Diffusion Maps for Group-Invariant Manifolds

TL;DR

This paper introduces a method to incorporate group symmetries into manifold learning by constructing a $K$-invariant graph Laplacian, improving convergence to the Laplace-Beltrami operator for data with symmetries.

Contribution

It extends the steerable graph Laplacian framework from SO(2) to arbitrary compact Lie groups, providing explicit formulas and convergence analysis.

Findings

The $K$-invariant Laplacian can be diagonalized using irreducible representations.

The normalized Laplacian converges to the Laplace-Beltrami operator with improved rate.

The method generalizes the steerable graph Laplacian to all compact Lie groups.

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.