Era of Big Data Processing: A New Approach via Tensor Networks and Tensor Decompositions

TL;DR

This paper explores tensor networks and tensor decompositions as innovative methods for efficiently analyzing massive multidimensional data across various scientific and engineering fields.

Contribution

It introduces fundamental tensor network models, their mathematical frameworks, and algorithms for large-scale tensor decompositions, highlighting their applications in big data analytics.

Findings

Tensor networks enable efficient representation of high-dimensional data.

Tensor decompositions facilitate discovery of hidden data structures.

Applications include anomaly detection, classification, and data fusion.

Abstract

Many problems in computational neuroscience, neuroinformatics, pattern/image recognition, signal processing and machine learning generate massive amounts of multidimensional data with multiple aspects and high dimensionality. Tensors (i.e., multi-way arrays) provide often a natural and compact representation for such massive multidimensional data via suitable low-rank approximations. Big data analytics require novel technologies to efficiently process huge datasets within tolerable elapsed times. Such a new emerging technology for multidimensional big data is a multiway analysis via tensor networks (TNs) and tensor decompositions (TDs) which represent tensors by sets of factor (component) matrices and lower-order (core) tensors. Dynamic tensor analysis allows us to discover meaningful hidden structures of complex data and to perform generalizations by capturing multi-linear and multi-aspect relationships. We will discuss some fundamental TN models, their mathematical and graphical descriptions and associated learning algorithms for large-scale TDs and TNs, with many potential applications including: Anomaly detection, feature extraction, classification, cluster analysis, data fusion and integration, pattern recognition, predictive modeling, regression, time series analysis and multiway component analysis. Keywords: Large-scale HOSVD, Tensor decompositions, CPD, Tucker models, Hierarchical Tucker (HT) decomposition, low-rank tensor approximations (LRA), Tensorization/Quantization, tensor train (TT/QTT) - Matrix Product States (MPS), Matrix Product Operator (MPO), DMRG, Strong Kronecker Product (SKP).