For random A with independent mean-zero unit-variance entries of bounded fourth moments and anti-concentration, s_min(MA) is of order 1/||M^{-1}||_HS with high probability for any fixed invertible M.
Litvak and Omar Rivasplata,Smallest singular value of sparse random matrices, Studia Math.212 (2012), no
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On the smallest singular value of the product of random and deterministic matrices
For random A with independent mean-zero unit-variance entries of bounded fourth moments and anti-concentration, s_min(MA) is of order 1/||M^{-1}||_HS with high probability for any fixed invertible M.