Spectral statistics of large dimensional Spearman's rank correlation matrix and its application

Let $\mathbf{Q}=(Q_1,\ldots,Q_n)$ be a random vector drawn from the uniform distribution on the set of all $n!$ permutations of $\{1,2,\ldots,n\}$. Let $\mathbf{Z}=(Z_1,\ldots,Z_n)$, where $Z_j$ is the mean zero variance one random variable obtained by centralizing and normalizing $Q_j$, $j=1,\ldots,n$. Assume that $\mathbf {X}_i,i=1,\ldots ,p$ are i.i.d. copies of $\frac{1}{\sqrt{p}}\mathbf{Z}$ and $X=X_{p,n}$ is the $p\times n$ random matrix with $\mathbf{X}_i$ as its $i$th row. Then $S_n=XX^*$ is called the $p\times n$ Spearman's rank correlation matrix which can be regarded as a high dimensional extension of the classical nonparametric statistic Spearman's rank correlation coefficient between two independent random variables. In this paper, we establish a CLT for the linear spectral statistics of this nonparametric random matrix model in the scenario of high dimension, namely, $p=p(n)$ and $p/n\to c\in(0,\infty)$ as $n\to\infty$. We propose a novel evaluation scheme to estimate the core quantity in Anderson and Zeitouni's cumulant method in [Ann. Statist. 36 (2008) 2553-2576] to bypass the so-called joint cumulant summability. In addition, we raise a two-step comparison approach to obtain the explicit formulae for the mean and covariance functions in the CLT. Relying on this CLT, we then construct a distribution-free statistic to test complete independence for components of random vectors. Owing to the nonparametric property, we can use this test on generally distributed random variables including the heavy-tailed ones.Comment: Published at http://dx.doi.org/10.1214/15-AOS1353 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Monovalent Ion Condensation at the Electrified Liquid/Liquid Interface

X-ray reflectivity studies demonstrate the condensation of a monovalent ion at the electrified interface between electrolyte solutions of water and 1,2-dichloroethane. Predictions of the ion distributions by standard Poisson-Boltzmann (Gouy-Chapman) theory are inconsistent with these data at higher applied interfacial electric potentials. Calculations from a Poisson-Boltzmann equation that incorporates a non-monotonic ion-specific potential of mean force are in good agreement with the data.Comment: 4 pages, 4 figure

Eigenvector overlaps in large sample covariance matrices and nonlinear shrinkage estimators

Consider a data matrix $Y = [\mathbf{y}_1, \cdots, \mathbf{y}_N]$ of size $M \times N$, where the columns are independent observations from a random vector $\mathbf{y}$ with zero mean and population covariance $\Sigma$. Let $\mathbf{u}_i$ and $\mathbf{v}_j$ denote the left and right singular vectors of $Y$, respectively. This study investigates the eigenvector/singular vector overlaps $\langle {\mathbf{u}_i, D_1 \mathbf{u}_j} \rangle$, $\langle {\mathbf{v}_i, D_2 \mathbf{v}_j} \rangle$ and $\langle {\mathbf{u}_i, D_3 \mathbf{v}_j} \rangle$, where $D_k$ are general deterministic matrices with bounded operator norms. We establish the convergence in probability of these eigenvector overlaps toward their deterministic counterparts with explicit convergence rates, when the dimension $M$ scales proportionally with the sample size $N$. Building on these findings, we offer a more precise characterization of the loss for Ledoit and Wolf's nonlinear shrinkage estimators of the population covariance $\Sigma$

Average Convergence Rate of Evolutionary Algorithms

In evolutionary optimization, it is important to understand how fast evolutionary algorithms converge to the optimum per generation, or their convergence rate. This paper proposes a new measure of the convergence rate, called average convergence rate. It is a normalised geometric mean of the reduction ratio of the fitness difference per generation. The calculation of the average convergence rate is very simple and it is applicable for most evolutionary algorithms on both continuous and discrete optimization. A theoretical study of the average convergence rate is conducted for discrete optimization. Lower bounds on the average convergence rate are derived. The limit of the average convergence rate is analysed and then the asymptotic average convergence rate is proposed

UniMSE: Towards Unified Multimodal Sentiment Analysis and Emotion Recognition

Multimodal sentiment analysis (MSA) and emotion recognition in conversation (ERC) are key research topics for computers to understand human behaviors. From a psychological perspective, emotions are the expression of affect or feelings during a short period, while sentiments are formed and held for a longer period. However, most existing works study sentiment and emotion separately and do not fully exploit the complementary knowledge behind the two. In this paper, we propose a multimodal sentiment knowledge-sharing framework (UniMSE) that unifies MSA and ERC tasks from features, labels, and models. We perform modality fusion at the syntactic and semantic levels and introduce contrastive learning between modalities and samples to better capture the difference and consistency between sentiments and emotions. Experiments on four public benchmark datasets, MOSI, MOSEI, MELD, and IEMOCAP, demonstrate the effectiveness of the proposed method and achieve consistent improvements compared with state-of-the-art methods.Comment: Accepted to EMNLP 2022 main conferenc

Finite element study of the biomechanical effects on the rotator cuff under load

Rotator cuff injuries account for 50% of shoulder disorders that can cause shoulder pain and reduced mobility. The occurrence of rotator cuff injury is related to the variation in shoulder load, but the mechanical changes in the rotator cuff caused by load remain unclear. Therefore, the mechanical results of the rotator cuff tissue during glenohumeral abduction and adduction were analyzed based on a finite element shoulder model under non-load (0 kg) and load (7.5 kg) conditions. The results showed that the maximum von Mises stress on the supraspinatus muscle was larger than that on the subscapularis, infraspinatus, and teres minor muscles during glenohumeral abduction. Compared with the non-load condition, the maximum von Mises stress on the supraspinatus muscle increased by 75% under the load condition at 30° abduction. Under the load condition, the supraspinatus joint side exhibited an average stress that was 32% greater than that observed on the bursal side. The von Mises stress on the infraspinatus muscle was higher than that in other rotator cuff tissues during adduction. The stress on the infraspinatus muscle increased by 36% in the load condition compared to the non-load condition at 30° adduction. In summary, the increased load changed the mechanical distribution of rotator cuff tissue and increased the stress differential between the joint aspect and the bursal aspect of the supraspinatus tendon