光滑嵌入空间中罚函数的梯度流
Gradient Flows of Penalty Functions in the Space of Smooth Embeddings
摘要：
受流形学习技术的启发，我们给出了${\mathbb R}^N$中光滑嵌入的紧致子流形在正常方向上移动并保持嵌入的明确下界。另外，给定嵌入空间上的罚函数$P : \text{Emb}(M,\mathbb{R}^N) \rightarrow \mathbb{R} $，我们给出了保证罚函数的梯度$\nabla P$在每一点都正规于$ϕ(M)$的条件.
英文摘要：
Motivated by manifold learning techniques, we give an explicit lower bound for how far a smoothly embedded compact submanifold in ${\mathbb R}^N$ can move in a normal direction and remain an embedding. In addition, given a penalty function $P : \text{Emb}(M,\mathbb{R}^N) \rightarrow \mathbb{R} $ on the space of embeddings, we give a condition which guarantees that the gradient $\nabla P$ of the penalty function is normal to $ϕ(M)$ at every point.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
