# Feature Qualification by Deep Nets: A Constructive Approach

arXiv: https://arxiv.org/abs/2503.18676 · HTML: https://mlsanity.com/paper/2503.18676/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 41 references that are in the map.
- First posted on arXiv 24 Mar 2025.

## Builds on (most cited first, 19 of 19)

- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Deep vs. shallow networks : An approximation theory perspective](https://mlsanity.com/paper/1608.03287/) (2016)
- [Provable approximation properties for deep neural networks](https://mlsanity.com/paper/1509.07385/) (2015)
- [The Global Landscape of Neural Networks: An Overview](https://mlsanity.com/paper/2007.01429/) (2020)
- [On ADMM in Deep Learning: Convergence and Saturation-Avoidance](https://mlsanity.com/paper/1902.02060/) (2019)
- [Depth Selection for Deep ReLU Nets in Feature Extraction and Generalization](https://mlsanity.com/paper/2004.00245/) (2020)
- [Generalization and Expressivity for Deep Nets](https://mlsanity.com/paper/1803.03772/) (2018)
- [Realization of spatial sparseness by deep ReLU nets with massive data](https://mlsanity.com/paper/1912.07464/) (2019)
- [Realizing data features by deep nets](https://mlsanity.com/paper/1901.00130/) (2019)
- [Deep Neural Networks Predicting Oil Movement in a Development Unit](https://mlsanity.com/paper/1901.02549/) (2019)
- [Generalization Performance of Empirical Risk Minimization on Over-parameterized Deep ReLU Nets](https://mlsanity.com/paper/2111.14039/) (2021)
- [Deep Convolutional Neural Networks with Zero-Padding: Feature Extraction and Learning](https://mlsanity.com/paper/2307.16203/) (2023)
- [Component-based Sketching for Deep ReLU Nets](https://mlsanity.com/paper/2409.14174/) (2024)

## Abstract (arXiv)

The great success of deep learning has stimulated avid research activities in verifying the power of depth in theory, a common consensus of which is that deep net are versatile in approximating and learning numerous functions. Such a versatility certainly enhances the understanding of the power of depth, but makes it difficult to judge which data features are crucial in a specific learning task. This paper proposes a constructive approach to equip deep nets for the feature qualification purpose. Using the product-gate nature and localized approximation property of deep nets with sigmoid activation (deep sigmoid nets), we succeed in constructing a linear deep net operator that possesses optimal approximation performance in approximating smooth and radial functions. Furthermore, we provide theoretical evidences that the constructed deep net operator is capable of qualifying multiple features such as the smoothness and radialness of the target functions.