Smooth Hard-Thresholding for Singular Values with Stein's Unbiased Risk Estimate
Abstract
Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.
Related
- A noncommutative weak L1-estimate for rough singular integrals
- On a pointwise estimate for a weighted rough singular integral operator in stratified Lie groups and applications
- Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators
- From estimate to proof: certified ground-state energy bounds for singular Schrödinger operators on R3
- Quantitative Volume Estimate on Singular Sets of One-phase Free Boundary Problems in Ricci Limit Spaces
BibTeX
@misc{yang2026smooth,
title = {{Smooth Hard-Thresholding for Singular Values with Stein's Unbiased Risk Estimate}},
author = {Guanzhong Yang and Zi Wang and Guang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.06632},
eprint = {2606.06632},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06632},
}
RIS
TY - UNPB
TI - Smooth Hard-Thresholding for Singular Values with Stein's Unbiased Risk Estimate
AU - Guanzhong Yang
AU - Zi Wang
AU - Guang Yang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2606.06632
UR - https://arxiv.org/abs/2606.06632
AB - Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.
ER -