arXiv subjects

Wang Zhou

Publications and source records attributed to Wang Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2025

On Convergence Rates of Spiked Eigenvalue Estimates: A General Study of Global and Local Laws in Sample Covariance Matrices

This paper investigates global and local laws for sample covariance matrices with general growth rates of dimensions. The sample size $N$ and population dimension $M$ can have the same order in logarithm, which implies that their ratio $M/N$ can approach zero, a constant, or infinity. These theories are utilized to determine the convergence rate of spiked eigenvalue estimates.

math.ST↗

arXiv2025

Two-dimensional Sparse Parallelism for Large Scale Deep Learning Recommendation Model Training

The increasing complexity of deep learning recommendation models (DLRM) has led to a growing need for large-scale distributed systems that can efficiently train vast amounts of data. In DLRM, the sparse embedding table is a crucial component for managing sparse categorical features. Typically, these tables in industrial DLRMs contain trillions of parameters, necessitating model parallelism strategies to address memory constraints. However, as training systems expand with massive GPUs, the traditional fully parallelism strategies for embedding table post significant scalability challenges, including imbalance and straggler issues, intensive lookup communication, and heavy embedding activation memory. To overcome these limitations, we propose a novel two-dimensional sparse parallelism approach. Rather than fully sharding tables across all GPUs, our solution introduces data parallelism on top of model parallelism. This enables efficient all-to-all communication and reduces peak memory consumption. Additionally, we have developed the momentum-scaled row-wise AdaGrad algorithm to mitigate performance losses associated with the shift in training paradigms. Our extensive experiments demonstrate that the proposed approach significantly enhances training efficiency while maintaining model performance parity. It achieves nearly linear training speed scaling up to 4K GPUs, setting a new state-of-the-art benchmark for recommendation model training.

cs.DC↗

arXiv2025

Few-Round Distributed Principal Component Analysis: Closing the Statistical Efficiency Gap by Consensus

Distributed algorithms and theories are called for in this era of big data. Under weaker local signal-to-noise ratios, we improve upon the celebrated one-round distributed principal component analysis (PCA) algorithm designed in the spirit of divide-and-conquer, by introducing a few additional communication rounds of consensus. The proposed shifted subspace iteration algorithm is able to close the local phase transition gap, reduce the asymptotic variance, and also alleviate the potential bias. Our estimation procedure is easy to implement and tuning-free. The resulting estimator is shown to be statistically efficient after an acceptable number of iterations. We also discuss extensions to distributed elliptical PCA for heavy-tailed data. Empirical experiments on synthetic and benchmark datasets demonstrate our method's statistical advantage over the divide-and-conquer approach.

stat.ME↗

arXiv2024

On Reward Transferability in Adversarial Inverse Reinforcement Learning: Insights from Random Matrix Theory

In the context of inverse reinforcement learning (IRL) with a single expert, adversarial inverse reinforcement learning (AIRL) serves as a foundational approach to providing comprehensive and transferable task descriptions. However, AIRL faces practical performance challenges, primarily stemming from the framework's overly idealized decomposability condition, the unclear proof regarding the potential equilibrium in reward recovery, or questionable robustness in high-dimensional environments. This paper revisits AIRL in \textbf{high-dimensional scenarios where the state space tends to infinity}. Specifically, we first establish a necessary and sufficient condition for reward transferability by examining the rank of the matrix derived from subtracting the identity matrix from the transition matrix. Furthermore, leveraging random matrix theory, we analyze the spectral distribution of this matrix, demonstrating that our rank criterion holds with high probability even when the transition matrices are unobservable. This suggests that the limitations on transfer are not inherent to the AIRL framework itself, but are instead related to the training variance of the reinforcement learning algorithms employed within it. Based on this insight, we propose a hybrid framework that integrates on-policy proximal policy optimization in the source environment with off-policy soft actor-critic in the target environment, leading to significant improvements in reward transfer effectiveness.

stat.ML↗

arXiv2024

$SLE_6$ and 2-d critical bond percolation on the square lattice

Through the rotational invariance of the 2-d critical bond percolation exploration path on the square lattice we express Smirnov's edge parafermionic observable as a sum of two new edge observables. With the help of these two new edge observables we can apply the discrete harmonic analysis and conformal mapping theory to prove the convergence of the 2-d critical bond percolation exploration path on the square lattice to the trace of $SLE_6$ as the mesh size of the lattice tends to zero.

math.PR↗

arXiv2024

Necessary and sufficient condition for CLT of linear spectral statistics of sample correlation matrices

In this paper, we establish the central limit theorem (CLT) for the linear spectral statistics (LSS) of sample correlation matrix $R$, constructed from a $p\times n$ data matrix $X$ with independent and identically distributed (i.i.d.) entries having mean zero, variance one, and infinite fourth moments in the high-dimensional regime $n/p\rightarrow ϕ\in \mathbb{R}_+\backslash \{1\}$. We derive a necessary and sufficient condition for the CLT. More precisely, under the assumption that the identical distribution $ξ$ of the entries in $X$ satisfies $\mathbb{P}(|ξ|>x)\sim l(x)x^{-α}$ when $x\rightarrow \infty$ for $α\in (2,4]$, where $l(x)$ is a slowly varying function, we conclude that: (i). When $α\in(3,4]$, the universal asymptotic normality for the LSS of sample correlation matrix holds, with the same asymptotic mean and variance as in the finite fourth moment scenario; (ii) We identify a necessary and sufficient condition $\lim_{x\rightarrow\infty}x^3\mathbb{P}(|ξ|>x)=0$ for the universal CLT; (iii) We establish a local law for $α\in (2, 4]$. Overall, our proof strategy follows the routine of the matrix resampling, intermediate local law, Green function comparison, and characteristic function estimation. In various parts of the proof, we are required to come up with new approaches and ideas to solve the challenges posed by the special structure of sample correlation matrix. Our results also demonstrate that the symmetry condition is unnecessary for the CLT of LSS for sample correlation matrix, but the tail index $α$ plays a crucial role in determining the asymptotic behaviors of LSS for $α\in (2, 3)$.

math.PR↗

arXiv2024

Knowledge Transfer across Multiple Principal Component Analysis Studies

Transfer learning has aroused great interest in the statistical community. In this article, we focus on knowledge transfer for unsupervised learning tasks in contrast to the supervised learning tasks in the literature. Given the transferable source populations, we propose a two-step transfer learning algorithm to extract useful information from multiple source principal component analysis (PCA) studies, thereby enhancing estimation accuracy for the target PCA task. In the first step, we integrate the shared subspace information across multiple studies by a proposed method named as Grassmannian barycenter, instead of directly performing PCA on the pooled dataset. The proposed Grassmannian barycenter method enjoys robustness and computational advantages in more general cases. Then the resulting estimator for the shared subspace from the first step is further utilized to estimate the target private subspace in the second step. Our theoretical analysis credits the gain of knowledge transfer between PCA studies to the enlarged eigenvalue gap, which is different from the existing supervised transfer learning tasks where sparsity plays the central role. In addition, we prove that the bilinear forms of the empirical spectral projectors have asymptotic normality under weaker eigenvalue gap conditions after knowledge transfer. When the set of informativesources is unknown, we endow our algorithm with the capability of useful dataset selection by solving a rectified optimization problem on the Grassmann manifold, which in turn leads to a computationally friendly rectified Grassmannian K-means procedure. In the end, extensive numerical simulation results and a real data case concerning activity recognition are reported to support our theoretical claims and to illustrate the empirical usefulness of the proposed transfer learning methods.

stat.ML↗

arXiv2023

Limiting behavior of bilinear forms for the resolvent of sample covariance matrices under elliptical distribution with applications

In this paper, we introduce a joint central limit theorem (CLT) for specific bilinear forms, encompassing the resolvent of the sample covariance matrix under an elliptical distribution. Through an exhaustive exploration of our theoretical findings, we unveil a phase transition in the limiting parameters that relies on the moments of the random radius in our derived CLT. Subsequently, we employ the established CLT to address two statistical challenges under elliptical distribution. The first task involves deriving the CLT for eigenvector statistics of the sample covariance matrix. The second task aims to ascertain the limiting properties of the spiked sample eigenvalues under a general spiked model. As a byproduct, we discover that the eigenmatrix of the sample covariance matrix under a light-tailed elliptical distribution satisfies the necessary conditions for asymptotic Haar, thereby extending the Haar conjecture to broader distributions.

math.ST↗

arXiv2023

Testing the number of common factors by bootstrapped sample covariance matrix in high-dimensional factor models

This paper studies the impact of bootstrap procedure on the eigenvalue distributions of the sample covariance matrix under a high-dimensional factor structure. We provide asymptotic distributions for the top eigenvalues of bootstrapped sample covariance matrix under mild conditions. After bootstrap, the spiked eigenvalues which are driven by common factors will converge weakly to Gaussian limits after proper scaling and centralization. However, the largest non-spiked eigenvalue is mainly determined by the order statistics of the bootstrap resampling weights, and follows extreme value distribution. Based on the disparate behavior of the spiked and non-spiked eigenvalues, we propose innovative methods to test the number of common factors. Indicated by extensive numerical and empirical studies, the proposed methods perform reliably and convincingly under the existence of both weak factors and cross-sectionally correlated errors. Our technical details contribute to random matrix theory on spiked covariance model with convexly decaying density and unbounded support, or with general elliptical distributions.

math.ST↗

arXiv2023

Extreme eigenvalues of sample covariance matrices under generalized elliptical models with applications

We consider the extreme eigenvalues of the sample covariance matrix $Q=YY^*$ under the generalized elliptical model that $Y=Σ^{1/2}XD.$ Here $Σ$ is a bounded $p \times p$ positive definite deterministic matrix representing the population covariance structure, $X$ is a $p \times n$ random matrix containing either independent columns sampled from the unit sphere in $\mathbb{R}^p$ or i.i.d. centered entries with variance $n^{-1},$ and $D$ is a diagonal random matrix containing i.i.d. entries and independent of $X.$ Such a model finds important applications in statistics and machine learning. In this paper, assuming that $p$ and $n$ are comparably large, we prove that the extreme edge eigenvalues of $Q$ can have several types of distributions depending on $Σ$ and $D$ asymptotically. These distributions include: Gumbel, Fréchet, Weibull, Tracy-Widom, Gaussian and their mixtures. On the one hand, when the random variables in $D$ have unbounded support, the edge eigenvalues of $Q$ can have either Gumbel or Fréchet distribution depending on the tail decay property of $D.$ On the other hand, when the random variables in $D$ have bounded support, under some mild regularity assumptions on $Σ,$ the edge eigenvalues of $Q$ can exhibit Weibull, Tracy-Widom, Gaussian or their mixtures. Based on our theoretical results, we consider two important applications. First, we propose some statistics and procedure to detect and estimate the possible spikes for elliptically distributed data. Second, in the context of a factor model, by using the multiplier bootstrap procedure via selecting the weights in $D,$ we propose a new algorithm to infer and estimate the number of factors in the factor model. Numerical simulations also confirm the accuracy and powerfulness of our proposed methods and illustrate better performance compared to some existing methods in the literature.

stat.ME↗

arXiv2023

Sampling without replacement from a high-dimensional finite population

It is well known that most of the existing theoretical results in statistics are based on the assumption that the sample is generated with replacement from an infinite population. However, in practice, available samples are almost always collected without replacement. If the population is a finite set of real numbers, whether we can still safely use the results from samples drawn without replacement becomes an important problem. In this paper, we focus on the eigenvalues of high-dimensional sample covariance matrices generated without replacement from finite populations. Specifically, we derive the Tracy-Widom laws for their largest eigenvalues and apply these results to parallel analysis. We provide new insight into the permutation methods proposed by Buja and Eyuboglu in [Multivar Behav Res. 27(4) (1992) 509--540]. Simulation and real data studies are conducted to demonstrate our results.

math.ST↗

arXiv2022

CLT for random quadratic forms based on sample means and sample covariance matrices

In this paper, we use the dimensional reduction technique to study the central limit theory (CLT) random quadratic forms based on sample means and sample covariance matrices. Specifically, we use a matrix denoted by $U_{p\times q}$, to map $q$-dimensional sample vectors to a $p$ dimensional subspace, where $q\geq p$ or $q\gg p$. Under the condition of $p/n\rightarrow 0$ as $(p,n)\rightarrow \infty$, we obtain the CLT of random quadratic forms for the sample means and sample covariance matrices.

math.ST↗

arXiv2022

Factor Modelling for Clustering High-dimensional Time Series

We propose a new unsupervised learning method for clustering a large number of time series based on a latent factor structure. Each cluster is characterized by its own cluster-specific factors in addition to some common factors which impact on all the time series concerned. Our setting also offers the flexibility that some time series may not belong to any clusters. The consistency with explicit convergence rates is established for the estimation of the common factors, the cluster-specific factors, the latent clusters. Numerical illustration with both simulated data as well as a real data example is also reported. As a spin-off, the proposed new approach also advances significantly the statistical inference for the factor model of Lam and Yao (2012).

math.ST↗

arXiv2022

Testing Kronecker Product Covariance Matrices for High-dimensional Matrix-Variate Data

Kronecker product covariance structure provides an efficient way to modeling the inter-correlations of matrix-variate data. In this paper, we propose testing statistics for Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. Central limit theorem is proved for the linear spectral statistics with explicit formulas for mean and covariance functions, which fills the gap in the literature. We then theoretically justify that the proposed testing statistics have well-controlled sizes and strong powers. To facilitate practical usefulness, we further propose a bootstrap resampling algorithm to approximate the limiting distributions of associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. A more general model which allows the existence of noises will also be discussed. In the simulations, the empirical sizes of the proposed testing procedure and its bootstrapped version are close to corresponding theoretical values, while the powers converge to one quickly as the dimension and sample size grow.

math.ST↗

arXiv2021

Quantification of Carbon Sequestration in Urban Forests

Vegetation, trees in particular, sequester carbon by absorbing carbon dioxide from the atmosphere. However, the lack of efficient quantification methods of carbon stored in trees renders it difficult to track the process. We present an approach to estimate the carbon storage in trees based on fusing multi-spectral aerial imagery and LiDAR data to identify tree coverage, geometric shape, and tree species -- key attributes to carbon storage quantification. We demonstrate that tree species information and their three-dimensional geometric shapes can be estimated from aerial imagery in order to determine the tree's biomass. Specifically, we estimate a total of $52,000$ tons of carbon sequestered in trees for New York City's borough Manhattan.

cs.CV↗

arXiv2021

Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let $X$ be an $M\times N$ random matrix consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ with general population covariance matrix $Σ$. In the literature, the quantity $XX^{*}$ is referred to as the sample covariance matrix after scaling, where $X^{*}$ is the transpose of $X$. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of $XX^{*}$ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ follow as $M,N\to\infty$ with $M/N\toϕ_0>0$ if the weak fourth moment of the radius of $\mathbf{x}_{1}$ exists . In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.

math.ST↗

arXiv2021

On eigenvalues of a high-dimensional spatial-sign covariance matrix

This paper investigates limiting properties of eigenvalues of multivariate sample spatial-sign covariance matrices when both the number of variables and the sample size grow to infinity. The underlying p-variate populations are general enough to include the popular independent components model and the family of elliptical distributions. A first result of the paper establishes that the distribution of the eigenvalues converges to a deterministic limit that belongs to the family of generalized Marcenko-Pastur distributions. Furthermore, a new central limit theorem is established for a class of linear spectral statistics. We develop two applications of these results to robust statistics for a high-dimensional shape matrix. First, two statistics are proposed for testing the sphericity. Next, a spectrum-corrected estimator using the sample spatial-sign covariance matrix is proposed. Simulation experiments show that in high dimension, the sample spatial-sign covariance matrix provides a valid and robust tool for mitigating influence of outliers.

math.ST↗

arXiv2020

Physics-Informed Neural Network Super Resolution for Advection-Diffusion Models

Physics-informed neural networks (NN) are an emerging technique to improve spatial resolution and enforce physical consistency of data from physics models or satellite observations. A super-resolution (SR) technique is explored to reconstruct high-resolution images ($4\times$) from lower resolution images in an advection-diffusion model of atmospheric pollution plumes. SR performance is generally increased when the advection-diffusion equation constrains the NN in addition to conventional pixel-based constraints. The ability of SR techniques to also reconstruct missing data is investigated by randomly removing image pixels from the simulations and allowing the system to learn the content of missing data. Improvements in S/N of $11\%$ are demonstrated when physics equations are included in SR with $40\%$ pixel loss. Physics-informed NNs accurately reconstruct corrupted images and generate better results compared to the standard SR approaches.

cs.CV↗