Conjecture on Peng–Wei exact recovery for the generalized stochastic ball model

Let a mixture be generated by the generalized stochastic ball model, with center separation parameter $\Delta$, dimension $m$, and total number of points $N$. The Peng–Wei relaxation is the semidefinite programming relaxation for the $k$-means clustering problem.

Exact-recovery conjecture. The Peng–Wei relaxation achieves exact recovery with high probability if

$$\Delta \geq 2 + \mathcal{O}\left(\frac{1}{m}\right),$$

provided that the total number of points $N$ is large enough.

The conjecture records the empirically observed dependence of the recovery threshold on the dimension, improving the previously established bounds in the regime of sufficiently many points. The source does not provide a resolution, so its status is open.