Controlling the least eigenvalue of a random Gram matrix

Consider a n×p random matrix X with i.i.d. rows. We show that the least eigenvalue of n−1X⊤X is bounded away from zero with high probability when p/n⩽y for some fixed y in (0,1) and normalized orthogonal projections of rows are not too close to zero. The principal difference from the previous results is that y can be arbitrarily close to one. Our results cover many cases of interest in high-dimensional statistics and random matrix theory.