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

TL;DR

This paper introduces tensorial neural networks (TNNs), a novel generalization that preserves multi-dimensional data structures, interprets existing architectures, and enhances neural network compression by exploiting tensor invariants, outperforming current methods.

Contribution

The paper presents TNNs as a new neural network framework that generalizes existing models, enabling better data structure preservation and more efficient compression through tensor algebra.

Findings

TNNs outperform low-rank approximation methods in neural network compression.

TNN-based compression improves test accuracy by 5% on CIFAR10.

TNNs achieve faster convergence rates in training.

Abstract

We propose tensorial neural networks (TNNs), a generalization of existing neural networks by extending tensor operations on low order operands to those on high order ones. The problem of parameter learning is challenging, as it corresponds to hierarchical nonlinear tensor decomposition. We propose to solve the learning problem using stochastic gradient descent by deriving nontrivial backpropagation rules in generalized tensor algebra we introduce. Our proposed TNNs has three advantages over existing neural networks: (1) TNNs naturally apply to high order input object and thus preserve the multi-dimensional structure in the input, as there is no need to flatten the data. (2) TNNs interpret designs of existing neural network architectures. (3) Mapping a neural network to TNNs with the same expressive power results in a TNN of fewer parameters. TNN based compression of neural network improves existing low-rank approximation based compression methods as TNNs exploit two other types of invariant structures, periodicity and modulation, in addition to the low rankness. Experiments on LeNet-5 (MNIST), ResNet-32 (CIFAR10) and ResNet-50 (ImageNet) demonstrate that our TNN based compression outperforms (5% test accuracy improvement universally on CIFAR10) the state-of-the-art low-rank approximation based compression methods under the same compression rate, besides achieving orders of magnitude faster convergence rates due to the efficiency of TNNs.