Efficient Estimation for Random Dot Product Graphs via a One-Step Procedure

Xie, Fangzheng; Xu, Yanxun

doi:10.1080/01621459.2021.1948419

Cited by 11 publications

(17 citation statements)

References 28 publications

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“…In a classical parametric model, under certain regularity conditions, the one-step update leads to an asymptotically efficient estimator when the initial guess is √ n-consistent (see, e.g., Section 5.7 in Van der Vaart, 2000). The same idea has also appeared in Xie and Xu (2019) for efficient estimation of a more general low-rank random graph. In what follows, we apply the one-step procedure to SBM when Σ 0 is potentially singular and obtain an efficient estimator.…”

Section: Stochastic Block Modelmentioning

confidence: 93%

Euclidean Representation of Low-Rank Matrices and Its Statistical Applications

Xie

2021

Preprint

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Low-rank matrices are pervasive throughout statistics, machine learning, signal processing, optimization, and applied mathematics. In this paper, we propose a novel and user-friendly Euclidean representation framework for low-rank matrices. Correspondingly, we establish a collection of technical and theoretical tools for analyzing the intrinsic perturbation of low-rank matrices in which the underlying referential matrix and the perturbed matrix both live on the same low-rank matrix manifold. Our analyses show that, locally around the referential matrix, the sine-theta distance between subspaces is equivalent to the Euclidean distance between two appropriately selected orthonormal basis, circumventing the orthogonal Procrustes analysis. We also establish the regularity of the proposed Euclidean representation function, which has a profound statistical impact and a meaningful geometric interpretation. These technical devices are applicable to a broad range of statistical problems. Specific applications considered in detail include Bayesian sparse spiked covariance model with non-intrinsic loss, efficient estimation in stochastic block models where the block probability matrix may be degenerate, and leastsquares estimation in biclustering problems. Both the intrinsic perturbation analysis of low-rank matrices and the regularity theorem may be of independent interest.

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Section: Stochastic Block Modelmentioning

confidence: 93%

Euclidean Representation of Low-Rank Matrices and Its Statistical Applications

Xie

2021

Preprint

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“…We now revisit Example 4 for illustration. In the context of random dot product graphs (Example 1), with the weight function being h n (s, t) = ρ n /{s(1 − s)}, the eigenvector-assisted Zestimator is the one-step estimator proposed in Xie and Xu (2021). Then it follows immediately from Theorem 4.1 that…”

Section: Large Sample Properties Of the Z-estimatormentioning

confidence: 99%

“…In the context of random graph inference, Athreya et al (2016), Tang and Priebe (2018), and Xie (2021) studied the central limit theorems for the rows of the eigenvector matrix. Xie and Xu (2021) and Xie (2021) proposed a one-step refinement for the eigenvectors and explored the corresponding entrywise limit theorem. Agterberg et al (2021) further extended the signal-plus-noise matrix framework to general rectangular matrices and allowed heteroskedasticity and dependence of the noise distributions.…”

Section: Related Workmentioning

confidence: 99%

“…Below, we provide two examples of the weight function h n (•, •) in the eigenvector-assisted estimating equation (3.1). These two choices of h n (•, •) lead to the spectral embedding defined in (2.2) and the one-step estimator for random dot product graphs (Xie and Xu, 2021).…”

Section: Eigenvector-assisted Estimating Equationmentioning

confidence: 99%

“…When A is the adjacency matrix of a random graph, the spectral embedding X is also referred to as the adjacency spectral embedding (ASE) (Sussman et al, 2012). Although the ASE is practically useful because of the numerical stability and the ease of implementation, as pointed out by Xie and Xu (2020) and Xie and Xu (2021), it is asymptotically sub-optimal because it does not incorporate the information of the Bernoulli likelihood. Instead, Xie and Xu (2021) proposed the following one-step estimator…”

Section: Eigenvector-assisted Estimating Equationmentioning

confidence: 99%

See 2 more Smart Citations

Eigenvector-Assisted Statistical Inference for Signal-Plus-Noise Matrix Models

Xie¹,

Wu²

2022

Preprint

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In this paper, we develop a generalized Bayesian inference framework for a collection of signal-plusnoise matrix models arising in high-dimensional statistics and many applications. The framework is built upon an asymptotically unbiased estimating equation with the assistance of the leading eigenvectors of the data matrix. The solution to the estimating equation coincides with the maximizer of an appropriate statistical criterion function. The generalized posterior distribution is constructed by replacing the usual log-likelihood function in the Bayes formula with the criterion function. The proposed framework does not require the complete specification of the sampling distribution and is convenient for uncertainty quantification via a Markov Chain Monte Carlo sampler, circumventing the inconvenience of resampling the data matrix. Under mild regularity conditions, we establish the large sample properties of the estimating equation estimator and the generalized posterior distributions. In particular, the generalized posterior credible sets have the correct frequentist nominal coverage probability provided that the socalled generalized information equality holds. The validity and usefulness of the proposed framework are demonstrated through the analysis of synthetic datasets and the real-world ENZYMES network datasets.

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Two‐sample testing for random graphs

Wen

2024

Statistical Analysis

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The employment of two‐sample hypothesis testing in examining random graphs has been a prevalent approach in diverse fields such as social sciences, neuroscience, and genetics. We advance a spectral‐based two‐sample hypothesis testing methodology to test the latent position random graphs. We propose two distinct asymptotic normal statistics, each optimally designed for two different models—the elementary Erdős–Rényi model and the more complex latent position random graph model. For the latter, the spectral embedding of the adjacency matrix was utilized to estimate the test statistic. The proposed method exhibited superior efficacy as it accomplished higher power than the conventional method of mean estimation. To validate our hypothesis testing procedure, we applied it to empirical biological data to discern structural variances in gene co‐expression networks between COVID‐19 patients and individuals who remained unaffected by the disease.

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Efficient Estimation for Random Dot Product Graphs via a One-Step Procedure

Cited by 11 publications

References 28 publications

Euclidean Representation of Low-Rank Matrices and Its Statistical Applications

Euclidean Representation of Low-Rank Matrices and Its Statistical Applications

Eigenvector-Assisted Statistical Inference for Signal-Plus-Noise Matrix Models

Two‐sample testing for random graphs

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