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

arXiv: https://arxiv.org/abs/1503.02817 · HTML: https://mlsanity.com/paper/1503.02817/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Mar 2015.

## Built on by (most cited first, 13 of 13)

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- [Distributed Learning with Discretely Observed Functional Data](https://mlsanity.com/paper/2410.02376/) (2024)
- [Generalized Sparse Additive Model with Unknown Link Function](https://mlsanity.com/paper/2410.06012/) (2024)
- [DDAC-SpAM: A Distributed Algorithm for Fitting High-dimensional Sparse Additive Models with Feature Division and Decorrelation](https://mlsanity.com/paper/2205.07932/) (2022)
- [Minimax Signal Detection in Sparse Additive Models](https://mlsanity.com/paper/2304.09398/) (2023)
- [Shallow Neural Networks Learn Low-Degree Spherical Polynomials with Feature Learning by Learnable Channel Attention](https://mlsanity.com/paper/2512.20562/) (2025)
- [Statistical Inference for Explainable Boosting Machines](https://mlsanity.com/paper/2601.18857/) (2026)
- [Gradient Descent with Projection Finds Over-Parameterized Neural Networks for Learning Low-Degree Polynomials with Nearly Minimax Optimal Rate](https://mlsanity.com/paper/2603.21062/) (2026)
- [Minimax Additive Regression under Unknown Dependent Designs](https://mlsanity.com/paper/2609.39212/) (2026)

## Abstract (arXiv)

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".