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).
翻译:应用量和范围都有显著的扩大,同时数据来源种类也随之增加。这些不断扩大的趋势突出表明,需要更加多用途的分析工具,为算法发展和计算速度比标准平面矩阵法更快的操作提供更多机会。Tansors或多路阵列提供了这样的代数框架,这种代数框架自然适合如此大的数量、多样性和真实性的数据。事实上,相关的 Exor 分解表明,它们有可能打破传统矩阵方法所伴随的尺寸曲线,而数据量的必要指数性增长导致计算复杂性的不利甚至棘手后果。一个支持多线性处理高压和高压网络的关键工具是标准的Tensor Contraction 产品(TCP ) 。 然而,根据底部电压、多样性和真实性数据,TCP也以高计算复杂性的价格出现。 在这项工作中,我们采用图表式的色素网络操纵方法,以高效和可计算的方式计算出这类产品,从而导致对计算复杂性产生不利甚至棘手的后果。Tensor decreal comal commall eral or oral oral oral errial adal or or or adlievational as lading as lading (Tegradude) lax the suder) sudududuceal to the subilding to the the the subild to the subilding the subildal decomlifal decommodal decommodal decomlibildal decomlifildaldaltition) sution) vicementaldaldaldaldaldaldaldaldald to vicildaldal vicildaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldal su su。在进行这种分析中, vicaldaldaldaldaldaldaldaldal vicaldaldal 也就是,