NS-RGS: Newton-Schulz based Riemannian gradient method for orthogonal group synchronization

2026-04-07

AI Summary

This work addresses the high computational cost in orthogonal group synchronization arising from the reliance on exact SVD or QR decompositions at each iteration. By formulating the problem as a constrained non-convex optimization task, the authors propose a Riemannian gradient method that leverages Newton–Schulz iterations to efficiently approximate the traditional projection step. Coupled with spectral initialization and a refined leave-one-out analysis, the algorithm effectively mitigates statistical dependencies and achieves linear convergence under near-optimal noise conditions. Experimental results demonstrate that the method matches the accuracy of the generalized power method on both synthetic and real-world global alignment tasks while offering nearly a two-fold speedup in computation time.

Abstract

Group synchronization is a fundamental task involving the recovery of group elements from pairwise measurements. For orthogonal group synchronization, the most common approach reformulates the problem as a constrained nonconvex optimization and solves it using projection-based methods, such as the generalized power method. However, these methods rely on exact SVD or QR decompositions in each iteration, which are computationally expensive and become a bottleneck for large-scale problems. In this paper, we propose a Newton-Schulz-based Riemannian Gradient Scheme (NS-RGS) for orthogonal group synchronization that significantly reduces computational cost by replacing the SVD or QR step with the Newton-Schulz iteration. This approach leverages efficient matrix multiplications and aligns perfectly with modern GPU/TPU architectures. By employing a refined leave-one-out analysis, we overcome the challenge arising from statistical dependencies, and establish that NS-RGS with spectral initialization achieves linear convergence to the target solution up to near-optimal statistical noise levels. Experiments on synthetic data and real-world global alignment tasks demonstrate that NS-RGS attains accuracy comparable to state-of-the-art methods such as the generalized power method, while achieving nearly a 2$\times$ speedup.

Problem

• orthogonal group synchronization
• constrained nonconvex optimization
• SVD
• computational bottleneck
• large-scale problems

Innovation

• Newton-Schulz iteration
• Riemannian optimization
• orthogonal group synchronization
• computational efficiency
• linear convergence