Singular vector and singular subspace distribution for the matrix denoising model

Abstract

In this paper, we study the matrix denosing model $Y=S+X$, where $S$ is a low-rank deterministic signal matrix and $X$ is a random noise matrix, and both are $M\times n$. In the scenario that $M$ and $n$ are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of $Y$. More specifically, we derive the limiting distribution of angles between the principal singular vectors of $Y$ and their deterministic counterparts, the singular vectors of $S$. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of $Y$ and that spanned by the singular vectors of $S$. It turns out that the limiting distributions depend on the structure of the singular vectors of $S$ and the distribution of $X$, and thus they are non-universal.

Final version. Accepted by the Annals of Statistics

References in corpus (22)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• On Asymptotic Properties of Large Random Matrices with Independent Entries
• The largest eigenvalue of rank one deformation of large Wigner matrices
• The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
• Limit theorems for sample eigenvalues in a generalized spiked population model
• Universality of covariance matrices
• Minimax risk of matrix denoising by singular value thresholding
• Averaging Fluctuations in Resolvents of Random Band Matrices
• Limits of spiked random matrices I
• Near-optimal bounds for phase synchronization
• Universality for the largest eigenvalue of sample covariance matrices with general population
• The outliers of a deformed Wigner matrix
• Normal approximation and concentration of spectral projectors of sample covariance
• Rate Optimal Denoising of Simultaneously Sparse and Low Rank Matrices
• Spectrum of deformed random matrices and free probability
• Principal component analysis for big data
• The Role of Principal Angles in Subspace Classification
• De-biased sparse PCA: Inference and testing for eigenstructure of large covariance matrices
• Optimal Subspace Estimation Using Overidentifying Vectors via Generalized Method of Moments
• Spiked covariances and principal components analysis in high-dimensional random effects models
• Confidence Region of Singular Subspaces for Low-rank Matrix Regression
• Principal components of spiked covariance matrices in the supercritical regime

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• Asymptotic Theory of Eigenvectors for Random Matrices with Diverging Spikes
• Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates
• Principal components of spiked covariance matrices in the supercritical regime
• Empirical Bayes PCA in high dimensions
• Eigenvector distribution in the critical regime of BBP transition