Recent results from linear algebra stating that any matrix can be decomposed into products of diagonal and circulant matrices has lead to the design of compact deep neural network architectures that perform well in practice. In this paper, we bridge the gap between these good empirical results 
 and the theoretical approximation capabilities of Deep diagonal-circulant ReLU networks. More precisely, we first demonstrate  that a Deep diagonal-circulant ReLU networks of
 bounded width and small depth can approximate a deep ReLU network in which the dense matrices are
 of low rank. Based on this result, we provide new bounds on the expressive power and universal approximativeness of this type of networks. We support our experimental results with thorough experiments on a large, real world video classification problem.
We provide a theoretical study of the properties of Deep circulant-diagonal ReLU Networks and demonstrate that they are bounded width universal approximators. The paper proposes using circulant and diagonal matrices to speed up computation and reduce memory requirements in eural networks. This paper proves that bounded width diagonal-circulent ReLU networks (DC-ReLU) are universal approximators.