# Optimal Approximation with Sparse Neural Networks and Applications

arXiv: https://arxiv.org/abs/2108.06467 · HTML: https://mlsanity.com/paper/2108.06467/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 14 Aug 2021.

## Builds on (most cited first, 4 of 4)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [SSD: Single Shot MultiBox Detector](https://mlsanity.com/paper/1512.02325/) (2015)
- [Optimal Approximation with Sparsely Connected Deep Neural Networks](https://mlsanity.com/paper/1705.01714/) (2017)

## Abstract (arXiv)

We use deep sparsely connected neural networks to measure the complexity of a function class in L2(mathbb Rd) by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system - a countable collection of functions to guide neural networks, since approximation theory with representation system has been well developed in Mathematics. We then prove the fundamental bound theorem, implying a quantity intrinsic to the function class itself can give information about the approximation ability of neural networks and representation system. We also provides a method for transferring existing theories about approximation by representation systems to that of neural networks, greatly amplifying the practical values of neural networks. Finally, we use neural networks to approximate B-spline functions, which are used to generate the B-spline curves. Then, we analyse the complexity of a class called β cartoon-like functions using rate-distortion theory and wedgelets construction.