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.
Cheng-Zhu Huang, An-Ru R. Zhang· Annals of Statistics· 0 citations
Generalized latent factor models provide a flexible framework for analyzing high-dimensional non-Gaussian data, but principled estimation and uncertainty quantification under missingness remain substantially less developed. We develop a theory that connects a computationally tractable nonconvex procedure directly to statistical inference for nonlinear latent factor models with exponential-family links and partially observed entries. Our procedure combines a link-aware double-SVD initialization, a unilateral refinement that achieves rowwise consistency, and vanilla gradient descent. We show that the refined initializer enters a region of incoherence and contraction and that gradient descent remains in this region through implicit regularization, contracting rapidly down to the statistical estimation error without explicit incoherence or balancing regularization. Our central result is a uniform rowwise linear approximation for the actual output of gradient descent that isolates the leading score fluctuations from higher-order estimation and optimization errors. These expansions yield asymptotically valid individual and Gaussian multiplier-bootstrap simultaneous inference for latent factors, together with simultaneous confidence bands for missing-entry means, without requiring an additional debiasing step. The resulting estimation rate matches a restricted-class minimax lower bound up to logarithmic factors, while the theory accommodates severe missingness, weak low-rank signals, and diminishing local curvature. Simulations support the theoretical findings, and an application to large language model evaluation illustrates uncertainty-aware estimation and ranking of latent model capabilities.
Cheng-Zhu Huang, Yu-Qi Gu· 0 citations
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