Gradient Flows of Penalty Functions in the Space of Smooth Embeddings

Abstract: Motivated by manifold learning techniques, we give an explicit lower bound for how far a smoothly embedded compact submanifold in

{m a t h b b R}^{N}

can move in a normal direction and remain an embedding. In addition, given a penalty function

P : e x t E m b (M, m a t h b b R^{N}) i g h t a r r o w m a t h b b R

on the space of embeddings, we give a condition which guarantees that the gradient

a b l a P

of the penalty function is normal to

p h i (M)

at every point.

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