Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
For researchers in machine learning and data mining, this is a survey paper that consolidates existing knowledge on tensor networks, but it is incremental as it does not introduce new methods or results.
This paper reviews tensor decompositions and tensor networks, focusing on Tucker and Tensor Train (TT) decompositions, as tools for large-scale data analysis and dimensionality reduction. It provides mathematical and graphical representations and discusses applications in optimization problems.
Machine learning and data mining algorithms are becoming increasingly important in analyzing large volume, multi-relational and multi--modal datasets, which are often conveniently represented as multiway arrays or tensors. It is therefore timely and valuable for the multidisciplinary research community to review tensor decompositions and tensor networks as emerging tools for large-scale data analysis and data mining. We provide the mathematical and graphical representations and interpretation of tensor networks, with the main focus on the Tucker and Tensor Train (TT) decompositions and their extensions or generalizations. Keywords: Tensor networks, Function-related tensors, CP decomposition, Tucker models, tensor train (TT) decompositions, matrix product states (MPS), matrix product operators (MPO), basic tensor operations, multiway component analysis, multilinear blind source separation, tensor completion, linear/multilinear dimensionality reduction, large-scale optimization problems, symmetric eigenvalue decomposition (EVD), PCA/SVD, huge systems of linear equations, pseudo-inverse of very large matrices, Lasso and Canonical Correlation Analysis (CCA) (This is Part 1)