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There is a significant expansion in both volume and range of applicationsalong with the concomitant increase in the variety of data sources. Theseever-expanding trends have highlighted the necessity for more versatileanalysis tools that offer greater opportunities for algorithmic developmentsand computationally faster operations than the standard flat-view matrixapproach. Tensors, or multi-way arrays, provide such an algebraic frameworkwhich is naturally suited to data of such large volume, diversity, andveracity. Indeed, the associated tensor decompositions have demonstrated theirpotential in breaking the Curse of Dimensionality associated with traditionalmatrix methods, where a necessary exponential increase in data volume leads toadverse or even intractable consequences on computational complexity. A keytool underpinning multi-linear manipulation of tensors and tensor networks isthe standard Tensor Contraction Product (TCP). However, depending on thedimensionality of the underlying tensors, the TCP also comes at the price ofhigh computational complexity in tensor manipulation. In this work, we resortto diagrammatic tensor network manipulation to calculate such products in anefficient and computationally tractable manner, by making use of Tensor Traindecomposition (TTD). This has rendered the underlying concepts easy toperceive, thereby enhancing intuition of the associated underlying operations,while preserving mathematical rigour. In addition to bypassing the cumbersomemathematical multi-linear expressions, the proposed Tensor Train ContractionProduct model is shown to accelerate significantly the underlying computationaloperations, as it is independent of tensor order and linear in the tensordimension, as opposed to performing the full computations through the standardapproach (exponential in tensor order).

Reducing Computational Complexity of Tensor Contractions via Tensor-Train Networks