🤖 AI Summary
This work addresses the computational intractability of the Fisher–Rao distance on statistical manifolds. Methodologically, it establishes the first systematic framework for computing tight, analytically tractable upper and lower bounds by integrating differential and information geometry—leveraging curvature constraints and parameterization invariance—and designing a low-complexity approximation algorithm via Taylor expansion and asymptotic analysis. Experimentally, the proposed bounds achieve over 40% improvement in tightness compared to state-of-the-art methods, while substantially reducing computational overhead. The contribution is twofold: (i) it advances the geometric understanding of the Fisher–Rao metric by revealing its curvature-dependent structural properties; and (ii) it delivers an efficient, robust metric tool applicable to statistical inference, model comparison, and generative learning—bridging theoretical insight with practical scalability.