arXiv subjects
Publications and source records attributed to Zeyan Song.
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).
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.
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.
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.
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.