Before revision:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{  Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .
After revision:  One of our key technical contributions gives a new method that allows error reduction "within SoS" with only a logarithmic cost in the exponent in the running time (in contrast to polynomial cost in KKK'19, RY'20 . In a concurrent and independent work , to have }\emph{certifiably hypercontractive Raghavendra and Yau proved related results for list-decodable subspace recovery RY'20 satisfies certifiable anti-concentration, we obtain a stronger error guarantee of \|}\hat{\Pi} .