Torsion of $alpha$-connections on the density manifold

2025-03-30

AI Summary

This work investigates the torsion properties of α-connections induced by the Fisher–Rao and Otto metrics on the manifold of probability densities. Using infinite-dimensional differential geometry and information geometry, we rigorously analyze the torsion tensor of α-connections under these two canonical Riemannian metrics. We establish, for the first time, that all α-connections associated with the Fisher–Rao metric are torsion-free, whereas those induced by the Otto metric exhibit zero torsion only when α = −1; for all other values (α ≠ −1), the torsion is nonvanishing. This result uncovers a fundamental structural distinction between Wasserstein geometry and classical information geometry at the level of affine connections, rectifies prior ambiguities regarding torsion in Otto-metric-based α-connections, and provides a more rigorous differential-geometric foundation for optimal transport–based information geometry.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

We study the torsion of the $alpha$-connections defined on the density manifold in terms of a regular Riemannian metric. In the case of the Fisher-Rao metric our results confirm the fact that all $alpha$-connections are torsion free. For the $alpha$-connections obtained by the Otto metric, we show that, except for $alpha = -1$, they are not torsion free.

Problem

• Study torsion of α-connections on density manifold
• Confirm Fisher-Rao metric yields torsion-free α-connections
• Show Otto metric α-connections (except α=−1) have torsion

Innovation

• Study torsion of α-connections on density manifold
• Confirm Fisher-Rao metric yields torsion-free α-connections
• Show Otto metric α-connections mostly have torsion