# Analysis of Deep Neural Networks with Quasi-optimal polynomial approximation rates

arXiv: https://arxiv.org/abs/1912.02302 · HTML: https://mlsanity.com/paper/1912.02302/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2019.

## Built on by (most cited first, 6 of 6)

- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning](https://mlsanity.com/paper/2208.12190/) (2022)
- [SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures](https://mlsanity.com/paper/2505.09572/) (2025)
- [Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics](https://mlsanity.com/paper/2406.01539/) (2024)
- [Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation](https://mlsanity.com/paper/2606.26705/) (2026)
- [Reconstruction of frequency-localized functions from pointwise samples via least squares and deep learning](https://mlsanity.com/paper/2502.09794/) (2025)

## Abstract (arXiv)

We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this network achieves an error rate that is sub-exponential in the number of polynomial functions, M, used in the polynomial approximation. The complexity of the network which achieves this sub-exponential rate is shown to be algebraic in M.