提出对称散度下的图拉普拉斯收敛性，扩展了流形学习的理论基础。
On the convergence of graph Laplacians with a symmetric divergence
- 用对称散度定义流形度量，统一处理多种距离形式
- 证明图拉普拉斯在点态意义下收敛，误差控制在四次方阶
- 适用于基于Sinkhorn散度的概率测度流形建模
在分析数据位于光滑、紧致、连通的黎曼子流形 $(\mathcal{M}, g)$ 时，关键估计表明存在 $K > 0$，使得对所有 $p, q \in \mathcal{M}$ 有 $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$。我们观察到，当 $\mathcal{M}$ 配备一个满足非退化条件的光滑对称散度 $D$，且度量 $g$ 由 $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ 给出时，存在 $K > 0$ 使得对所有 $p, q \in \mathcal{M}$ 有 $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$。我们证明这一估计足以保证以 $D$ 构造的图拉普拉斯的点态收敛，并讨论了 $D$ 为参数化于流形上的概率测度族的Sinkhorn散度的情形。
原文摘要 · Abstract (English)
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 $p, q \in \mathcal{M}$. We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with $D$ and discuss examples where $D$ is given by the Sinkhorn divergence on a family of probability measures parametrized by a manifold.
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