Synthetic aperture radar (SAR) despeckling is an inverse-recovery problem in which multiplicative non-Gaussian noise must be suppressed without erasing scattering structures. We revisit a nonlocal sparse estimator that applies a log--Yeo--Johnson transformation, stacks similar patches into groups, codes each group on its own left singular basis, and shrinks the resulting coefficients. Three quantities usually treated as tunable are shown to be fixed by this construction. First, the group dictionary is orthonormal, so the weighted Lasso admits an exact coefficient-wise soft-threshold solution: the iterative inner solver is unnecessary, and the two apparent weighting matrices are the numerator and denominator of a single threshold field rather than independent modules. Second, because the dictionary is estimated from the noisy group itself, its retained subspace absorbs speckle in proportion to the group aspect ratio $γ=p^2/K$; a random-matrix argument converts the corresponding regularization constant into a geometry-calibrated correction and collapses patch size, group size, and shrinkage scale into one analytically determined degree of freedom. Third, singular projection makes the coefficient noise nearly Gaussian at every tested look number, which locates the point at which an exact speckle likelihood ceases to be informative. The resulting estimator is deterministic, training-free, and applies one set of analytically determined settings to every image and sensor. It ranks first in 18 of 24 PSNR/SSIM comparisons against twelve published methods on three synthetic benchmarks, and attains the lowest mean deviation of the ratio image from the theoretical speckle model over six real-SAR configurations from five sensors. Code is available \href{https://github.com/Teriri1999/Geometry-Calibrated-Closed-Form-Shrinkage-for-SAR-Despeckling}{here}.
We develop an approximate risk minimization framework for shrinkage-thresholding estimation in normal mean problems. In the canonical multivariate normal mean model, we introduce a general functional class of estimators that contains classical shrinkage and thresholding behavior, including James-Stein-type and lasso-type rules. We express quadratic risk as a functional over this class, derive optimality conditions for both oracle risk and data-driven approximate risk minimization, and construct a feasible approximate risk criterion from the observed data when the oracle risk is unavailable. The resulting estimator, NOMAD, is obtained by minimizing this approximate risk over the proposed class. For the canonical model, we develop an approximate risk minimization theory that includes optimizer characterization, sieve-based consistency under regularity conditions, and approximate-risk inequalities relative to benchmark procedures in the admissible class. We then extend the framework to multivariate normal mean estimation with correlated observations, develop both MLE-based and conditional MLE-based constructions, and establish consistency results under regularity conditions. We further apply the framework to linear regression and derive an equivalent penalized regression representation in which the shrinkage-thresholding map induces a data-adaptive penalty, recovering ridge-type and lasso-type behavior as special cases or limiting forms. The results provide a unified risk-based framework for shrinkage, thresholding, and regularization across canonical and correlated normal mean estimation and linear regression.
Wayne Yuan Gao, Zhiheng Youstat.ME econ.EM stat.ML
We study how the choice of default prior for a common Gaussian scale affects high-dimensional shrinkage risk, highlighting the role played by high-dimensional geometry. Formally, we consider a high-dimensional setting in which the near-zero behavior of the common scale prior has first-order consequences for shrinkage risk, and show that priors that are flat on the variance and those flat on the standard deviation allocate markedly different mass near the zero-scale boundary, leading to distinct shrinkage behavior and informing principled default prior selection. Specifically, under a radial-power benchmark, we establish that the SD-flat benchmark has a one-unit asymptotic risk advantage near the origin, crosses over in the critical regime, and is second-order equivalent to the variance-flat benchmark for strong signals. Proper single global-scale hyperpriors and bounded coordinate-multiplier mixtures inherit these limits through the near-zero exponent of their SD-scale density. For heavier-tailed or sparse priors, that exponent still classifies the common global-scale component, while local-scale tails, model-size priors, or allocation priors can also affect risk.