Bootstrap Inference for Riemannian Optimization Problems

Xiaotian Chang; Qirui Hu; Yangdi Jiang; Nan Qiao

stat.MLOct 31, 2026

Bootstrap Inference for Riemannian Optimization Problems

Authors: Xiaotian Chang, Qirui Hu, Yangdi Jiang, Nan Qiao

Organizations: Division of Mathematical Sciences, Nanyang Technological University. · School of Statistics and Data Science, Shanghai University of Finance and Economics. · School of Mathematics, Statistics and Mechanics, Beijing University of Technology, China.

Abstract

We develop an online multiplier bootstrap for manifold-valued parameters estimated by averaging the iterates of Riemannian stochastic approximation. The method runs weighted iterations alongside the original algorithm, with each path using the same observations while updating its own state and intrinsic average. The weighted averages provide uncertainty assessments without a plug-in asymptotic covariance estimator. Under local stability and regularity conditions, we establish bootstrap consistency and show that a fixed number of paths suffices for asymptotically valid Student intervals and intrinsic confidence regions. The theory accommodates samplewise nonsmooth scores and identifies a critical oracle-bias regime in which the estimator's limiting distribution shifts, but the centered bootstrap distribution does not. Applications include spherical means, positive-definite matrix means and medians, and principal-subspace estimation. Simulations show coverage close to nominal in regular settings, and an analysis of North Atlantic storm locations illustrates descriptive online inference for a spherical center.

Figures & tables

	Symmetric		Skew
Method	$n=250$	$n=500$	$n=100$	$n=250$
Proposed ( $B=1$ )	0.940	0.938	0.954	0.950
Proposed ( $B=3$ )	0.956	0.954	0.942	0.938
Proposed ( $B=5$ )	0.960	0.950	0.938	0.942
Proposed ( $B=10$ )	0.952	0.946	0.936	0.932
Proposed (quantile)	0.968	0.944	0.932	0.924

Table 1: Nominal 95% coverage in localized spherical designs. Proposed Student and quantile intervals use ( 7 ) and ( 35 ); the quantile and offline methods each use 200 replicates.

Proposed (Student)				Proposed (quantile)	Sandwich	Offline
$B=1$	$B=3$	$B=5$	$B=10$	$B=200$		$B=200$
0.948	0.958	0.950	0.970	0.964	0.948	0.900

Table 2: Nominal 95% coverage in the wider-cap spherical design (

q=2

n=1000

). Proposed (quantile) and offline basic intervals each use 200 replicates.

Figure 1: Proposed (

B=10

) coverage in regular designs; the dashed line marks 95%. Spherical laws differ between horizons, so connecting segments are visual guides rather than same-law sample-size trends.

Contamination probability	Fréchet mean	Geometric median
0	0.930	0.936
0.05	0.952	0.948
0.15	0.952	0.956

Table 3: Nominal 95% coverage of Proposed (

B=10

) for the largest log eigenvalue of the common positive-definite mean and median (

3\times 3

matrices,

n=5000

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	Symmetric		Skew		Wider cap
Method	$n=250$	$n=500$	$n=100$	$n=250$	$n=1000$
Proposed ( $B=1$ )	0.3280	0.2316	0.8337	0.4994	0.2814
Proposed ( $B=3$ )	0.1075	0.0720	0.2782	0.1775	0.0963
Proposed ( $B=5$ )	0.0927	0.0634	0.2384	0.1515	0.0818
Proposed ( $B=10$ )	0.0841	0.0573	0.2217	0.1370	0.0737
Proposed (quantile)	0.0764	0.0526	0.1996	0.1256	0.0662

Appendix

Table 4: Median 95% interval lengths for the spherical coverage comparisons. Proposed (quantile) and offline methods each use 200 replicates; all methods share the base estimator within a configuration.

Quantity	Estimate	95% fixed $B=10$	95% large $B=200$
Longitude	$-59.88$	$[-61.77,-57.98]$	$[-61.57,-58.37]$
Latitude	$21.92$	$[21.06,22.78]$	$[21.11,22.49]$

Appendix

Table 5: North Atlantic first recorded tropical/subtropical locations, 1980–2025. Coordinates and interval endpoints are in degrees; west longitudes are negative. All displayed intervals are descriptive working-independence multiplier intervals.

Figure 2: Retained storm-level locations and the chronological online intrinsic center. The plotted points are first archive records satisfying the tropical/subtropical filter, not physical genesis estimates.

Confidence Region Estimation

Confidence region estimation builds sets around unknown parameters or predictions that quantify uncertainty while controlling coverage.

See all 44 papers

Riemannian Newton Methods with Statistical Inference
Oct 31, 2026Xiaotian Chang, Qirui Hu, Yangdi Jiang +1Covariance Estimation Riemannian Optimization
Division of Mathematical Sciences, Nanyang Technological University. · School of Statistics and Data Science, Shanghai University of Finance and Economics. · School of Mathematics, Statistics and Mechanics, Beijing University of Technology, China.
Locally Private Inference for Riemannian Stochastic Optimization
Sep 18, 2026Xiaotian Chang, Yangdi Jiang, Qirui HuRiemannian Optimization Confidence Region Estimation
School of Physical and Mathematical Sciences, Nanyang Technological University · School of Statistics and Data Science, Shanghai University of Finance and Economics.
Online Covariance Estimation for Averaged Stochastic Gradient Descent on Riemannian Manifolds
Oct 24, 2026Jingrong Guan, Liwei Jiang, Sen NaCovariance Estimation Riemannian Optimization
Department of Industrial Engineering and Management Sciences, Northwestern University · Edwardson School of Industrial Engineering, Purdue University · School of Industrial and Systems Engineering, Georgia Institute of Technology

Oct 31, 2026J/K move · Enter open · S save

Riemannian Newton Methods with Statistical Inference

Xiaotian Chang, Qirui Hu, Yangdi Jiang, Nan Qiao

Division of Mathematical Sciences, Nanyang Technological University. · School of Statistics and Data Science, Shanghai University of Finance and Economics. · School of Mathematics, Statistics and Mechanics, Beijing University of Technology, China.

Covariance Estimation Riemannian Optimization Stochastic Approximation Polyak-Ruppert Averaging

Open paper AskPDF arXiv

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.

Abstract

Confidence Region Estimation

Explore similar work

Riemannian Newton Methods with Statistical Inference

Locally Private Inference for Riemannian Stochastic Optimization

Online Covariance Estimation for Averaged Stochastic Gradient Descent on Riemannian Manifolds