When Will a Sequence of Points in a Riemannian Submanifold Converge?

Tuyen Trung Truong
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Tuyen Trung Truong: Department of Mathematics, University of Oslo, Blindern, 0851 Oslo, Norway

Mathematics, 2020, vol. 8, issue 11, 1-5

Abstract: Let X be a Riemannian manifold and x n a sequence of points in X. Assume that we know a priori some properties of the set A of cluster points of x n . The question is under what conditions that x n will converge. An answer to this question serves to understand the convergence behaviour for iterative algorithms for (constrained) optimisation problems, with many applications such as in Deep Learning. We will explore this question, and show by some examples that having X a submanifold (more generally, a metric subspace) of a good Riemannian manifold (even in infinite dimensions) can greatly help.

Keywords: compact metric space; deep neural networks; random dynamical systems; global convergence of Gradient Descent; iterative optimisation; Nash conjecture; Nash’s embedding theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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