High-order Accurate Inference on Manifolds

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 an appropriate coordinate representation induced by a second-order retraction, 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 matrix manifolds, and rank-one tensor manifolds; for Euclidean submanifolds, we also introduce a class of projection-like coordinate charts with strong consistency properties. Finally, numerical studies confirm the practical merits of the proposed procedure.

💡 Research Summary

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The paper tackles the problem of performing statistically accurate inference when the parameter of interest lies on a Riemannian manifold, a setting that increasingly appears in modern data‑driven applications. While classical results provide a first‑order asymptotic normality for manifold‑valued M‑estimators, they ignore curvature‑induced second‑order bias, leading to confidence regions whose coverage deviates from the nominal level by an order of (n^{-1/2}).
To overcome this limitation, the authors introduce a framework that achieves high‑order accuracy (i.e., error of order (o(n^{-1/2}))) for both hypothesis testing and confidence region construction. The key technical device is the use of a second‑order retraction (R_\theta), a smooth map from a neighbourhood of the origin in (\mathbb R^p) to the manifold that coincides with the exponential map up to first order and whose second derivative vanishes. By fixing a chart (\phi_{\theta_0}) induced by such a retraction, the estimator (\hat\theta_n) can be expressed in Euclidean coordinates and expanded via an Edgeworth series that explicitly incorporates curvature terms.
The authors then design a bootstrap‑studentization procedure: given the original sample ({X_i}_{i=1}^n), they generate resampled datasets, compute the corresponding M‑estimator (\hat\theta_n^{*}) using the same retraction‑based algorithm, and form the studentized statistic
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