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Modern ConvNets continue to achieve state-of-the-art results over a vastarray of vision and image classification tasks, but at the cost of increasingparameters. One strategy for compactifying a network without sacrificing muchexpressive power is to reshape it into a tensorial neural network (TNN), whichis a higher-order tensorization of its layers, followed by a factorization,such as a CP-decomposition, which strips a weight down to its critical basiscomponents. Passes through TNNs can be represented as sequences of multilinearoperations (MLOs), where the evaluation path can greatly affect the number offloating point operations (FLOPs) incurred. While functions such as the populareinsum can evaluate simple MLOs such as contractions, existing implementationscannot process multi-way convolutions, resulting in scant assessments of howoptimal evaluation paths through tensorized convolutional layers can improvetraining speed. In this paper, we develop a unifying framework for representingtensorial convolution layers as einsum-like strings and a meta-algorithmconv_einsum which is able to evaluate these strings in a FLOPs-minimizingmanner. Comprehensive experiments, using our open-source implementation, over awide range of models, tensor decompositions, and diverse tasks, demonstratethat conv_einsum significantly increases both computational andmemory-efficiency of convolutional TNNs.

conv_einsum: A Framework for Representation and Fast Evaluation of Multilinear Operations in Convolutional Tensorial Neural Networks