arXiv subjects

Zeyan Song

Publications and source records attributed to Zeyan Song.

10 recordsLinked to original sources

arXiv2026

An upper bound on the smallest singular value of dense signed random combinatorial matrices

Let $M_n$ be an $n\times n$ random matrix with entries in $\{-1, 0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{-1, 0, 1\}^n$ containing exactly $d$ nonzero entries, with $d=pn$ for some fixed constant $p\in(0,1/2]$. A recent result by one of the authors gives a lower bound of order $n^{-1/2}$ for the smallest singular value $s_n(M_n)$ of this model with high probability. In this paper, we establish a matching upper bound for $s_n(M_n)$. Specifically, there exists a constant $C_p>0$ depending only on $p$ such that \[ \forall \varepsilon >0 \qquad \mathbb{P} \left( s_n(M_n) \le \frac{\sqrt{d}}{\varepsilon^2 n} \right) \ge 1-C_p\left(\varepsilon + \frac{1}{\sqrt{d}}\right). \] Consequently, in the dense regime, the smallest singular value $s_n(M_n)$ of signed random combinatorial matrices is typically of order $n^{-1/2}$.

math.PR↗

arXiv2026

Least singular values of Gaussian matrices with deterministic shifts

Let $A$ be a deterministic real square matrix and $G$ a standard Gaussian matrix. We determine $\E\smin(A+G)$, up to universal constants, in terms of the singular values of $A$. The expectation is comparable to the square root of the smallest zero of the expected characteristic polynomial of $(A+G)^{\top}(A+G)$. This resolves Problem~3 in Tikhomirov's ICM 2022 survey. We also prove that the zeros of the expected characteristic polynomial are majorized by the expected ordered eigenvalues of $(A+G)^{\top}(A+G)$.

math.PR↗

arXiv2026

Sharp singularity asymptotics for discrete random matrices

We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality $q\ge2$, we prove $\mathbb{P}(s_n(M_n)\le z/\sqrt{n}) \le Cz+a_n(S)+C'n^{1+\varepsilon}q^{-n}$ for all $n\ge1$, $z\ge0$, and $\varepsilon>0$, with $C=C(S)$ and $C'=C'(S,\varepsilon)$. Here $s_n$ denotes the least singular value, and $a_n(S)$ is the sum of the probabilities of the events involving zero rows or columns and equal or opposite pairs of rows or columns.

math.PR↗

arXiv2026

Frontend-Assisted Large Audio-Language Models for SLT2026 SmartGlasses Challenge

This paper describes our systems for the Track 1 time-stamped speaker-attributed ASR task of the SmartGlasses Challenge. To handle overlapping speech, background noise and far-field conditions in two-person conversations, we employ SoulX-Transcriber, a multimodal large language model that takes full-length audio as input and generates structured transcriptions. In addition, we investigate the effect of speech frontend processing, including a rule-based Guided Source Separation method and a data-driven speech separation model, both of which improve performance on the validation set. Considering inference stability, our final submission uses the fine-tuned SoulX-Transcriber system, achieving a tcpCER of 6.22 on the test set.

eess.AS↗

arXiv2026

The singularity probability of dense random regular graphs

Let $G_{n,d}$ be a uniformly random simple $d$-regular graph on $n$ vertices, and let $A_n$ be its adjacency matrix. For every fixed $λ\in(0,1/2)$, we prove that for any $λ(n-1)\le d\le(1-λ)(n-1)$, $\mathbb P(A_n\text{ is singular})\le e^{-cn}$, where $c>0$ depends only on $λ$, and $n$ is sufficiently large with $nd$ even.

math.PR↗

arXiv2026

Rank deficiency of Bernoulli random matrices for growing corank

Let A be an n x n Bernoulli random matrix whose entries are i.i.d. Bernoulli(p) random variables. In this paper, we determine the probability that the corank of A is at least k when k is of order o(sqrt(log n)): P(corank A >= k) = (1-p+o_n(1))^(kn).

math.PR↗

arXiv2026

The eigenvalue gap of inhomogeneous symmetric discrete random matrix

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and eigenvectors. Firstly, we prove that for k <= n / log n, 1 <= i <= n - k and epsilon >= 0, P(the gap between the (i+k)-th and i-th eigenvalues is at most epsilon n^(-1/2)) <= (C epsilon)^((k^2 + k)/2) + exp(-c n), where the eigenvalues are ordered increasingly. Secondly, combining the recent result of Yi Han, we give a quantitative estimate of the singular values of A. For c log n <= k <= sqrt(n) and epsilon >= 0, we have P(the (n-k+1)-th smallest singular value of A is at most k epsilon n^(-1/2)) <= (C epsilon)^(c k^2) + exp(-c k n), where the singular values are ordered increasingly. Finally, based on the distance analytical framework developed for the eigenvalue gap, we further derive quantitative bounds for singular values and delocalization of eigenvectors. In particular, we establish a quantitative bound for the probability that some eigenvector of A exhibits no-gap delocalization, which improves the result of Rudelson and Vershynin.

math.PR↗

arXiv2025

Singularity of biased discrete random matrix

We study the singularity probability of n*n random matrices with i.i.d. entries from highly biased discrete distributions. We obtain sharp non-asymptotic bounds for this probability and derive estimates on the least singular values. Our method combines combinatorial, geometric, and probabilistic techniques such as sphere decomposition and anticoncentration inequalities. The results extend classical invertibility theory to biased discrete settings and resolve an open problem by characterizing the dominant causes of singularity in biased discrete random matrices, namely the presence of zero columns or linearly dependent column pairs.

math.PR↗

arXiv2025

Auditory Attention Decoding from Ear-EEG Signals: A Dataset with Dynamic Attention Switching and Rigorous Cross-Validation

Recent promising results in auditory attention decoding (AAD) using scalp electroencephalography (EEG) have motivated the exploration of cEEGrid, a flexible and portable ear-EEG system. While prior cEEGrid-based studies have confirmed the feasibility of AAD, they often neglect the dynamic nature of attentional states in real-world contexts. To address this gap, a novel cEEGrid dataset featuring three concurrent speakers distributed across three of five distinct spatial locations is introduced. The novel dataset is designed to probe attentional tracking and switching in realistic scenarios. Nested leave-one-out validation-an approach more rigorous than conventional single-loop leave-one-out validation-is employed to reduce biases stemming from EEG's intricate temporal dynamics. Four rule-based models are evaluated: Wiener filter (WF), canonical component analysis (CCA), common spatial pattern (CSP) and Riemannian Geometry-based classifier (RGC). With a 30-second decision window, WF and CCA models achieve decoding accuracies of 41.5% and 41.4%, respectively, while CSP and RGC models yield 37.8% and 37.6% accuracies using a 10-second window. Notably, both WF and CCA successfully track attentional state switches across all experimental tasks. Additionally, higher decoding accuracies are observed for electrodes positioned at the upper cEEGrid layout and near the listener's right ear. These findings underscore the utility of dynamic, ecologically valid paradigms and rigorous validation in advancing AAD research with cEEGrid.

eess.AS↗

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↗