Circulant matrices and the stability of a class 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 discuss conditions for a CNN to be contractive, ensuring global asymptotic stability.