Neural network approaches to point lattice decoding

TL;DR

This paper analyzes the complexity of lattice decoding using neural networks, showing how network depth and structure influence decoding efficiency and complexity, especially for different lattice types.

Contribution

It introduces the Voronoi-reduced basis concept and links lattice decoding complexity to neural network CPWL functions, highlighting the benefits of deep networks and lattice structure.

Findings

Decoding complexity is exponential in space dimension for general lattices.

Deep neural networks can reduce complexity from exponential to polynomial for structured lattices.

Shallow networks are more effective for unstructured MIMO lattices in low to moderate dimensions.

Abstract

We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown to be equivalent to computing a continuous piecewise linear (CPWL) function restricted to the fundamental parallelotope. On the other hand, it is known that any function computed by a ReLU feed-forward neural network is CPWL. As a result, we count the number of affine pieces in the CPWL decoding function to characterize the complexity of the decoding problem. It is exponential in the space dimension $n$, which induces shallow neural networks of exponential size. For structured lattices we show that folding, a technique equivalent to using a deep neural network, enables to reduce this complexity from exponential in $n$ to polynomial in $n$. Regarding unstructured MIMO lattices, in contrary to dense lattices many pieces in the CPWL decoding function can be neglected for quasi-optimal decoding on the Gaussian channel. This makes the decoding problem easier and it explains why shallow neural networks of reasonable size are more efficient with this category of lattices (in low to moderate dimensions).