The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

We show the existence of a net near sphere, such that values any matrix on sphere and are compared via regularized Hilbert-Schmidt norm, which we introduce. This allows to construct an efficient controls length Ax for random A with independent columns (no other assumptions required). As consequence smallest singular value σn (A) N × n i.i.d. mean zero, variance one entries enjoys following small ball estimate, ϵ > 0 $$P({\sigma _n}(A) < \epsilon (\sqrt {N + 1} - \sqrt )) \le {(C\epsilon \,\log \,1/\epsilon )^{N 1}} {e^{ cN}}.$$ The proof this result requires working matrices whose rows not independent, and, therefore, fact theorem about discretization works dependent rows, is crucial. Furthermore, in case square n×n having concentration function separated from 1, $$\mathbb{E}\left\| \right\|_{HS}^2 c{n^2}$$ , has {\epsilon \over {\sqrt }} C\epsilon cn}},$$ 0. In addition, $$\epsilon {c }}$$ assumption required. Our estimates generalize previous results Rudelson Vershynin [29], [30], required sub-gaussian zero assumptions, as well work Rebrova Tikhomirov [25], where 1 were