Diffusion Maps for Group-Invariant Manifolds

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 orbits under the action of $K$ of the existing data points. We prove that this $K$-invariant Laplacian operator $L$ can be diagonalized by using the unitary irreducible representation matrices of $K$, and we provide an explicit formula for computing the eigenvalues and eigenvectors of $L$. Moreover, 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.