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

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).

arXiv admin note: text overlap with arXiv:1403.2048

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• Tensor Analysis and Fusion of Multimodal Brain Images
• Machine Learning meets Quantum Foundations: A Brief Survey
• Fundamental Tensor Operations for Large-Scale Data Analysis in Tensor Train Formats
• A literature survey of matrix methods for data science
• A Multi-Scale Tensor Network Architecture for Classification and Regression
• Efficient Low Rank Tensor Ring Completion
• Numerical Stability of Generalized Entropies
• Sparse Hierarchical Tucker Factorization and its Application to Healthcare