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arXiv · 2210.00301

Learning Globally Smooth Functions on Manifolds

Juan Cervino·Luiz F. O. Chamon·Benjamin D. Haeffele·Rene Vidal·Alejandro Ribeiro
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Abstract

Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. This work combines techniques from semi-infinite constrained learning and manifold regularization to learn representations that are globally smooth on a manifold. To do so, it shows that under typical conditions the problem of learning a Lipschitz continuous function on a manifold is equivalent to a dynamically weighted manifold regularization problem. This observation leads to a practical algorithm based on a weighted Laplacian penalty whose weights are adapted using stochastic gradient techniques. It is shown that under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Experiments on real world data illustrate the advantages of the proposed method relative to existing alternatives.

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Juan Cervino, Luiz F. O. Chamon, Benjamin D. Haeffele, Rene Vidal, Alejandro Ribeiro. 2022-10-01. Learning Globally Smooth Functions on Manifolds. https://arxiv.org/abs/2210.00301

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