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Guozheng Dai

Publications and source records attributed to Guozheng Dai.

13 recordsLinked to original sources
arXiv2026

Randomized second order Riesz projections on the Hamming cube

In this paper, we improve the arbitrary Banach space \(n \log n\) bound of Ivanisvili--Volberg \cite{IvanisviliVolberg2022} for the second order projection bound to the order \(\sqrt{n}\) bound. Moreover, we study the lower Riesz estimate with the pointwise square gradient, and prove a fixed chaos characterization: on every fixed homogeneous Walsh chaos $H_k$, the dimension free estimate \[ \|Δ^{1/2}f\|_{L^p(Ω_n;X)} \lesssim_{p,k,X} \||\nabla f|_X\|_{L^p(Ω_n)} \] holds for all $n$ if and only if $X$ has Rademacher type $2$. We also consider an exact tail space norm of the analytic paraproduct $T_φg(z)=\int_0^z g(ζ)φ'(ζ)\,dζ$ on Banach valued \(H^\infty\) spaces. A matching lower bound of Volberg \cite{Volberg2024} \[ \|T_φ:H_d^\infty(\mathbb D;Y)\to H^\infty(\mathbb D;Y)\| \asymp_{α,φ} d^{-α} \] under a nondegenerate boundary singularity assumption is established.

math.PR↗
arXiv2026

Sparsity-adaptive concentration inequalities for random polynomials

We prove concentration inequalities for polynomials of independent, sparse $α$-sub-exponential random variables. Specifically, we consider $X_i=δ_iξ_i$, where the Bernoulli selectors $δ_i$ are independent with parameters $p_i$, and the variables $ξ_i$ are independent \(α\)-sub-exponential random variables (not necessarily centered). For any polynomial $f:\mathbb R^n\to\mathbb R $ of degree at most $D$ and any $0<α\le 1 $, we establish an $L_r$-moment bound for \(f(X)-\mathbb E f(X)\) in terms of partition norms of sparsity-weighted expected derivative tensors. The weights count distinct coordinates rather than multiplicities and therefore distinguish diagonal, partially diagonal, and off-diagonal contributions. This captures the sparse scaling in both collective fluctuation regimes and extreme-coordinate regimes. When all sparsity parameters are equal to one, our result recovers the polynomial concentration inequality of Götze, Sambale, and Sinulis. In degree two, it recovers sparse Hanson-Wright bounds. As applications, we derive deviation inequalities for the distance between a sparse simple random tensor and a fixed subspace, and obtain lower bounds for the smallest singular value of matrices whose columns are independent sparse simple random tensors.

math.PR↗
arXiv2026

The exact group-sparse recovery for block diagonal matrices with subexponential entries

We study block-diagonal random matrices with i.i.d. subexponential entries and show that, despite their highly structured form, they already guarantee exact sparse recovery from a nearly optimal number of measurements. When the matrix reduces to a single block, our framework collapses to the classical i.i.d. subexponential ensemble, and our bounds recover the well-known optimal rates previously established for unstructured random matrices.

math.PR↗
arXiv2026

A note on correlation inequalities for regular increasing families

This paper establishes quantitative correlation inequalities between monotone events and structured threshold objects in both the discrete cube and Gaussian space. We prove that for any increasing balanced family, there exists a linear threshold function yielding a covariance lower bound of $c \frac{\log n}{\sqrt{n}}$, and extend this principle to halfspaces in Gaussian space. These results verify the conjectures of Kalai, Keller, and Mossel regarding optimal correlation bounds for linear threshold functions and their Gaussian analogues.

math.PR↗
arXiv2025

A note on the improved sparse Hanson-Wright inequalities

We establish sparse Hanson-Wright inequalities for quadratic forms of sparse $α$-sub-exponential random vectors with exponent parameter $α\in(0, 2]$. In the regime $0< α\le 1$ we derive a refined inequality that is optimal in several canonical models. These results extend the classical Hanson-Wright bound to the sparse setting. Illustrative applications include covariance matrix estimation with incomplete observations, low-rank matrix approximation under the maximum norm with sparsified sketches, and concentration inequalities for sparse $α$-sub-exponential random vectors.

math.PR↗
arXiv2025

Quantitative estimates of the spectral norm of random matrices with independent columns

This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an $m\times n$ random matrix $BA$, where $A$ is an $N\times n$ random matrix with independent mean-zero subexponential entries, and $B$ is an $m\times N$ deterministic matrix. We prove that the $L_{p}$ norm of the spectral norm of $BA$ is upper bounded by $(\sqrt{m}+\sqrt{n})p$. It is remarkable that this result is independent of the dimension $N$.

math.PR↗
arXiv2025

The Rank and Singular Values of the Inhomogeneous Subgaussian Random Matrices

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.

math.PR↗
arXiv2024

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

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 $γ\in (0, 1/2), \varepsilon>0$ 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)^{γk^{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 $(C\varepsilon/k)^{γk^{2}}+e^{-cn}$ decay.

math.PR↗
arXiv2024

Deviation Inequalities for the Spectral Norm of Structured Random Matrices

We study the deviation inequality for the spectral norm of structured random matrices with non-gaussian entries. In particular, we establish an optimal bound for the $p$-th moment of the spectral norm by transfering the spectral norm into the suprema of canonical processes. A crucial ingredient of our proof is a comparison of weak and strong moments. As an application, we show a deviation inequality for the smallest singular value of a rectangular random matrix.

math.PR↗
arXiv2024

On Log-Concave-Tailed Chaoses and the Restricted Isometry Property

In this paper, we obtain a $p$-th moment bound for the suprema of a log-concave-tailed nonhomogeneous chaos process, which is optimal in some special cases. A crucial ingredient of the proof is a novel decoupling inequality, which may be of independent interest. With this $p$-th moment bound, we show two uniform Hanson-Wright type deviation inequalities for $α$-subexponential entries ($1\le α\le 2$), which recover some known results. As applications, we prove the restricted isometry property of partial random circulant matrices and time-frequency structured random matrices induced by standard $α$-subexponential vectors ($1\le α\le 2$), which extends the previously known results for the subgaussian case.

math.PR↗
arXiv2024

Uniform Hanson-Wright Type Deviation Inequalities for $α$-Subexponential Random Vectors

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector with independent centered $α$-subexponential entries, $0<α\le 1$. Our method relies upon a novel decoupling inequality and a comparison of weak and strong moments. As an application, we use the derived inequality to prove the restricted isometry property of partial random circulant matrices generated by standard $α$-subexponential random vectors, $0<α\le 1$.

math.PR↗
arXiv2023

Tail Bounds on the Spectral Norm of Sub-Exponential Random Matrices

Let $X$ be an $n\times n$ symmetric random matrix with independent but non-identically distributed entries. The deviation inequalities of the spectral norm of $X$ with Gaussian entries have been obtained by using the standard concentration of Gaussian measure results. This paper establishes an upper tail bound of the spectral norm of $X$ with sub-Exponential entries. Our method relies upon a crucial ingredient of a novel chaining argument that essentially involves both the particular structure of the sets used for the chaining and the distribution of coordinates of a point on the unit sphere.

math.PR↗