Minimax Optimal Rates of Estimation in High Dimensional Additive Models: Universal Phase Transition

Abstract

We establish minimax optimal rates of convergence for estimation in a high dimensional additive model assuming that it is approximately sparse. Our results reveal an interesting phase transition behavior universal to this class of high dimensional problems. In the {\it sparse regime} when the components are sufficiently smooth or the dimensionality is sufficiently large, the optimal rates are identical to those for high dimensional linear regression, and therefore there is no additional cost to entertain a nonparametric model. Otherwise, in the so-called {\it smooth regime}, the rates coincide with the optimal rates for estimating a univariate function, and therefore they are immune to the "curse of dimensionality".

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Cited by in corpus (4)

• Nonparametric Regression with Adaptive Truncation via a Convex Hierarchical Penalty
• Minimax density estimation for growing dimension
• Minimax Optimal Estimation in Partially Linear Additive Models under High Dimension
• Adaptive Estimation In High-Dimensional Additive Models With Multi-Resolution Group Lasso