Learning Globally Smooth Functions on Manifolds

• URL: http://arxiv.org/abs/2210.00301v1
• Date: Sat, 1 Oct 2022 15:45:35 GMT
• Title: Learning Globally Smooth Functions on Manifolds
• Authors: Juan Cervino, Luiz Chamon, Benjamin D. Haeffele, Rene Vidal, and Alejandro Ribeiro
• Abstract summary: Learning smooth functions is generally challenging, except in simple cases such as learning linear or kernel models. This work proposes to overcome these obstacles by combining techniques from semi-infinite constrained learning and manifold regularization. We prove that, under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct.
• Score: 94.22412028413102
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Smoothness and low dimensional structures play central roles in improving generalization and stability in learning and statistics. The combination of these properties has led to many advances in semi-supervised learning, generative modeling, and control of dynamical systems. However, learning smooth functions is generally challenging, except in simple cases such as learning linear or kernel models. Typical methods are either too conservative, relying on crude upper bounds such as spectral normalization, too lax, penalizing smoothness on average, or too computationally intensive, requiring the solution of large-scale semi-definite programs. These issues are only exacerbated when trying to simultaneously exploit low dimensionality using, e.g., manifolds. This work proposes to overcome these obstacles by combining techniques from semi-infinite constrained learning and manifold regularization. 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. We prove that, under mild conditions, this method estimates the Lipschitz constant of the solution, learning a globally smooth solution as a byproduct. Numerical examples illustrate the advantages of using this method to impose global smoothness on manifolds as opposed to imposing smoothness on average.

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