Dihedral Multi-Reference Alignment

We study the dihedral multi-reference alignment problem of estimating orbit a signal from multiple noisy observations signal, acted on by random elements group. show that if group are drawn generic distribution, is uniquely determined second moment observations. This implies optimal estimation rate in high noise regime proportional to square variance noise. first result this type for over non-abelian with non-uniform distribution elements. Based tools invariant theory and algebraic geometry, we also delineate conditions unique recovery models finite groups (namely, when replaced general group) distribution. Finally, design numerically three computational frameworks based synchronization, expectation-maximization, method moments.