Circulant matrices and the stability theory of CNNs

In this paper we show that feedback matrices of ring CNNs are block circulants; as special cases, for example, feedback matrices of one-dimensional ring CNNs are circulant matrices. Circulants and their close relations the block circulants possess many pleasant properties which allow one to describe their spectrum completely. After deriving the spectrum of the feedback operator we present the main theorem of this paper which gives a parameter range for which convergence of the CNN dynamical system is assured.<>