The smallest singular value of inhomogenous random rectangular matrices

Let $A\in {\mathbb{R}}^{N\times n}$ ($N\ge n$) be a random matrix with with independent entries that have mean 0 variance 1 and bounded $2+\beta$ moment. We show that the smallest singular value ${\sigma }_n(A)$ satisfies $$\mathrm{Pr}({\sigma }_n(A)\le \epsilon (\sqrt{N+1}-\sqrt{n}))\le (C\epsilon {)}^{N-n+1}+e^{-cN},$$ for all $\epsilon >0$, where $c,C$ depend only on $\beta$ and the $2+\beta$ 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 $2+\beta$ moment assumption is replaced with a uniform anti-concentration assumption, ${\sup }_z\mathrm{Pr}(\mathrm{\mid }X-z\mathrm{\mid }<a)<b$, we show that $$\mathrm{Pr}({\sigma }_n(A)\le \epsilon (\sqrt{N+1}-\sqrt{n}))\le (C\epsilon \log (1\mathrm{/}\epsilon ){)}^{N-n+1}+e^{-cN},$$ 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.