Quantitative estimates of the singular values of random i.i.d. matrices

TL;DR

This paper provides new deviation inequalities for the smallest singular values of random i.i.d. matrices with subgaussian entries, improving previous bounds and offering probabilistic estimates for their behavior.

Contribution

It introduces sharper deviation bounds for the lower singular values of i.i.d. matrices, extending the range and accuracy of probabilistic estimates compared to prior work.

Findings

Derived deviation inequality for the k-th smallest singular value

Improved bounds over previous Nguyen's results

Applicable to subgaussian i.i.d. matrices with specific parameter ranges

Abstract

Let $M$ be an $n\times n$ random i.i.d. matrix. This paper studies the deviation inequality of $s_{n-k+1}(M)$, the $k$-th smallest singular value of $M$. In particular, when the entries of $M$ are subgaussian, we show that for any $\gamma\in (0, 1/2), \varepsilon>0$ and $\log n\le k\le c\sqrt{n}$ \begin{align} \textsf{P}\{s_{n-k+1}(M)\le \frac{\varepsilon}{\sqrt{n}} \}\le \Big( \frac{C\varepsilon}{k}\Big)^{\gamma k^{2}}+e^{-c_{1}kn}.\nonumber \end{align} This result improves an existing result of Nguyen, which obtained a deviation inequality of $s_{n-k+1}(M)$ with $(C\varepsilon/k)^{\gamma k^{2}}+e^{-cn}$ decay.