State space representations of the Roesser type for convolutional layers
This work addresses a theoretical bottleneck for researchers in control theory and neural network analysis by offering a new representation method, though it is incremental as it adapts existing control theory concepts to convolutional layers.
The paper tackles the problem of analyzing convolutional layers in neural networks by providing a state space representation of the Roesser type, which enables the use of control theory tools like linear matrix inequalities, with a minimal representation for equal input and output channels.
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.