An elementary proof of anti-concentration of polynomials in Gaussian variables

By Shachar Lovett. 2010

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Beating the Perils of Non-Convexity: Guaranteed Training of Neural Networks using Tensor Methods (2017, methodology): “Using the anti-concentration of Gaussian random variables, we have Pr[|uj | ≤ ǫ] ≤ Cǫ for some constant C; see Lovett (2010). Now if we choose ǫ = O(1/k), and apply the union bound on all entries of u, we have umin ≥ 1 k with some constant probability.”
When can Wasserstein GANs minimize Wasserstein Distance? (2020, unknown): “According to the anti-concentration property of Gaussian polynomials introduced by [23], assume that the ground truth parameters α∗(1), α∗(2), · · · , α∗(r) are chosen from a Gaussian distribution with positive integer mean M and standard deviation σ, then: Pr [ |det(T ∗)| 6 · √ C(det(T ∗)) · σ 2 r/”
Making Method of Moments Great Again? -- How can GANs learn the target distribution (2020, background): “s an integercoecient homogeneous polynomial with rvariables ((1);(2); ; r) and degree 1 + 3 + 5 + + (2K r 1) = K2. According to the anti-concentration property of Gaussian polynomials introduced by [56], assume that the ground truth parameters ((1);(2); ; r) are chosen from a Gaussian distribution w”