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We propose tensorial neural networks (TNNs), a generalization of existingneural networks by extending tensor operations on low order operands to thoseon high order ones. The problem of parameter learning is challenging, as itcorresponds to hierarchical nonlinear tensor decomposition. We propose to solvethe learning problem using stochastic gradient descent by deriving nontrivialbackpropagation rules in generalized tensor algebra we introduce. Our proposedTNNs has three advantages over existing neural networks: (1) TNNs naturallyapply to high order input object and thus preserve the multi-dimensionalstructure in the input, as there is no need to flatten the data. (2) TNNsinterpret designs of existing neural network architectures. (3) Mapping aneural network to TNNs with the same expressive power results in a TNN of fewerparameters. TNN based compression of neural network improves existing low-rankapproximation based compression methods as TNNs exploit two other types ofinvariant structures, periodicity and modulation, in addition to the lowrankness. Experiments on LeNet-5 (MNIST), ResNet-32 (CIFAR10) and ResNet-50(ImageNet) demonstrate that our TNN based compression outperforms (5% testaccuracy improvement universally on CIFAR10) the state-of-the-art low-rankapproximation based compression methods under the same compression rate,besides achieving orders of magnitude faster convergence rates due to theefficiency of TNNs.

Tensorial Neural Networks: Generalization of Neural Networks and Application to Model Compression