  Orthogonal group synchronization is the problem of estimating $n$ elements
$Z_1, \ldots, Z_n$ from the $r \times r$ orthogonal group given some relative
measurements $R_{ij} \approx Z_i^{}Z_j^{-1}$. The least-squares formulation is
nonconvex. To avoid its local minima, a Shor-type convex relaxation squares the
dimension of the optimization problem from $O(n)$ to $O(n^2)$. Alternatively,
Burer--Monteiro-type nonconvex relaxations have generic landscape guarantees at
dimension $O(n^{3/2})$. For smaller relaxations, the problem structure matters.
It has been observed in the robotics literature that, for SLAM problems, it
seems sufficient to increase the dimension by a small constant multiple over
the original. We partially explain this. This also has implications for
Kuramoto oscillators.
  Specifically, we minimize the least-squares cost function in terms of
estimators $Y_1, \ldots, Y_n$. For $p \geq r$, each $Y_i$ is relaxed to the
Stiefel manifold $\mathrm{St}(r, p)$ of $r \times p$ matrices with orthonormal
rows. The available measurements implicitly define a (connected) graph $G$ on
$n$ vertices. In the noiseless case, we show that, for all connected graphs
$G$, second-order critical points are globally optimal as soon as $p \geq r+2$.
(This implies that Kuramoto oscillators on $\mathrm{St}(r, p)$ synchronize for
all $p \geq r + 2$.) This result is the best possible for general graphs; the
previous best known result requires $2p \geq 3(r + 1)$. For $p > r + 2$, our
result is robust to modest amounts of noise (depending on $p$ and $G$). Our
proof uses a novel randomized choice of tangent direction to prove
(near-)optimality of second-order critical points. Finally, we partially extend
our noiseless landscape results to the complex case (unitary group); we show
that there are no spurious local minima when $2p \geq 3r$.
