The smallest singular value of inhomogenous random rectangular matrices

Let A ∈ RN × n (N ≥ n) be a random matrix with columns that are mean 0 and isotropic and with independent entries having bounded 2+β moments. We show that the smallest singular value σₙ(A) satisfies \[ (σ_n(A) ≤ ε(√N+1 - √n)) ≤ (Cε)N-n+1 + e-cN, \] for all ε > 0, where $c,C$ depend only on β and the 2+β moments. 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, z (|X-z| < a) < b, we show that \[ (σ_n(A) ≤ ε(√N+1 - √n)) ≤ (Cεlog(1/ε))N-n+1 + e-cN, \] where c₂,C₃ 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.