The reflection invariant bispectrum: signal recovery in the dihedral model

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 has shown that cyclic invariants of degree three (the bispectrum) suffice to recover generic signals up to circular shift, the dihedral setting introduces new challenges due to the groups 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 the generic signal in the $n$-dimensional standard representation of the then $2n$-element dihedral group $D_{n}$ is uniquely determined by invariant tensors of degree at most three. This resolves an open question in the literature and establishes that the sample complexity for dihedral MRA with uniform distribution is $\omega(\sigma^6)$ matching the cyclic case. 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.