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

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

doi:10.1109/ISIT.2019.8849501

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

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in reals ⁿ, 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 neural 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 neural network.

Original language	English
Title of host publication	2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1622-1626
Number of pages	5
ISBN (Electronic)	9781538692912
DOIs	https://doi.org/10.1109/ISIT.2019.8849501
Publication status	Published - 1 Jul 2019
Event	2019 IEEE International Symposium on Information Theory, ISIT 2019 - Paris, France Duration: 7 Jul 2019 → 12 Jul 2019

Publication series

Name	IEEE International Symposium on Information Theory - Proceedings
Volume	2019-July
ISSN (Print)	2157-8095

Conference

Conference	2019 IEEE International Symposium on Information Theory, ISIT 2019
Country/Territory	France
City	Paris
Period	7/07/19 → 12/07/19

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10.1109/ISIT.2019.8849501

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Corlay, V., Boutros, J. J., Ciblat, P., & Brunel, L. (2019). On the CVP for the root lattices via folding with deep ReLU neural networks. In 2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings (pp. 1622-1626). Article 8849501 (IEEE International Symposium on Information Theory - Proceedings; Vol. 2019-July). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ISIT.2019.8849501

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title = "On the CVP for the root lattices via folding with deep ReLU neural networks",

abstract = "Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in reals n, 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 neural 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 neural network.",

author = "Vincent Corlay and Boutros, \{Joseph J.\} and Philippe Ciblat and Loic Brunel",

note = "Publisher Copyright: {\textcopyright} 2019 IEEE.; 2019 IEEE International Symposium on Information Theory, ISIT 2019 ; Conference date: 07-07-2019 Through 12-07-2019",

year = "2019",

month = jul,

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doi = "10.1109/ISIT.2019.8849501",

language = "English",

series = "IEEE International Symposium on Information Theory - Proceedings",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

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Corlay, V, Boutros, JJ, Ciblat, P & Brunel, L 2019, On the CVP for the root lattices via folding with deep ReLU neural networks. in 2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings., 8849501, IEEE International Symposium on Information Theory - Proceedings, vol. 2019-July, Institute of Electrical and Electronics Engineers Inc., pp. 1622-1626, 2019 IEEE International Symposium on Information Theory, ISIT 2019, Paris, France, 7/07/19. https://doi.org/10.1109/ISIT.2019.8849501

On the CVP for the root lattices via folding with deep ReLU neural networks. / Corlay, Vincent; Boutros, Joseph J.; Ciblat, Philippe et al.
2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings. Institute of Electrical and Electronics Engineers Inc., 2019. p. 1622-1626 8849501 (IEEE International Symposium on Information Theory - Proceedings; Vol. 2019-July).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in reals n, 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 neural 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 neural network.

AB - Point lattices and their decoding via neural networks are considered in this paper. Lattice decoding in reals n, 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 neural 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 neural network.

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Corlay V, Boutros JJ, Ciblat P, Brunel L. On the CVP for the root lattices via folding with deep ReLU neural networks. In 2019 IEEE International Symposium on Information Theory, ISIT 2019 - Proceedings. Institute of Electrical and Electronics Engineers Inc. 2019. p. 1622-1626. 8849501. (IEEE International Symposium on Information Theory - Proceedings). doi: 10.1109/ISIT.2019.8849501

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

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