Convolution neural networks typically consist of many convolutional layers followed by several fully-connected layers.   While convolutional layers map between high-order activation tensors, the fully-connected layers operate on flattened activation vectors.   Despite its success, this approach has notable drawbacks. Flattening discards the multi-dimensional structure of the activations, and the fully-connected layers require a large number of parameters. 
 We present two new techniques to address these problems.   First, we introduce tensor contraction layers which can replace the ordinary fully-connected layers in a neural network. Second, we introduce tensor regression layers, which express the output of a neural network as a low-rank multi-linear mapping from a high-order activation tensor to the softmax layer.   Both the contraction and regression weights are learned end-to-end by backpropagation. By imposing low rank on both, we use significantly fewer parameters.   Experiments on the ImageNet dataset show that applied to the popular VGG and ResNet architectures, our methods significantly reduce the number of parameters in the fully connected layers (about 65% space savings) while negligibly impacting accuracy.
We propose tensor contraction and low-rank tensor regression layers to preserve and leverage the multi-linear structure throughout the network, resulting in huge space savings with little to no impact on performance. This paper proposes new layer architectures of neural networks using a low-rank representation of tensors This paper incorporates tensor decomposition and tensor regression into CNN by using a new tensor regression layer.