Structural adaptation via $$\mathbb{L}_p$$ -norm oracle inequalities

In this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption. We propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function. The problem of structural adaptation is stated as the problem of selection from a given collection of estimators. We develop a general selection rule and establish for it global oracle inequalities under arbitrary $${\mathbb{L}}_p$$ -losses. These results are applied for adaptive estimation in the additive multi-index model.

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