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About this paper
Abigail Hickok, Andrew J. Blumberg
Scalar Ollivier-Ricci Curvature (SORC), a new discrete measure for weighted graphs and point clouds, is introduced with formal proof of its convergence in probability to the scalar curvature of an underlying Riemannian manifold. This work from Columbia University provides the first theoretical guarantee for a discrete scalar curvature definition derived from Ollivier-Ricci curvature to reliably approximate its continuous counterpart.
Key insights
- The w(x,y)^2 weighting in SORC is identified as the correct scaling factor for convergence to scalar curvature, addressing the specific behavior of ORC.
- Ollivier-Ricci curvature's approximation to Ricci curvature becomes less accurate for very short edges, necessitating a weighting scheme that assigns higher importance to longer edges.
- Proving the convergence of the mean "epsilon_N / d_M(x,y)" ratio, even in challenging low-dimensional cases with infinite variance, was critical for bounding error terms in the main convergence proof.
Summary by alphaXiv.
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