Structural adaptation via $L_p$-norm oracle inequalities

TL;DR

This paper introduces a new adaptive estimation method for multivariate functions that automatically adjusts to unknown structure and smoothness, providing theoretical guarantees under various loss functions.

Contribution

It proposes a novel selection rule for estimators that achieves structural adaptation with oracle inequalities, applied to additive multi-index models.

Findings

Establishes oracle inequalities under arbitrary Lp-losses.

Develops a general selection rule for adaptive estimation.

Demonstrates effectiveness in additive multi-index models.

Abstract

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 $\rL_p$--losses. These results are applied for adaptive estimation in the additive multi--index model.