  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.
