Testability of high-dimensional linear models with non-sparse structures

Som e notable recent advances include proposals based on the ridge estimate ( B uhlmann, 2013; Nickl and van de Geer, 2013), on the lasso estimate ( Van de Geer et al., 2014; Zhang and Zhang, 2014), score and orthogonal moments methods ( Belloni et al., 2014a; Goeman et al., 2006), as well as combinations thereof (see for example Belloni et al. ( 2014b); Javanmard and Montanari ( 2014)). Although this line of work has led to many promising methods, the literature, however, does not provide an answer as to how these meth ods should be adapted for the lack of sparse structures in the underlying mod els. First, there is no guidance on how to check whether a model is sparse or no t in high-dimensional settings; the majority of methods construct co nfidence intervals under a set of assumptions describing how sparse the underlyin g model is. The process of developing algorithms that detect model sparsity is still fairly "unattainable", therefore in practice effectively rendering a prior i belief in the sparsity. Second, no comprehensive answer is able to confirm or de ny the ability to perform a hypothesis test (or to construct optimal confidenc e intervals) with formal guarantees that do not rely on model sparsity.