Tail Bounds for <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <msub> <mrow> <mi>ℓ</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </math> Norm of Gaussian Random Matrices with Applications

As major components of the random matrix theory, Gaussian random matrices have been playing an important role in many fields, because they are both unitary invariant and have independent entries and can be used as models for multivariate data or multivariate phenomena. Tail bounds for eigenvalues of Gaussian random matrices are one of the hot study problems. In this paper, we present tail and expectation bounds for the ℓ 1 norm of Gaussian random matrices, respectively. Moreover, the tail and expectation bounds for the ℓ 1 norm of the Gaussian Wigner matrix are calculated based on the resulting bounds. Compared with existing results, our results are more suitable for the high-dimensional matrix case. Finally, we study the tail bounds for the parameter vector of some existing regularization algorithms.