Neural network approaches to point lattice decoding
This work characterizes the complexity of lattice decoding for researchers and practitioners in signal processing and communications, offering insights into why deep learning can be effective for structured lattices and why shallow networks work for unstructured MIMO lattices.
This paper analyzes the complexity of the lattice decoding problem using a neural network perspective, showing it is equivalent to computing a continuous piecewise linear (CPWL) function. The complexity of this function is exponential in space dimension 'n', but for structured lattices, a technique called 'folding' (equivalent to deep neural networks) reduces this to polynomial in 'n'.
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).