Signed random combinatorial matrices with exactly n/2 zeros per row have smallest singular value at least order 1/sqrt(n) with probability 1 minus O(ε) minus exp(-cn).
Singularity of random Bernoulli matrices.Ann
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For n x n i.i.d. Bernoulli(p) matrices, P(corank >= k) = (1-p + o_n(1))^{k n} when k = O(sqrt(log n)).
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The smallest singular value of signed random combinatorial matrices
Signed random combinatorial matrices with exactly n/2 zeros per row have smallest singular value at least order 1/sqrt(n) with probability 1 minus O(ε) minus exp(-cn).
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Rank deficiency of Bernoulli random matrices for growing corank
For n x n i.i.d. Bernoulli(p) matrices, P(corank >= k) = (1-p + o_n(1))^{k n} when k = O(sqrt(log n)).