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Problems involving the simultaneous estimation of multiple parameters arise in many areas of theoretical and applied statistics. A canonical example is the estimation of a vector of normal means. Frequently, structural information about relationships between the parameters of interest is available. For example, in a gene expression denoising problem, genes with similar functions may have similar expression levels. Despite its importance, structural information has not been well-studied in the simultaneous estimation literature, perhaps in part because it poses challenges to the usual geometric or empirical Bayes shrinkage estimation paradigms. This article proposes that some of these challenges can be resolved by adopting an alternate paradigm, based on regression modeling. This approach can naturally incorporate structural information and also motivates new shrinkage estimation and inference procedures. As an illustration, this regression paradigm is used to develop a class of estimators with asymptotic risk optimality properties that perform well in simulations and in denoising gene expression data from a single cell RNA-sequencing experiment. [ABSTRACT FROM AUTHOR]