# Lattice-based Deep Neural Networks: Regularity and Tailored Regularization

arXiv: https://arxiv.org/abs/2603.02809 · HTML: https://mlsanity.com/paper/2603.02809/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 89 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Real-time Neural Radiance Caching for Path Tracing](https://mlsanity.com/paper/2106.12372/) (2021)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [Random-Access Neural Compression of Material Textures](https://mlsanity.com/paper/2305.17105/) (2023)
- [Enhancing accuracy of deep learning algorithms by training with low-discrepancy sequences](https://mlsanity.com/paper/2005.12564/) (2020)
- [Higher-order Quasi-Monte Carlo Training of Deep Neural Networks](https://mlsanity.com/paper/2009.02713/) (2020)
- [Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks](https://mlsanity.com/paper/2404.03761/) (2024)
- [Langevin Quasi-Monte Carlo](https://mlsanity.com/paper/2309.12664/) (2023)
- [Integral Equations and Machine Learning](https://mlsanity.com/paper/1712.06115/) (2017)
- [Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration](https://mlsanity.com/paper/2208.02767/) (2022)
- [Artificial Neural Networks generated by Low Discrepancy Sequences](https://mlsanity.com/paper/2103.03543/) (2021)
- [Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/2210.16196/) (2022)
- [Quasi-Monte Carlo Algorithms (not only) for Graphics Software](https://mlsanity.com/paper/2307.15584/) (2023)
- [Using Low-Discrepancy Points for Data Compression in Machine Learning: An Experimental Comparison](https://mlsanity.com/paper/2407.07450/) (2024)

## Built on by (most cited first, 2 of 2)

- [Fourier Neural Operators with rank-1 lattice points and hyperbolic cross](https://mlsanity.com/paper/2606.08871/) (2026)
- [High Probability Derivative Bounds for Random tanh Neural Networks on a Hypercube](https://mlsanity.com/paper/2608.26526/) (2026)

## Abstract (arXiv)

This survey article is concerned with the application of lattice rules to Deep Neural Networks (DNNs), lattice rules being a family of quasi-Monte Carlo methods. They have demonstrated effectiveness in various contexts for high-dimensional integration and function approximation. They are extremely easy to implement thanks to their very simple formulation -- all that is required is a good integer generating vector of length matching the dimensionality of the problem. In recent years there has been a burst of research activities on the application and theory of DNNs. We review our recent article on using lattice rules as training points for DNNs with a smooth activation function, where we obtained explicit regularity bounds of the DNNs. By imposing restrictions on the network parameters to match the regularity features of the target function, we prove that DNNs with tailored lattice training points can achieve good theoretical generalization error bounds, with implied constants independent of the input dimension. We also demonstrate numerically that DNNs trained with our tailored regularization perform significantly better than with standard ℓ2 regularization.