Benign-landscape conjecture for rank-two and rank-three synchronization factorizations

Let $p$ denote the factorization rank in the Burer–Monteiro factorization

. The optimization landscape is **benign** when it has no spurious local minimizers. **Benign-landscape conjecture.** For $p=2$ and $p=3$, the optimization landscape of

is benign with high probability in each of the following settings: the high-dimensional Kuramoto model, provided

$$\frac{1}{2}-\theta \gtrsim \sqrt{\frac{\log n}{n}},\qquad \theta<\frac{1}{2};$$

$\mathbb{Z}_2$-synchronization with additive Gaussian noise, provided

$$\sigma\lesssim\sqrt{\frac{n}{\log n}};$$

and community detection under the stochastic block model with $p_{in}=an^{-1}\log n$ and $p_{out}=bn^{-1}\log n$, provided

$$\frac{a-b}{\sqrt{a+b}}\geq C_0$$

for some constant $C_0$. These cases are open precisely where the paper describes the state-of-the-art bounds as suboptimal, and would extend near-information-theoretic benign-landscape results to the practically important ranks $p=2,3$.