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The Rank and Singular Values of the Inhomogeneous Subgaussian Random Matrices

Guozheng Dai, Zeyan Song, Hanchao Wang

TL;DR

The paper addresses the rank and conditioning of $n\times n$ random matrices with independent inhomogeneous subgaussian entries (mean zero), proving that for $k< c\sqrt{n}$ the rank deficit satisfies $\Pr(\mathrm{rank}(A)\le n-k) \le \exp(-ckn)$ and establishing a deviation bound for small singular values. Central to the approach is the randomized log-least common denominator (RLCD), which handles non-identically distributed entries and enables small-ball control on linear images; the authors couple RLCD with randomized rounding to discretize subspaces and control the arithmetic structure of kernels. The analysis partitions the unit sphere into sparse, compressible, and incompressible regions, builds nets for the incompressible part, and proves a key Proposition 3.1 to bound kernel events. The main results extend Rudelson’s 2024 i.i.d. subgaussian framework to inhomogeneous ensembles, yielding sharp rank-deficiency and singular-value deviation bounds with broad applicability to non-iid random matrices in numerical and probabilistic contexts.

Abstract

Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we get a deviation inequality for the singular values of A. This extends earlier results of Rudelson's paper in 2024 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 the past advance, 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 can overcome this problem using a randomized least common denominator (RLCD) from Livshyts in 2021.

The Rank and Singular Values of the Inhomogeneous Subgaussian Random Matrices

TL;DR

The paper addresses the rank and conditioning of random matrices with independent inhomogeneous subgaussian entries (mean zero), proving that for the rank deficit satisfies and establishing a deviation bound for small singular values. Central to the approach is the randomized log-least common denominator (RLCD), which handles non-identically distributed entries and enables small-ball control on linear images; the authors couple RLCD with randomized rounding to discretize subspaces and control the arithmetic structure of kernels. The analysis partitions the unit sphere into sparse, compressible, and incompressible regions, builds nets for the incompressible part, and proves a key Proposition 3.1 to bound kernel events. The main results extend Rudelson’s 2024 i.i.d. subgaussian framework to inhomogeneous ensembles, yielding sharp rank-deficiency and singular-value deviation bounds with broad applicability to non-iid random matrices in numerical and probabilistic contexts.

Abstract

Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we get a deviation inequality for the singular values of A. This extends earlier results of Rudelson's paper in 2024 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 the past advance, 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 can overcome this problem using a randomized least common denominator (RLCD) from Livshyts in 2021.

Paper Structure

This paper contains 16 sections, 20 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Decomposition of the sphere
  5. Concentration and tensorization
  6. Randomized least common denominators
  7. The number of integer points inside a ball
  8. Almost orthogonal systems of vectors
  9. Restricted invertibility phenomenon
  10. Estimating of compressible case
  11. Incompressible vectors
  12. Size of the net
  13. Approximating
  14. The proof of Proposition 3.1
  15. The proof of Theorem \ref{['Theo_main1']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k,n\in \mathbb{N}$ be numbers such that $k<d_{1.1}\sqrt{n}$ and A be an $n\times n$ matrix with independent entries with satisfying Eq_condition. Then where $d_{1.1}$, $c_{Theo_main1}>0$ are some constants depending only on $K$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • ...and 29 more