Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems

TL;DR

This paper reviews tensor network models, especially Tensor Train decompositions, and demonstrates their effectiveness in compressing and solving large-scale data optimization problems through tensorization and low-rank approximations.

Contribution

It introduces novel mathematical and graphical representations for tensor networks and shows how they enable efficient solutions to big data optimization challenges.

Findings

Tensor train decompositions enable super compression of data.

Tensorization allows creation of high-order tensors from lower-dimensional data.

Tensor networks facilitate solving large-scale optimization problems efficiently.

Abstract

In this paper we review basic and emerging models and associated algorithms for large-scale tensor networks, especially Tensor Train (TT) decompositions using novel mathematical and graphical representations. We discus the concept of tensorization (i.e., creating very high-order tensors from lower-order original data) and super compression of data achieved via quantized tensor train (QTT) networks. The purpose of a tensorization and quantization is to achieve, via low-rank tensor approximations "super" compression, and meaningful, compact representation of structured data. The main objective of this paper is to show how tensor networks can be used to solve a wide class of big data optimization problems (that are far from tractable by classical numerical methods) by applying tensorization and performing all operations using relatively small size matrices and tensors and applying iteratively optimized and approximative tensor contractions. Keywords: Tensor networks, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, tensorization, distributed representation od data optimization problems for very large-scale problems: generalized eigenvalue decomposition (GEVD), PCA/SVD, canonical correlation analysis (CCA).