State space representations of the Roesser type for convolutional layers
Patricia Pauli Dennis Gramlich Fran Allgöwer
From the perspective of control theory, convolutional layers (of neural
networks) are 2-D (or N-D) linear time-invariant dynamical systems. The usual
representation of convolutional layers by the convolution kernel corresponds to
the representation of a dynamical system by its impulse response. However, many
analysis tools from control theory, e.g., involving linear matrix inequalities,
require a state space representation. For this reason, we explicitly provide a
state space representation of the Roesser type for 2-D convolutional layers
with $c_\mathrm{in}r_1 + c_\mathrm{out}r_2$ states, where
$c_\mathrm{in}$/$c_\mathrm{out}$ is the number of input/output channels of the
layer and $r_1$/$r_2$ characterizes the width/length of the convolution kernel.
This representation is shown to be minimal for $c_\mathrm{in} =
c_\mathrm{out}$. We further construct state space representations for dilated,
strided, and N-D convolutions.