The net argument for the spectral norm

ProofGraduate⏱ 3 hoursRandom Matricesoperator normnetsub-gaussianunion boundflagship

Problem

Let $A\in\R^{n\times p}$ have independent, mean-zero entries with $\|A_{ij}\|_{\psi_2}\le K$ . Prove that there is an absolute constant $C$ with

\P\Big(\|A\| \ge CK\big(\sqrt n+\sqrt p+t\big)\Big) \le 2e^{-t^2}

for all $t\ge0$ . Your proof must construct the nets, bound their sizes, control a single bilinear form, and perform the exponent count explicitly.

Write a complete argument. Every inequality you use should be one you could state precisely.

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Problem	Field	Type	Difficulty	Time
When is the union bound tight?	Probability	Proof	Foundation	15 min
Generalisation for a finite class	Learning Theory	Proof	Beginner	30 min
Bounding the spectral norm by row and column sums	Foundations	Proof	Intermediate	30 min
Frobenius versus operator norm for random matrices	Random Matrices	Numerical	Intermediate	30 min
Measure the covariance estimation rate run	High-Dim Statistics	Coding	Intermediate	1 hour