The small ball probability for singular values of the inhomogeneous subgaussian random matrices
Abstract
Let $A$ be an $n\times n$ random matrix with independent centered nonconstant subgaussian entries. We give the small ball probability for the $k$th smallest singular value of $A$ when $k \in ( c \log n, c\sqrt{n})$. Furthermore, we prove that for any $k \lt c\sqrt{n}$, the probability of the matrix having a rank lower than $n-k$ is sub-exponential. This extends the earlier results of Rudelson’s 2024 paper by removing the assumption of the identical distribution of the entries across the matrix. Our model covers inhomogeneous matrices, allowing different subgaussian moments for the entries as long as their subgaussian moments have a standard upper bound. In past work, the assumption of i.i.d. entries was required due to the lack of least common denominators of the non-i.i.d. random matrix. We overcome this problem using the randomized least common denominator (RLCD) of Livshyts (2021).
