  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.
