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

By Ming Yuan, Ding-Xuan Zhou. 2015

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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".

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Local Inference in Additive Models with Decorrelated Local Linear Estimator (2019, background): “Much progress has been made to understand the prediction performance of various proposals, including [23,26,30,31,37,38,42,43].”