# Neural Lattice Reduction: A Self-Supervised Geometric Deep Learning Approach

arXiv: https://arxiv.org/abs/2311.08170 · HTML: https://mlsanity.com/paper/2311.08170/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 14 Nov 2023.

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

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Knowledge Distillation: A Survey](https://mlsanity.com/paper/2006.05525/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Pointer Networks](https://mlsanity.com/paper/1506.03134/) (2015)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Deep Learning with Limited Numerical Precision](https://mlsanity.com/paper/1502.02551/) (2015)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [Understanding Straight-Through Estimator in Training Activation Quantized Neural Nets](https://mlsanity.com/paper/1903.05662/) (2019)
- [Learning to Solve NP-Complete Problems - A Graph Neural Network for Decision TSP](https://mlsanity.com/paper/1809.02721/) (2018)
- [Graph Colouring Meets Deep Learning: Effective Graph Neural Network Models for Combinatorial Problems](https://mlsanity.com/paper/1903.04598/) (2019)
- [Edge Representation Learning with Hypergraphs](https://mlsanity.com/paper/2106.15845/) (2021)
- [Augment with Care: Contrastive Learning for Combinatorial Problems](https://mlsanity.com/paper/2202.08396/) (2022)
- [On the approximation capability of GNNs in node classification/regression tasks](https://mlsanity.com/paper/2106.08992/) (2021)

## Built on by (most cited first, 3 of 3)

- [Local Look-Ahead Guidance via Verifier-in-the-Loop for Automated Theorem Proving](https://mlsanity.com/paper/2503.09730/) (2025)
- [Reducing Formal Context Extraction: A Newly Proposed Framework from Big Corpora](https://mlsanity.com/paper/2504.06285/) (2025)
- [Discovering Lattice Reduction Strategies via Self-Play](https://mlsanity.com/paper/2606.15301/) (2026)

## Abstract (arXiv)

Lattice reduction is a combinatorial optimization problem aimed at finding the most orthogonal basis in a given lattice. The Lenstra-Lenstra-Lov'asz (LLL) algorithm is the best algorithm in the literature for solving this problem. In light of recent research on algorithm discovery, in this work, we would like to answer this question: is it possible to parametrize the algorithm space for lattice reduction problem with neural networks and find an algorithm without supervised data? Our strategy is to use equivariant and invariant parametrizations and train in a self-supervised way. We design a deep neural model outputting factorized unimodular matrices and train it in a self-supervised manner by penalizing non-orthogonal lattice bases. We incorporate the symmetries of lattice reduction into the model by making it invariant to isometries and scaling of the ambient space and equivariant with respect to the hyperocrahedral group permuting and flipping the lattice basis elements. We show that this approach yields an algorithm with comparable complexity and performance to the LLL algorithm on a set of benchmarks. Additionally, motivated by certain applications for wireless communication, we extend our method to a convolutional architecture which performs joint reduction of spatially-correlated lattices arranged in a grid, thereby amortizing its cost over multiple lattices.