Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators

Abstract

In an increasing number of applications, it is of interest to recover an approximately low-rank data matrix from noisy observations. This paper develops an unbiased risk estimate---holding in a Gaussian model---for any spectral estimator obeying some mild regularity assumptions. In particular, we give an unbiased risk estimate formula for singular value thresholding (SVT), a popular estimation strategy which applies a soft-thresholding rule to the singular values of the noisy observations. Among other things, our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems. In particular, we demonstrate the utility of the unbiased risk estimation for SVT-based denoising of real clinical cardiac MRI series data. We also give new results concerning the differentiability of certain matrix-valued functions.

29 pages, 8 figures

References in corpus (1)

• Risk estimation for matrix recovery with spectral regularization

Cited by in corpus (40)

• Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography
• Group-Sparse Signal Denoising: Non-Convex Regularization, Convex Optimization
• Different Approaches for Human Activity Recognition: A Survey
• Minimax risk of matrix denoising by singular value thresholding
• Navigator-free EPI Ghost Correction with Structured Low-Rank Matrix Models: New Theory and Methods
• The Optimal Hard Threshold for Singular Values is 4/sqrt(3)
• Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery
• Operator-Lipschitz estimates for the singular value functional calculus
• Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics
• Approximate Leave-One-Out for Fast Parameter Tuning in High Dimensions
• Near-optimal matrix recovery from random linear measurements
• Empirical Risk Minimization as Parameter Choice Rule for General Linear Regularization Methods
• Proximal Markov chain Monte Carlo algorithms
• Approximate Leave-One-Out for High-Dimensional Non-Differentiable Learning Problems
• TARM: A Turbo-type Algorithm for Affine Rank Minimization
• Multi-sample Estimation of Bacterial Composition Matrix in Metagenomics Data
• Risk estimation for matrix recovery with spectral regularization
• Quantile universal threshold for model selection
• Regularized and Smooth Double Core Tensor Factorization for Heterogeneous Data
• Multinomial Multiple Correspondence Analysis
• Robust Cardiac Motion Estimation using Ultrafast Ultrasound Data: A Low-Rank-Topology-Preserving Approach
• Ridge-type Linear Shrinkage Estimation of the Matrix Mean of High-dimensional Normal Distribution
• High dimensional regression and matrix estimation without tuning parameters
• Low-Rank Modeling and Its Applications in Image Analysis
• Adaptive Higher-order Spectral Estimators
• Spectral State Compression of Markov Processes
• Matrix Completion with Nonconvex Regularization: Spectral Operators and Scalable Algorithms
• Imputation and low-rank estimation with Missing Not At Random data
• Tensor SVD: Statistical and Computational Limits
• Empirical Bayes Estimates for a 2-Way Cross-Classified Additive Model
• Going off the Grid: Iterative Model Selection for Biclustered Matrix Completion
• Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile
• Adaptive Shrinkage of singular values
• A Fast Data Driven Shrinkage of Singular Values for Arbitrary Rank Signal Matrix Denoising
• Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?
• Computing the degrees of freedom of rank-regularized estimators and cousins
• Denoising-based Turbo Compressed Sensing
• MultiResolution Low-Rank Regularization of Dynamic Imaging Problems
• A comment on Stein's unbiased risk estimate for reduced rank estimators
• Degrees of Freedom and Model Selection for k-means Clustering