Sparsity and the truncated `-norm

Sparsity is a fundamental topic in highdimensional data analysis. Perhaps the most common measures of sparsity are the `norms, for 0 ≤ p < 2. In this paper, we study an alternative measure of sparsity, the truncated `-norm, which is related to other `-norms, but appears to have some unique and useful properties. Focusing on the ndimensional Gaussian location model, we derive exact asymptotic minimax results for estimation over truncated `-balls, which complement existing results for `-balls. We then propose simple new adaptive thresholding estimators that are inspired by the truncated `-norm and are adaptive asymptotic minimax over `-balls (0 ≤ p < 2), as well as truncated `-balls. Finally, we derive lower bounds on the Bayes risk of an estimator, in terms of the parameter’s truncated `-norm. These bounds provide necessary conditions for Bayes risk consistency in certain problems that are relevant for high-dimensional Bayesian modeling.