  In this work we study orbit recovery over $SO(3)$, where the goal is to
recover a function on the sphere from noisy, randomly rotated copies of it. We
assume that the function is a linear combination of low-degree spherical
harmonics. This is a natural abstraction for the problem of recovering the
three-dimensional structure of a molecule through cryo-electron tomography. For
provably learning the parameters of a generative model, the method of moments
is the standard workhorse of theoretical machine learning. It turns out that
there is a natural incarnation of the method of moments for orbit recovery
based on invariant theory. Bandeira et al. [BBSK+18] used invariant theory to
give tight bounds on the sample complexity in terms of the noise level. However
many of the key challenges remain: Can we prove bounds on the sample complexity
that are polynomial in $n$, the dimension of the signal? The bounds in
[BBSK+18] hide constants that have an unspecified dependence on $n$ and only
hold in the limit as $\sigma^2 \rightarrow \infty$ where $\sigma^2$ is the
variance of the noise. Moreover can we give efficient algorithms? We revisit
these challenges from the perspective of smoothed analysis, where we assume
that the coefficients of the signal, in the basis of spherical harmonics, are
subject to small random perturbations. Our main result is a quasi-polynomial
time algorithm for orbit recovery over $SO(3)$ in this model. Our approach is
based on frequency marching, which sequentially solves linear systems to find
higher degree coefficients. Our main technical contribution is to show that
these linear systems have unique solutions, are well-conditioned, and that the
error can be made to compound over at most a logarithmic number of rounds. We
believe that our work takes an important first step towards uncovering the
algorithmic implications of invariant theory.
