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   https://doi.org/10.1016/j.aim.2022.108612  

   Lott, John (October 2022, Advances in Mathematics)
2. Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

   Bhaskar, Dhananjay; MacDonald, Kincaid; Fasina, Oluwadamilola; Thomas, Dawson; Rieck, Bastian; Adelstein, Ian; Krishnaswamy, Smita (January 2022, Advances in Neural Information Processing Systems 35 (NeurIPS 2022))

   We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes. 

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3. Mean curvature flow

   https://doi.org/10.1090/S0273-0979-2015-01468-0  

   Colding, Tobias; Minicozzi, William; Pedersen, Erik (January 2015, Bulletin of the American Mathematical Society)

   Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in low-dimensional topology. 

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4. Curvature Graph Network

   Ye Ze, Kin Sum (April 2020, Proceedings of the 8th International Conference on Learning Representations (ICLR 2020))
5. Curvature Graph Network

   Ze, Ye; Liu, Kin Sum; Ma, Tengfei; Gao, Jie; Chen, Chao (April 2020, Proceedings of the 8th International Conference on Learning Representations (ICLR 2020))