We study the problem of signal recovery in the dihedral multi-reference alignment (MRA) model, where a signal is observed under random actions of the dihedral group and corrupted by additive noise. While previous work has shown that cyclic invariants of degree three (the bispectrum) suffice to recover generic signals up to circular shift [11], the dihedral setting introduces new challenges due to the group’s non-abelian structure. In particular, reflections prevent the diagonalization of the third moment tensor in the Fourier basis, making classical bispectrum techniques inapplicable. In this work, we prove that the orbit of a generic signal in the n-dimensional standard representation of the 2n-element dihedral group \(D_{n}\) is uniquely determined by its invariant tensors of degree at most three. This resolves an open question posed in [12], and establishes that the sample complexity for dihedral MRA with uniform distribution is \(\omega (\sigma ^6)\) , matching the cyclic case. Along the way we prove a result of independent interest (Theorem 3.2), namely that invariants of degree at most three separate generic real orbits in band-limited representations of the orthogonal group O(2). While frequency marching becomes computationally impractical in the dihedral setting, we show numerically that a simple optimization algorithm reliably recovers the signal from third-order moments, even with random initialization. Our results establish the dihedral model as both a challenging and viable alternative to the cyclic model–one that better reflects practical symmetry constraints and deserves further exploration in both theoretical and applied settings.
