Least singular values of independent random tensor columns

Let \(A\) have \(m=\lfloor(1-\varepsilon)n^d\rfloor\) independent columns, each the tensor product of \(d\) random vectors in \(\R^n\). All scalar coordinates are independent, centered, of variance one, and have subgaussian norm at most \(K\). We prove that, for every fixed \(\varepsilon\in(0,1/2]\) and every integer sequence \(d=o(\sqrt{n/\log n})\), the least singular value is at least \(c_K\varepsilon n^{d/2}\) with probability tending to one. The constant depends only on \(K\); neither identical distributions nor symmetry of the coordinates is required. The proof conditions on all but the last tensor factor, estimates quadratic Gram moments of projected column blocks, and lifts the block estimate by averaging over subsets of columns.