Well-invertible column subsets of sparse matrices are rare

A random $n\times k$ matrix $S$ is an \emph{$(r,\alpha)$-oblivious subspace injection} (OSI) if $\Exp\|S^\top x\|_2^2=\|x\|_2^2$ for every $x\in\R^n$, and for every fixed $r$-dimensional subspace $V\subset\R^n$, with probability close to one, one has $\alpha\|x\|_2^2\le\|S^\top x\|_2^2$ for all $x\in V$. In this work, we show that in the regime $r=\Omega(k)$ and $\alpha=\Omega(1)$, and under a mild additional structural assumption, no constant-row-sparsity matrix $S$ is OSI, thereby answering, in a strong form, a question raised by Camaño, Epperly, Meyer, and Tropp.
We show that the failure of the OSI property for sparse random matrices stems from a general deterministic phenomenon, thereby reducing a probabilistic problem to a non-probabilistic one. This phenomenon is related to the restricted invertibility principle introduced in the seminal work of Bourgain--Tzafriri. Let $(n_k)_{k\in\N}$ be a sequence of integers satisfying $\frac{n_k}{k}\to\infty$. For each $k$, let $S^{(k)}$ be a $n_k\times k$ non-random matrix with $O(1)$ nonzero entries per row, whose nonzero entries have average magnitude $O(1)$, and such that the total number of pairs of rows with supports overlapping at two or more indices is $o({n_k}^2/k)$. We prove that for every constant $\varepsilon>0$, as $k\to\infty$, the overwhelming majority of $k\times \lfloor\varepsilon k\rfloor$ submatrices of $(S^{(k)})^\top$ have the smallest singular value $o(1)$. Thus, the well-invertible submatrices whose existence is guaranteed by the Bourgain--Tzafriri theorem are rare. The proof is itself based on probabilistic tools.