On the spectral norm of Rademacher matrices

We discuss two-sided nonasymptotic bounds for the mean spectral norm of nonhomogenous weighted Rademacher matrices. We show that the recently formulated conjecture holds up to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log log log n"> <mml:semantics> <mml:mrow> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>log</mml:mi> <mml:mo> ⁡ </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\log \log \log n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> factor for arbitrary <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> × </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Rademacher matrices and the triple logarithm may be eliminated for matrices with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet 0 comma 1 EndSet"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\{0,1\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -coefficients.