# Mathematical theory of deep learning

arXiv: https://arxiv.org/abs/2407.18384 · HTML: https://mlsanity.com/paper/2407.18384/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jul 2024.

## Built on by (most cited first, 23 of 23)

- [Online Learning of Neural Networks](https://mlsanity.com/paper/2505.09167/) (2025)
- [On Universality Classes of Equivariant Networks](https://mlsanity.com/paper/2506.02293/) (2025)
- [Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity](https://mlsanity.com/paper/2605.07097/) (2026)
- [Causal pieces: analysing and improving spiking neural networks piece by piece](https://mlsanity.com/paper/2504.14015/) (2025)
- [Minimum Width of Deep Narrow Networks for Universal Approximation](https://mlsanity.com/paper/2511.06837/) (2025)
- [Feature Attribution from First Principles](https://mlsanity.com/paper/2505.24729/) (2025)
- [Autonomous-Flow-Based Generation](https://mlsanity.com/paper/2511.09902/) (2025)
- [Equivalence of approximation by networks of single- and multi-spike neurons](https://mlsanity.com/paper/2603.13478/) (2026)
- [Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning](https://mlsanity.com/paper/2605.04995/) (2026)
- [Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy](https://mlsanity.com/paper/2606.16975/) (2026)
- [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)
- [New Approximation Results and Optimal Estimation for Fully Connected Deep Neural Networks](https://mlsanity.com/paper/2512.09853/) (2025)
- [Classification Fields: Arbitrarily Fine Recursive Hierarchical Clustering From Few Examples](https://mlsanity.com/paper/2605.07119/) (2026)
- [Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples](https://mlsanity.com/paper/2605.08559/) (2026)
- [Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks](https://mlsanity.com/paper/2607.01266/) (2026)
- [Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One](https://mlsanity.com/paper/2607.04597/) (2026)
- [Local large deviations for linear-region growth in random piecewise-linear networks](https://mlsanity.com/paper/2607.07014/) (2026)
- [Mathematics of Data Science](https://mlsanity.com/paper/2607.11938/) (2026)
- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)
- [Learning piecewise-smooth dynamical systems](https://mlsanity.com/paper/2608.19785/) (2026)
- [Neural operators approximate strongly continuous convex monotone semigroups](https://mlsanity.com/paper/2609.02727/) (2026)
- [Neural Approximation by Function Composition: Rigidity and Doubly Exponential Convergence](https://mlsanity.com/paper/2609.25874/) (2026)

## Abstract (arXiv)

This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural network theory. Serving as a guide for students and researchers in mathematics and related fields, the book aims to equip readers with foundational knowledge on the topic. It prioritizes simplicity over generality, and presents rigorous yet accessible results to help build an understanding of the essential mathematical concepts underpinning deep learning.