Riemannian Newton Methods with Statistical Inference
Abstract
We develop online statistical inference for Polyak-Ruppert averages of inexact Newton iterates on Riemannian manifolds. A local asymptotic representation separates observation noise, solver randomization, and accumulated implementation bias, and yields one-pass covariance estimators and asymptotically valid Wald regions under local regularity and centering conditions. Independent fresh solves add a positive semidefinite term to the score-sandwich covariance; averaging solver replicas reduces this term in inverse proportion to their number. Slowly varying predictable gains instead cancel from the first order law. Product torus simulations show how Newton scaling shortens an ill-conditioned transient and how retaining solver variation restores coverage. Intrinsic means and principal subspaces provide geometric applications, with a streaming data illustration.
Related
- Inference for Newton Methods with Accelerated Sketch-and-Project via Random Scaling
- Rank-Optimal Sample Complexity of Riemannian Gauss-Newton for Low-Rank Matrix Recovery
- Grad-div stabilized finite element Picard-Newton iterative methods for the Navier-Stokes equations
- Self-Normalized Inference from Riemannian Stochastic Gradient Paths
- Global 𝒪(k−2) Convergence of Multilevel Gradient-Regularized Newton Methods
BibTeX
@misc{chang2026riemannian,
title = {{Riemannian Newton Methods with Statistical Inference}},
author = {Xiaotian Chang and Qirui Hu and Yangdi Jiang and Nan Qiao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00949},
eprint = {2611.00949},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00949},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Riemannian Newton Methods with Statistical Inference
AU - Xiaotian Chang
AU - Qirui Hu
AU - Yangdi Jiang
AU - Nan Qiao
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00949
UR - https://arxiv.org/abs/2611.00949
AB - We develop online statistical inference for Polyak-Ruppert averages of inexact Newton iterates on Riemannian manifolds. A local asymptotic representation separates observation noise, solver randomization, and accumulated implementation bias, and yields one-pass covariance estimators and asymptotically valid Wald regions under local regularity and centering conditions. Independent fresh solves add a positive semidefinite term to the score-sandwich covariance; averaging solver replicas reduces this term in inverse proportion to their number. Slowly varying predictable gains instead cancel from the first order law. Product torus simulations show how Newton scaling shortens an ill-conditioned transient and how retaining solver variation restores coverage. Intrinsic means and principal subspaces provide geometric applications, with a streaming data illustration.
ER -