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

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