Gradient Flows of Penalty Functions in the Space of Smooth Embeddings

TL;DR

This paper establishes bounds on how much a smooth embedded manifold can be deformed normally while remaining an embedding, and provides conditions ensuring the gradient of a penalty function is normal to the manifold.

Contribution

It introduces explicit bounds on normal deformations of embedded manifolds and characterizes when the gradient of a penalty function is normal to the embedding.

Findings

Derived explicit lower bounds for normal deformations of embeddings.

Provided conditions under which the gradient of penalty functions is normal to the embedded manifold.

Enhanced understanding of the geometry of embedding spaces in manifold learning.

Abstract

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 $\phi(M)$ at every point.