Before revision:  It outputs a list of O(1/\alpha) projection matrices guaranteed to contain a projection matrix that\hat{\Pi}  satisfies certifiable anti-concentration.
After revision:  It outputs a list of O(1/\alpha) projection matrices guaranteed to contain a projection matrix that\hat{\Pi} close to \Pi_*. Two recent independent works (Raghavendra-Yau, Bakshi-Kothari (2020)) gave algorithms for this problem that work whenever \mathcal{D satisfies certifiable anti-concentration.