Constant Metric Scaling in Riemannian Computation

2026-01-16

AI Summary

This study systematically investigates the effects of a global constant rescaling of the metric on a Riemannian manifold, clarifying the common misconception that such a rescaling alters curvature or the underlying manifold structure. Through a self-contained differential-geometric analysis, it precisely distinguishes between quantities affected by the rescaling—such as distances, norms, and volume forms—and geometric objects that remain invariant, including the Levi-Civita connection, geodesics, and the exponential map. The work demonstrates that this global metric rescaling is equivalent to a uniform step-size adjustment in Riemannian optimization, leaving the intrinsic geometry unchanged. This provides a rigorous theoretical foundation for introducing a global scale parameter in optimization algorithms and, for the first time, clearly delineates the geometric boundaries of constant metric rescaling.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

Constant rescaling of a Riemannian metric appears in many computational settings, often through a global scale parameter that is introduced either explicitly or implicitly. Although this operation is elementary, its consequences are not always made clear in practice and may be confused with changes in curvature, manifold structure, or coordinate representation. In this note we provide a short, self-contained account of constant metric scaling on arbitrary Riemannian manifolds. We distinguish between quantities that change under such a scaling, including norms, distances, volume elements, and gradient magnitudes, and geometric objects that remain invariant, such as the Levi--Civita connection, geodesics, exponential and logarithmic maps, and parallel transport. We also discuss implications for Riemannian optimization, where constant metric scaling can often be interpreted as a global rescaling of step sizes rather than a modification of the underlying geometry. The goal of this note is purely expository and is intended to clarify how a global metric scale parameter can be introduced in Riemannian computation without altering the geometric structures on which these methods rely.

Problem

• Riemannian metric
• constant scaling
• geometric invariance
• Riemannian computation
• metric rescaling

Innovation

• Riemannian metric scaling
• geometric invariance
• Riemannian optimization
• Levi-Civita connection
• geodesics