# On the CVP for the root lattices via folding with deep ReLU neural networks

## TL;DR

This paper demonstrates how folding techniques with deep ReLU neural networks significantly reduce the complexity of solving the closest vector problem (CVP) in root lattices, from exponential to polynomial growth in network size.

## Contribution

It introduces a folding method that simplifies the boundary function in lattice decoding, enabling deep ReLU networks to efficiently solve CVP with polynomial complexity.

## Key findings

- Folding reduces boundary complexity from exponential to linear.
- Deep ReLU networks require polynomial neurons for CVP.
- Two-layer networks need exponential neurons for the same task.

## Abstract

Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in Rn, known as the closest vector problem (CVP), becomes a classification problem in the fundamental parallelotope with a piecewise linear function defining the boundary. Theoretical results are obtained by studying root lattices. We show how the number of pieces in the boundary function reduces dramatically with folding, from exponential to linear. This translates into a two-layer ReLU network requiring a number of neurons growing exponentially in n to solve the CVP, whereas this complexity becomes polynomial in n for a deep ReLU network.