arXiv subjects

Wanjie Wang

Publications and source records attributed to Wanjie Wang.

At least 19 recordsLinked to original sources

arXiv2026

Geometric bias in eigenspace perturbation under random heterogeneous noise

Spectral methods rely on the stability of principal eigenspaces under random perturbations. Classically, this is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error via the operator norm of the noise and the relevant spectral gaps. While sharp for arbitrary deterministic perturbations, these worst-case bounds can be wasteful in the low-rank signal-plus-noise setting, as they fail to capture the interaction between the signal geometry and the noise distribution. We study the spectral perturbation of signal-plus-noise matrices corrupted by sparse random noise with an arbitrary, inhomogeneous variance profile. Under heterogeneous variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias invisible to classical bounds. Leveraging the Quadratic Vector Equation (QVE) and fine-grained isotropic local laws, we derive near-optimal, non-asymptotic bounds for the leading eigenspaces in the operator and 2-to-infinity norms. These separate the usual signal-to-noise contribution, stochastic fluctuations, and structured geometric bias terms determined by the alignment between the signal eigenspaces and the row-wise variance profile. We further develop refined rowwise bounds that adapt to the variance-weighted leverage of the signal space, yielding sharper guarantees in delocalized regimes. As applications, we establish strong consistency of adjacency spectral clustering for degree-corrected stochastic block models with heterogeneous degrees and unbalanced communities, recovering the logarithmic expected-degree scale in the regular balanced case. We also study spectral embedding for generalized random dot product graphs, showing that the full signal embedding admits sharp rowwise control, whereas spectral truncation can retain a systematic geometric bias determined by the omitted signal directions and the variance profile.

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arXiv2026

Fixed-order PCA: Theory for Overestimated Factor Models

We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimension $R$ is fixed and only required to satisfy $R \ge r$, where $r$ is the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the ``extra'' empirical eigencomponents beyond the $r$-th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expanded $r\times R$ map $H'$ and a compressed $R\times r$ map $H^{+}$, and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove $\sqrt{T}$-asymptotic normality for every fixed $R \ge r$. These results provide a theoretical underpinning for the common empirical practice of adopting a conservative upper bound on the number of factors, and shift the analytical burden from consistent dimension selection to the milder requirement of bounding $r$ from above.

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arXiv2026

i-IF-Learn: Iterative Feature Selection and Unsupervised Learning for High-Dimensional Complex Data

Unsupervised learning of high-dimensional data is challenging due to irrelevant or noisy features obscuring underlying structures. It's common that only a few features, called the influential features, meaningfully define the clusters. Recovering these influential features is helpful in data interpretation and clustering. We propose i-IF-Learn, an iterative unsupervised framework that jointly performs feature selection and clustering. Our core innovation is an adaptive feature selection statistic that effectively combines pseudo-label supervision with unsupervised signals, dynamically adjusting based on intermediate label reliability to mitigate error propagation common in iterative frameworks. Leveraging low-dimensional embeddings (PCA or Laplacian eigenmaps) followed by $k$-means, i-IF-Learn simultaneously outputs influential feature subset and clustering labels. Numerical experiments on gene microarray and single-cell RNA-seq datasets show that i-IF-Learn significantly surpasses classical and deep clustering baselines. Furthermore, using our selected influential features as preprocessing substantially enhances downstream deep models such as DeepCluster, UMAP, and VAE, highlighting the importance and effectiveness of targeted feature selection. Code is available at: [https://github.com/mc25800852/i_if_learn].

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arXiv2026

Efficient Propose-Test-Release for Optimal Differentially Private Estimation

Differential privacy (DP) is a rigorous framework that protects the participation of individuals in a dataset by controlling information leakage through released estimators. It brings a challenge for statisticians: DP uniformly considers all possible datasets, whereas statistical practice often downweights atypical or rare outcomes. The conceptual challenge is especially pronounced in sensitivity analysis, where atypical datasets introduces markedly high sensitivity, even for a basic estimator such as ordinary least square. Standard DP recipe adds a noise governed by this large overall sensitivity, which causes excessive loss in accuracy. We introduce an efficient Propose-Test Release (ePTR) pipeline, which tests the dataset via a user-designed Safety Lower Bound, and then probabilistically releases the estimator based on local sensitivity level. This flexible pipeline enables substantially simple DP mechanisms for many problems. To illustrate, we study basic estimators for Bayes classification, linear regression, and kernel regression. Each estimator can be highly sensitive to atypical datasets, yet admits simple ePTR-based algorithms that achieve minimax optimality. In numerical studies, these ePTR estimators demonstrate improved accuracy against popular DP baselines under privacy guarantees.

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arXiv2026

Permutation Recovery on Manifold Data via Spectral Seriation

Data points in many scientific experiments originate from an ordered structure, yet this ordering is often unavailable.We consider noisy data points with the correct ordering to be recovered. The underlying structure naturally places the data on a 1-dimensional manifold. Because eigenfunctions of 1-dimensional manifold Laplacian are trigonometric functions, and the manifold Laplacian can be approximated by the graph data Laplacian, the data ordering can be recovered by inverting the data Laplacian eigenvectors.We propose two spectral algorithms, one for the periodic structure (closed loop) and one for the non-periodic structure (open curve). We have derived the uniform error bound for the algorithms, which is composed of two parts: the discretization error between the manifold eigenfunctions and the noiseless graph Laplacian eigenvectors, and the eigenvectors error caused by data noise. In numerical studies, our spectral seriation algorithms outperform other manifold learning methods. The superior performance of our algorithms is demonstrated further on a biomolecule data example.

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arXiv2026

NetPTR: Optimal Differentially Private Spectral Community Detection on Sparse Networks

Spectral community detection estimates latent labels from the leading eigenspace of a network adjacency matrix, but releasing the resulting labels can disclose sensitive relational information. We consider this problem under differential privacy for both ordinary and bipartite networks. For ordinary networks, the protected unit is a single edge, leading to edge differential privacy (edge-DP). For bipartite networks, the inferential target is the community structure of the left-side nodes, while the protected unit is an entire right-side incidence profile, leading to column-node-DP. We propose NetPTR, a private spectral clustering procedure that releases a noisy empirical spectral embedding after a stability test. The algorithm requires perturbation bounds for empirical eigenspaces under neighboring-network changes, which yield computable stability certificates and local sensitivity bounds. For ordinary networks, we establish edge-DP and the error bound under the degree-corrected stochastic blockmodel, which separates the non-private spectral clustering error from the additional privacy-induced error. It therefore guarantees weak consistency in sparse networks and exact recovery in moderate sparse networks. A matching lower bound shows that the required privacy budget is sharp up to logarithmic factors. We further develop a column-node-DP algorithm for bipartite networks and prove consistency under a bipartite degree-corrected block model. Simulations and real-data examples illustrate the resulting privacy--accuracy tradeoff.

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arXiv2026

Nonparametric undirected graphical model selection using diffusion models

Undirected graphical models provide a fundamental framework for representing conditional independence structures among high-dimensional random variables. While undirected graphical model selection has become a central problem in high-dimensional statistics, most existing methods are restricted to parametric settings. In this paper, we develop a nonparametric approach to undirected graphical model selection based on diffusion models. Recent work has shown that diffusion models can adapt to the unknown graph structure of the underlying distribution, yet utilizing these models for explicit graph estimation remains unexplored. To bridge this gap, we introduce a novel diffusion-based method for nonparametric undirected graphical model selection. We establish the model selection consistency of the proposed method and demonstrate its empirical performance through extensive simulations and two real data analyses.

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arXiv2026

Inlier Recovery for Robust Registration via Gram-Matrix Overlap

Robust point-set registration in the presence of noise and outliers is challenging because the matched points (inliers) must be identified before reliable alignment can be performed. Existing robust registration methods typically optimize over the transformation space and are often designed for regimes with a nonvanishing fraction of inliers. In this paper, we study the inlier recovery problem arising in robust registration by comparing two datasets through the Hadamard product of their Gram matrices. This formulation converts the inlier identification into a structured recovery problem and avoids direct optimization over the rotation group. Based on this idea, we develop two methods: an eigenvector matching method based on the leading eigenvector of the Gram-matrix overlap, and a row-sum matching method based on aggregated entrywise comparison. We show that the eigenvector method achieves weak recovery when the dimension and sample size are of the same order, while the row-sum method achieves exact recovery under a broader range of dimensional scalings. In particular, when the dimension is comparable to the sample size, exact recovery is possible even when the inlier fraction vanishes, with the number of inliers as small as order $\sqrt{n}$, up to logarithmic factors. We also discuss a parallel implementation for large-scale settings. Numerical experiments on brain imaging data and image examples demonstrate that the proposed methods effectively identify matched structure under substantial corruption.

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arXiv2026

Community Detection on Inhomogeneous Multilayer Networks with Extreme Sparsity

We study layer-specific community detection in an $L$-layer network $\{A^{(l)}\}_{l\in[L]}$ on a common set of $n$ nodes. Because modern networks are constructed from multi-modal data or with different contexts, the community labels $π^{(l)}\in[K]^n$ are layer-dependent and the degree heterogeneity parameters $θ_i^{(l)}$ vary widely across nodes and layers. The inhomogeneity and extreme sparsity raise a challenge for classical community detection methods. We propose a multilayer-assisted regularized spectral method (MARS-CD) to address this challenge. For layer $l$, MARS-CD first constructs $X^{(l)}$ from the remaining layers, so that the problem is transformed into a network-with-covariates clustering problem on $(A^{(l)}, X^{(l)})$. Then we recover $π^{(l)}$ by NAC in Hu and Wang (2024) that allows misalignment. The key component is to construct $X^{(l)}$, where we stack regularized embeddings. Building upon this, we establish the first theoretical guarantees for the quality of $X^{(l)}$ under multilayer networks with extreme sparsity. These further lead to weak and strong consistency for recovering $π^{(l)}$. We further develop an optional label alignment step to interpret the shared community structure across layers. Simulations demonstrate the superior performance of our MARS-CD method. Applying MARS-CD to international food trading networks provides an interpretable product-specific community structure.

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arXiv2025

Optimal Network-Guided Covariate Selection for High-Dimensional Data Integration

Modern data often arises with multiple modalities. For example, covariates and a network are observed on the same subjects, and both contain useful information. Effectively integrating these modalities is important and challenging, especially when the response is unavailable. We study the fundamental covariate selection problem for high-dimensional data by leveraging network information. We propose the Network-Guided Covariate Selection (NGCS) algorithm. NGCS exploits the spectral structure of the network to construct a network-guided screening statistic, and employs data-driven Higher Criticism Thresholding for covariate recovery. We establish consistency guarantees for NGCS under general networks. In particular, under two commonly used network models, we relate the projected signal strength to the individual signal strength, and demonstrate that NGCS is optimal for covariate selection. It could achieve the same rate as supervised learning. We further consider a two-study setting for downstream applications, where the network is observed only in Study 1. For clustering and regression, we propose NG-clu and NG-reg algorithms. NG-clu accurately clusters all subjects, while NG-reg improves prediction by using the post-selection covariate matrix. Experiments on synthetic and real datasets demonstrate the robustness and superior performance of our algorithms across various network models, noise distributions, and signal strengths.

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arXiv2025

Spectral Clustering on Multilayer Networks with Covariates

The community detection problem on multilayer networks have drawn much interest. When the nodal covariates ar also present, few work has been done to integrate information from both sources. To leverage the multilayer networks and the covariates, we propose two new algorithms: the spectral clustering on aggregated networks with covariates (SCANC), and the spectral clustering on aggregated Laplacian with covariates (SCALC). These two algorithms are easy to implement, computationally fast, and feature a data-driven approach for tuning parameter selection. We establish theoretical guarantees for both methods under the Multilayer Stochastic Blockmodel with Covariates (MSBM-C), demonstrating their consistency in recovering community structure. Our analysis reveals that increasing the number of layers, incorporating covariate information, and enhancing network density all contribute to improved clustering accuracy. Notably, SCANC is most effective when all layers exhibit similar assortativity, whereas SCALC performs better when both assortative and disassortative layers are present. On the simulation studies and a primary school contact data analysis, our method outperforms other methods. Our results highlight the advantages of spectral-based aggregation techniques in leveraging both network structure and nodal attributes for robust community detection.

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arXiv2024

Uniform error bound for PCA matrix denoising

Principal component analysis (PCA) is a simple and popular tool for processing high-dimensional data. We investigate its effectiveness for matrix denoising. We consider the clean data are generated from a low-dimensional subspace, but masked by independent high-dimensional sub-Gaussian noises with standard deviation $σ$. Under the low-rank assumption on the clean data with a mild spectral gap assumption, we prove that the distance between each pair of PCA-denoised data point and the clean data point is uniformly bounded by $O(σ\log n)$. To illustrate the spectral gap assumption, we show it can be satisfied when the clean data are independently generated with a non-degenerate covariance matrix. We then provide a general lower bound for the error of the denoised data matrix, which indicates PCA denoising gives a uniform error bound that is rate-optimal. Furthermore, we examine how the error bound impacts downstream applications such as clustering and manifold learning. Numerical results validate our theoretical findings and reveal the importance of the uniform error.

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arXiv2024

Network-Adjusted Covariates for Community Detection

Community detection is a crucial task in network analysis that can be significantly improved by incorporating subject-level information, i.e. covariates. However, current methods often struggle with selecting tuning parameters and analyzing low-degree nodes. In this paper, we introduce a novel method that addresses these challenges by constructing network-adjusted covariates, which leverage the network connections and covariates with a unique weight to each node based on the node's degree. Spectral clustering on network-adjusted covariates yields an exact recovery of community labels under certain conditions, which is tuning-free and computationally efficient. We present novel theoretical results about the strong consistency of our method under degree-corrected stochastic blockmodels with covariates, even in the presence of mis-specification and sparse communities with bounded degrees. Additionally, we establish a general lower bound for the community detection problem when both network and covariates are present, and it shows our method is optimal up to a constant factor. Our method outperforms existing approaches in simulations and a LastFM app user network, and provides interpretable community structures in a statistics publication citation network where $30\%$ of nodes are isolated.

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arXiv2023

Covariate-Assisted Community Detection on Sparse Networks

Community detection is an important problem when processing network data. Traditionally, this is done by exploiting the connections between nodes, but connections can be too sparse to detect communities in many real datasets. Node covariates can be used to assist community detection; see Binkiewicz et al. (2017); Weng and Feng (2022); Yan and Sarkar (2021); Yang et al. (2013). However, how to combine covariates with network connections is challenging, because covariates may be high-dimensional and inconsistent with community labels. To study the relationship between covariates and communities, we propose the degree corrected stochastic block model with node covariates (DCSBM-NC). It allows degree heterogeneity among communities and inconsistent labels between communities and covariates. Based on DCSBM-NC, we design the adjusted neighbor-covariate (ANC) data matrix, which leverages covariate information to assist community detection. We then propose the covariate-assisted spectral clustering on ratios of singular vectors (CA-SCORE) method on the ANC matrix. We prove that CA-SCORE successfully recovers community labels when 1) the network is relatively dense; 2) the covariate class labels match the community labels; 3) the data is a mixture of 1) and 2). CA-SCORE has good performance on synthetic and real datasets. The algorithm is implemented in the R(R Core Team (2021)) package CASCORE.

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arXiv2023

High Dimensional Quadratic Discriminant Analysis: Optimality and Phase Transitions

Consider a two-class classification problem where we observe samples $(X_i, Y_i)$ for i = 1, ..., n, $X_i \in R^p$ and $Y_i$ in {0, 1}. Given $Y_i = k$, $X_i$ is assumed to follow a multivariate normal distribution with mean $μ_k \in R^k$ and covariance matrix $Σ_k$, k=0,1. Supposing a new sample X from the same mixture is observed, our goal is to estimate its class label Y. Such a high-dimensional classification problem has been studied thoroughly when Sigma_0 = Sigma_1. However, the discussions over the case $Σ_0 \neq Σ_1$ are much less over the years. This paper presents the quadratic discriminant analysis (QDA) for the weak signals (QDAw) algorithm, and the QDA with feature selection (QDAfs) algorithm. QDAfs applies Partial Correlation Screening to estimate $\hatΩ_0$ and $\hatΩ_1$, and then applies a hard-thresholding on the diagonals of $\hatΩ_0 - \hatΩ_1$. QDAfs further includes the linear term $d^T X$, where d is achieved by a hard-thresholding on $\hatΩ_1\hatμ_1 - \hatΩ_0\hatμ_0$. We further propose the rare and weak model to model the signals in $Ω_0 - Ω_1$ and $μ_0 - μ_1$. Based on the signal weakness and sparsity in $μ_0 - μ_1$, we propose two ways to estimate labels: 1) QDAw for weak but dense signals; 2) QDAfs for relatively strong but sparse signals. We figure out the classification boundary on the 4-dim parameter space: 1) Region of possibility, where either QDAw or QDAfs will achieve a mis-classification error rate of 0; 2) Region of impossibility, where all classifiers will have a constant error rate. The numerical results from real datasets support our theories and demonstrate the necessity and superiority of using QDA over LDA for classification.

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arXiv2022

Truncated Rank-Based Tests for Two-Part Models with Excessive Zeros and Applications to Microbiome Data

High-throughput sequencing technology allows us to test the compositional difference of bacteria in different populations. One important feature of human microbiome data is that it often includes a large number of zeros. Such data can be treated as being generated from a two-part model that includes a zero point-mass. Motivated by analysis of such non-negative data with excessive zeros, we introduce several truncated rank-based two-group and multi-group tests for such data, including a truncated rank-based Wilcoxon rank-sum test for two-group comparison and two truncated Kruskal-Wallis tests for multi-group comparison. We show both analytically through asymptotic relative efficiency analysis and by simulations that the proposed tests have higher power than the standard rank-based tests, especially when the proportion of zeros in the data is high. The tests can also be applied to repeated measurements of compositional data via simple within-subject permutations. In a simple before-and-after treatment experiment, the within-subject permutation is similar to the paired rank test. However, the proposed tests handle the excessive zeros, which leads to a better power. We apply the tests to the analysis of a gut microbiome data set to compare the microbiome compositions of healthy and pediatric Crohn's disease patients and to assess the treatment effects on microbiome compositions. We identify several bacterial genera that are missed by the standard rank-based tests.

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arXiv2020

Graph matching beyond perfectly-overlapping Erdős--Rényi random graphs

Graph matching is a fruitful area in terms of both algorithms and theories. In this paper, we exploit the degree information, which was previously used only in noiseless graphs and perfectly-overlapping Erdős--Rényi random graphs matching. We are concerned with graph matching of partially-overlapping graphs and stochastic block models, which are more useful in tackling real-life problems. We propose the edge exploited degree profile graph matching method and two refined varations. We conduct a thorough analysis of our proposed methods' performances in a range of challenging scenarios, including a zebrafish neuron activity data set and a coauthorship data set. Our methods are proved to be numerically superior than the state-of-the-art methods.

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arXiv2016

Phase Transitions for High Dimensional Clustering and Related Problems

Consider a two-class clustering problem where we observe $X_i = \ell_i μ+ Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature vector $μ\in R^p$ is unknown but is presumably sparse. The class labels $\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate them. We are interested in the statistical limits. In the two-dimensional phase space calibrating the rarity and strengths of useful features, we find the precise demarcation for the Region of Impossibility and Region of Possibility. In the former, useful features are too rare/weak for successful clustering. In the latter, useful features are strong enough to allow successful clustering. The results are extended to the case of colored noise using Le Cam's idea on comparison of experiments. We also extend the study on statistical limits for clustering to that for signal recovery and that for hypothesis testing. We compare the statistical limits for three problems and expose some interesting insight. We propose classical PCA and Important Features PCA (IF-PCA) for clustering. For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all columns of $X$ with an $L^2$-norm larger than $t$. We also propose two aggregation methods. For any parameter in the Region of Possibility, some of these methods yield successful clustering. We find an interesting phase transition for IF-PCA. Our results require delicate analysis, especially on post-selection Random Matrix Theory and on lower bound arguments.

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