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@ARTICLE{Dahmen:874632,
      author       = {Dahmen, David and Gilson, Matthieu and Helias, Moritz},
      title        = {{C}apacity of the covariance perceptron},
      journal      = {Journal of physics / A},
      volume       = {53},
      number       = {35},
      issn         = {0022-3689},
      address      = {Bristol},
      publisher    = {IOP Publ.},
      reportid     = {FZJ-2020-01552},
      pages        = {354002},
      year         = {2020},
      abstract     = {The classical perceptron is a simple neural network that
                      performs a binary classification by a linear mapping between
                      static inputs and outputs and application of a threshold.
                      For small inputs, neural networks in a stationary state also
                      perform an effectively linear input-output transformation,
                      but of an entire time series. Choosing the temporal mean of
                      the time series as the feature for classification, the
                      linear transformation of the network with subsequent
                      thresholding is equivalent to the classical perceptron. Here
                      we show that choosing covariances of time series as the
                      feature for classification maps the neural network to what
                      we call a 'covariance perceptron'; a mapping between
                      covariances that is bilinear in terms of weights. By
                      extending Gardner's theory of connections to this bilinear
                      problem, using a replica symmetric mean-field theory, we
                      compute the pattern and information capacities of the
                      covariance perceptron in the infinite-size limit.
                      Closed-form expressions reveal superior pattern capacity in
                      the binary classification task compared to the classical
                      perceptron in the case of a high-dimensional input and
                      low-dimensional output. For less convergent networks, the
                      mean perceptron classifies a larger number of stimuli.
                      However, since covariances span a much larger input and
                      output space than means, the amount of stored information in
                      the covariance perceptron exceeds the classical counterpart.
                      For strongly convergent connectivity it is superior by a
                      factor equal to the number of input neurons. Theoretical
                      calculations are validated numerically for finite size
                      systems using a gradient-based optimization of a
                      soft-margin, as well as numerical solvers for the NP hard
                      quadratically constrained quadratic programming problem, to
                      which training can be mapped.},
      cin          = {INM-6 / IAS-6 / INM-10},
      ddc          = {530},
      cid          = {I:(DE-Juel1)INM-6-20090406 / I:(DE-Juel1)IAS-6-20130828 /
                      I:(DE-Juel1)INM-10-20170113},
      pnm          = {571 - Connectivity and Activity (POF3-571) / 574 - Theory,
                      modelling and simulation (POF3-574) / MSNN - Theory of
                      multi-scale neuronal networks (HGF-SMHB-2014-2018) / HBP
                      SGA2 - Human Brain Project Specific Grant Agreement 2
                      (785907) / neuroIC002 - Recurrence and stochasticity for
                      neuro-inspired computation (EXS-SF-neuroIC002)},
      pid          = {G:(DE-HGF)POF3-571 / G:(DE-HGF)POF3-574 /
                      G:(DE-Juel1)HGF-SMHB-2014-2018 / G:(EU-Grant)785907 /
                      G:(DE-82)EXS-SF-neuroIC002},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000561625100001},
      doi          = {10.1088/1751-8121/ab82dd},
      url          = {https://juser.fz-juelich.de/record/874632},
}