Sparsity and the truncated l2-norm

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