Mathematics > Differential Geometry
[Submitted on 25 Jan 2019 (v1), last revised 2 May 2024 (this version, v5)]
Title:Discretized Gradient Flow for Manifold Learning in the Space of Embeddings
View PDF HTML (experimental)Abstract:Gradient descent, or negative gradient flow, is a standard technique in optimization to find minima of functions. Many implementations of gradient descent rely on discretized versions, i.e., moving in the gradient direction for a set step size, recomputing the gradient, and continuing. In this paper, we present an approach to manifold learning where gradient descent takes place in the infinite dimensional space $\mathcal{E} = {\rm Emb}(M,\mathbb{R}^N)$ of smooth embeddings $\phi$ of a manifold $M$ into $\mathbb{R}^N$. Implementing a discretized version of gradient descent for $P:\mathcal{E}\to {\mathbb R}$, a penalty function that scores an embedding $\phi \in \mathcal{E}$, requires estimating how far we can move in a fixed direction -- the direction of one gradient step -- before leaving the space of smooth embeddings. Our main result is to give an explicit lower bound for this step length in terms of the Riemannian geometry of $\phi(M)$. In particular, we consider the case when the gradient of $P$ is pointwise normal to the embedded manifold $\phi(M)$. We prove this case arises when $P$ is invariant under diffeomorphisms of $M$, a natural condition in manifold learning.
Submission history
From: Steven Rosenberg [view email][v1] Fri, 25 Jan 2019 19:25:33 UTC (22 KB)
[v2] Wed, 6 Feb 2019 21:42:57 UTC (22 KB)
[v3] Wed, 23 Dec 2020 23:30:12 UTC (28 KB)
[v4] Tue, 30 Apr 2024 20:16:28 UTC (25 KB)
[v5] Thu, 2 May 2024 12:49:24 UTC (25 KB)
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