2024/08/26 by Dabagia, Max, Fernandez, Manuel · 1 citation
#FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.2408.14389
Let A ∈ ℝN × n (N ≥ n) be a random matrix with with independent entries that have mean 0 variance 1 and bounded 2+β moment. We show that the smallest singular value σn(A) satisfies Pr (σn(A) ≤ ε(√(N+1) - √(n))) ≤ (Cε)N-n+1 + e-cN, for all ε > 0, where c,C depend only on β and the 2+β moment. This extends earlier results of Rudelson and Vershynin, who showed that such lower tail estimates held for rectangular matrices with i.i.d. mean 0 subgaussian entries. When the 2+β moment assumption is replaced with a uniform anti-concentration assumption, supz Pr(|X-z| < a) < b, we show that Pr(σn(A) ≤ ε(√(N+1) - √(n))) ≤ (Cεlog(1/ε))N-n+1 + e-cN, where c,C now depend only on a and b. This extends more recent work of Livshyts, whose showed that such lower tail estimates held for rectrangular matrices with i.i.d. rows. To prove these results we employ a number of new technical ingredients, including a new deviation inequality for the regularized Hilbert-Schmidt norm and a recently proven small ball estimate for the distance between a random vector and a subspace spanned by an inhomogeneous rectangular matrix.