Estimators on Manifolds
  4. Constrained Parameter Space.
  5. Nonconvex Optimization.
  6. Bootstrap and Studentization on Manifolds
  7. Our Contributions
  8. Related Literature
  9. Preliminaries on Riemannian Manifolds
  10. Riemannian Manifold.
  11. Levi-Civita Connection.
  12. Parallel Transport.
  13. Retraction.
  14. Derivatives.
  15. Notation
  16. ...and 55 more sections

Key Result

Proposition 1

Suppose the following assumptions hold: If $\theta_0 = \theta_1$, then it holds for every $\xi \in (0,\frac{1}{2}]$ that where $z^*_{T_{\phi,j},k, \alpha}$ is defined as a value that minimizes $|\mathbb{P}[T_{\phi,j}^* \leq z |\mathcal{X}_n] - (1- \alpha) |$ over $z\in \mathbb{R}$ and $z^*_{W_\phi,2,\alpha}$ is defined analogously.

Figures (6)

  • Figure 1: A Counterexample: Irregular Empirical Distribution Using an Irregular Chart. We consider the fixed-rank matrices manifold with the sample size $n = 1000$ and the ground truth parameter $\boldsymbol \theta_0 = \textrm{diag}(1,1,0,0)$. The histogram represents the studentized version of $(R_{\widehat{\theta}_n}^{-1}(\theta_0))_1$ while the blue curve depicts the probability density function of the standard normal distribution.
  • Figure 2: Tangent spaces and coordinates of a dimension-$2$ submanifold $\mathcal{M}$ in $\mathbb{R}^3$.
  • Figure 3: Double exponential mapping
  • Figure 4: Scaled Cumulative Distribution Function Difference. column $1$: Stiefel Manifold; column $2$: fixed-rank matrices manifold; column $3$: Rank-One Tensor Manifold; row $1$: log-scaled Wald Statistic; row $2$: $\sqrt{n}$-scaled Wald Statistic; row $3$: log-scaled intrinsic $t$-Statistic; row $4$: $\sqrt{n}$-scaled intrinsic $t$-Statistic; Black: resampled statistic (our method); Red: non-studentized resampled statistic; Blue: chi-square distribution / standard normal distribution.
  • Figure 5: Scaled Cumulative Distribution Function Difference for Extrinsic $t$-statistic for fixed-rank matrices manifold. Black: the empirical distribution of the resampled statistic; Red: the empirical distribution of the non-studentized resampled statistic; Blue: the distribution of the standard normal distribution.
  • ...and 1 more figures

Theorems & Definitions (40)

  • Definition 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • Theorem 1: Convergence guarantees: Algorithm \ref{['algorithm: update']} with full data
  • Theorem 2: Convergence guarantees: Algorithm \ref{['algorithm: update']} with bootstrapped data
  • Remark 3
  • Remark 4
  • Theorem 3
  • Proposition 2
  • ...and 30 more