Orbit recovery for band-limited functions

We study the third moment for functions on arbitrary compact Lie groups. We use techniques of representation theory to generalize the notion of band-limited functions in classical Fourier theory to functions on the compact groups $\SU(n), \SO(n), \Sp(n)$. We then prove that for generic band-limited functions the third moment or, its Fourier equivalent, the bispectrum determines the function up to translation by a single unitary matrix. Moreover, if $G=\SU(n)$ or $G=\SO(2n+1)$ we prove that the third moment determines the $G$-orbit of a band-limited function. As a corollary we obtain a large class of finite-dimensional representations of these groups for which the third moment determines the orbit of a generic vector. When $G=\SO(3)$ this gives a result relevant to cryo-EM which was our original motivation for studying this problem.