  We propose a nonlinear, wavelet based signal representation that is
translation invariant and robust to both additive noise and random dilations.
Motivated by the multi-reference alignment problem and generalizations thereof,
we analyze the statistical properties of this representation given a large
number of independent corruptions of a target signal. We prove the nonlinear
wavelet based representation uniquely defines the power spectrum but allows for
an unbiasing procedure that cannot be directly applied to the power spectrum.
After unbiasing the representation to remove the effects of the additive noise
and random dilations, we recover an approximation of the power spectrum by
solving a convex optimization problem, and thus reduce to a phase retrieval
problem. Extensive numerical experiments demonstrate the statistical robustness
of this approximation procedure.
