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

Vincent Corlay; Joseph J. Boutros; Philippe Ciblat; Loic Brunel

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

Machine Learning 2019-03-01 v2 Information Theory math.IT

Authors: Vincent Corlay , Joseph J. Boutros , Philippe Ciblat , Loic Brunel

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.

Keywords

point lattices closest vector problem root lattices folding deep relu network piecewise linear boundary

Cite

@article{arxiv.1902.05146,
  title  = {On the CVP for the root lattices via folding with deep ReLU neural networks},
  author = {Vincent Corlay and Joseph J. Boutros and Philippe Ciblat and Loic Brunel},
  journal= {arXiv preprint arXiv:1902.05146},
  year   = {2019}
}

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Submitted to 2019 IEEE International Symposium on Information Theory (ISIT)

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

Abstract

Keywords

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