Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions

Modern applications in engineering and data science are increasinglybased on multidimensional data of exceedingly high volume, variety,and structural richness. However, standard machine learning algorithmstypically scale exponentially with data volume and complexityof cross-modal couplings - the so called curse of dimensionality -which is prohibitive to the analysis of large-scale, multi-modal andmulti-relational datasets. Given that such data are often efficientlyrepresented as multiway arrays or tensors, it is therefore timely andvaluable for the multidisciplinary machine learning and data analyticcommunities to review low-rank tensor decompositions and tensor networksas emerging tools for dimensionality reduction and large scaleoptimization problems. Our particular emphasis is on elucidating that,by virtue of the underlying low-rank approximations, tensor networkshave the ability to alleviate the curse of dimensionality in a numberof applied areas. In Part 1 of this monograph we provide innovativesolutions to low-rank tensor network decompositions and easy to interpretgraphical representations of the mathematical operations ontensor networks. Such a conceptual insight allows for seamless migrationof ideas from the flat-view matrices to tensor network operationsand vice versa, and provides a platform for further developments, practicalapplications, and non-Euclidean extensions. It also permits theintroduction of various tensor network operations without an explicitnotion of mathematical expressions, which may be beneficial for manyresearch communities that do not directly rely on multilinear algebra.Our focus is on the Tucker and tensor train TT decompositions andtheir extensions, and on demonstrating the ability of tensor networksto provide linearly or even super-linearly e.g., logarithmically scalablesolutions, as illustrated in detail in Part 2 of this monograph.