  There is a significant expansion in both volume and range of applications
along with the concomitant increase in the variety of data sources. These
ever-expanding trends have highlighted the necessity for more versatile
analysis tools that offer greater opportunities for algorithmic developments
and computationally faster operations than the standard flat-view matrix
approach. Tensors, or multi-way arrays, provide such an algebraic framework
which is naturally suited to data of such large volume, diversity, and
veracity. Indeed, the associated tensor decompositions have demonstrated their
potential in breaking the Curse of Dimensionality associated with traditional
matrix methods, where a necessary exponential increase in data volume leads to
adverse or even intractable consequences on computational complexity. A key
tool underpinning multi-linear manipulation of tensors and tensor networks is
the standard Tensor Contraction Product (TCP). However, depending on the
dimensionality of the underlying tensors, the TCP also comes at the price of
high computational complexity in tensor manipulation. In this work, we resort
to diagrammatic tensor network manipulation to calculate such products in an
efficient and computationally tractable manner, by making use of Tensor Train
decomposition (TTD). This has rendered the underlying concepts easy to
perceive, thereby enhancing intuition of the associated underlying operations,
while preserving mathematical rigour. In addition to bypassing the cumbersome
mathematical multi-linear expressions, the proposed Tensor Train Contraction
Product model is shown to accelerate significantly the underlying computational
operations, as it is independent of tensor order and linear in the tensor
dimension, as opposed to performing the full computations through the standard
approach (exponential in tensor order).
