When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We observe that more generally, when $\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ for all $p \in \mathcal{M}$, there exists $K > 0$ such that $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$.
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By Xiuyuan Cheng, Yixuan Tan, Nan Wu
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By Vu Khac Ky
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By Razvan-Andrei Lascu, Mateusz B. Majka, David \v{S}i\v{s}ka, {\L}ukasz Szpruch
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By Srinivas Nambirajan
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By Vu Khac Ky
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By Charles C. Margossian, Isaac E. Rankin, Lawrence K. Saul
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By Xinliang Liu, Tong Mao, Jinchao Xu
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By Hoang-Son Tran, Pranav Gupta, Subhroshekhar Ghosh
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By Yunfei Yang, Jun Fan
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By Jingyi Zhang, Cheng Mao, Debankur Mukherjee
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By Yeari Vigder, Paulina Hoyos, David Thong, Joakim and\'en, Joe Kileel, Amit Moscovich