We study a discrete analogue of the classical multivariate Gaussian distribution. It is supported on the integer lattice and is parametrized by the Riemann theta function. Over the reals, the discrete Gaussian is characterized by the property of maximizing entropy, just as its continuous counterpart. We capitalize on the theta function representation to derive statistical properties. Throughout, we exhibit strong connections to the study of Abelian varieties in algebraic geometry.


  1. discrete Gaussians
  2. entropy
  3. Abelian varieties
  4. theta functions

MSC codes

  1. 14K20
  2. 14K25
  3. 94A17 28D20
  4. 60E99
  5. 62B10

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Information & Authors


Published In

cover image SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
Pages: 1 - 30
ISSN (online): 2470-6566


Submitted: 12 January 2018
Accepted: 12 November 2018
Published online: 30 January 2019


  1. discrete Gaussians
  2. entropy
  3. Abelian varieties
  4. theta functions

MSC codes

  1. 14K20
  2. 14K25
  3. 94A17 28D20
  4. 60E99
  5. 62B10



Funding Information

Berlin Mathematical School
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659
Deutscher Akademischer Austauschdienst https://doi.org/10.13039/501100001655
Einstein Stiftung Berlin https://doi.org/10.13039/501100006188

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