Diffusion Maps for Group-Invariant Manifolds

• URL: http://arxiv.org/abs/2303.16169v2
• Date: Mon, 3 Apr 2023 16:55:54 GMT
• Title: Diffusion Maps for Group-Invariant Manifolds
• Authors: Paulina Hoyos and Joe Kileel
• Abstract summary: We consider the manifold learning problem when the data set is invariant under the action of a compact Lie group $K$. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the $K$-orbits of the existing data points. We show that the normalized Laplacian operator $L_N$ converges to the Laplace-Beltrami operator of the data manifold with an improved convergence rate.
• Score: 1.90365714903665
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this article, we consider the manifold learning problem when the data set is invariant under the action of a compact Lie group $K$. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the $K$-orbits of the existing data points, which yields a $K$-invariant graph Laplacian $L$. We prove that $L$ can be diagonalized by using the unitary irreducible representation matrices of $K$, and we provide an explicit formula for computing its eigenvalues and eigenfunctions. In addition, we show that the normalized Laplacian operator $L_N$ converges to the Laplace-Beltrami operator of the data manifold with an improved convergence rate, where the improvement grows with the dimension of the symmetry group $K$. This work extends the steerable graph Laplacian framework of Landa and Shkolnisky from the case of $\operatorname{SO}(2)$ to arbitrary compact Lie groups.

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