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低秩随机矩阵特征向量的逐元素高阶分析：偏差校正、Edgeworth 展开与自助法AI 解读

Understanding the distributions of spectral estimators in low-rank random matrix models, also known as signal-plus-noise matrix models, is fundamentally important in various statistical learning problems, including network analysis, matrix denoising and matrix completion. This paper investigates the entrywise eigenvector distributions in a broad range of low-rank signal-plus-noise matrix models by establishing their higher-order accurate stochastic expansions. At a high…

信号加噪声矩阵模型的特征向量辅助估计框架AI 解读

Summary In this paper, we develop an eigenvector-assisted estimation framework for a collection of signal-plus-noise matrix models arising in high-dimensional statistics and many applications. The framework is built upon a novel asymptotically unbiased estimating equation using the leading eigenvectors of the data matrix. However, the estimator obtained by directly solving the estimating equation could be numerically unstable in practice and lacks robustness against model…

随机点积图潜在位置的高效一步估计AI 解读

We propose a one-step procedure to estimate the latent positions in random dot product graphs efficiently. Unlike the classical spectral-based methods, the proposed one-step procedure takes advantage of both the low-rank structure of the expected adjacency matrix and the Bernoulli likelihood information of the sampling model simultaneously. We show that for each vertex, the corresponding row of the one-step estimator (OSE) converges to a multivariate normal distribution…

计算机模型的贝叶斯投影校准AI 解读

We develop a Bayesian approach called the Bayesian projected calibration to address the problem of calibrating an imperfect computer model using observational data from an unknown complex physical system. The calibration parameter and the physical system are parameterized in an identifiable fashion via the L2-projection. The physical system is imposed a Gaussian process prior distribution, which naturally induces a prior distribution on the calibration parameter through the…

随机点积图潜在位置的最优贝叶斯估计AI 解读

Summary We propose and prove the optimality of a Bayesian approach for estimating the latent positions in random dot product graphs, which we call posterior spectral embedding. Unlike classical spectral-based adjacency, or Laplacian spectral embedding, posterior spectral embedding is a fully likelihood-based graph estimation method that takes advantage of the Bernoulli likelihood information of the observed adjacency matrix. We develop a minimax lower bound for estimating…

贝叶斯排斥型高斯混合模型AI 解读

We develop a general class of Bayesian repulsive Gaussian mixture models that encourage well-separated clusters, aiming at reducing potentially redundant components produced by independent priors for locations (such as the Dirichlet process). The asymptotic results for the posterior distribution of the proposed models are derived, including posterior consistency and posterior contraction rate in the context of nonparametric density estimation. More importantly, we show that…