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

TL;DR

This work introduces a self-supervised neural approach to lattice reduction that outputs unimodular base changes to minimize the orthogonality defect, while respecting geometric symmetries through invariance to isometries and scaling and equivariance to a hyperoctahedral subgroup. It advances a Gauss-move–based construction (including extended Gauss moves) to generate a unimodular matrix $Q$ with $B' = BQ$, and analyzes computational complexity relative to the classical LLL algorithm. The method is extended to joint reduction on grids via a convolutional design to exploit cross-lattice correlations, with applications demonstrated in MIMO detection scenarios. While not universally outperforming LLL, the framework demonstrates competitiveness, enables amortized reduction over correlated lattices, and opens pathways for data-driven lattice algorithms tuned to specific domains.

Abstract

Lattice reduction is a combinatorial optimization problem aimed at finding the most orthogonal basis in a given lattice. The Lenstra-Lenstra-Lovász (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.

Figures (8)

• Figure 1: Our neural network maps a basis $B$ of a lattice to a reduced basis $B'=BQ$, where $Q$ is a unimodular base change.
• Figure 2: Two equivalent bases of a two-dimensional lattice.
• Figure 3: Overview of our method.
• Figure 4: The hyperoctahedral group permutes and flips the sign of the basis vector.
• Figure 5: Performance of our method at different steps vs LLL on $n$ dimensional lattices from the Uniform distribution. The trends indicate that the performance of our models starts to saturate after about $n$ steps; this is in line with the theoretical upper-bound of $O(n)$ steps in Proposition \ref{['prop:extgauss']}.

Theorems & Definitions (10)

• Definition 3.1
• Definition 3.4
• Theorem 4.1: carter1983bounded
• Proposition 4.2
• Definition A.1
• Lemma A.2: lenstra1982factoring
• Proposition B.1
• proof
• Proposition B.2
• proof