Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives

TL;DR

This paper explores tensor network models like TT and HT for data compression and large-scale optimization, highlighting their applications in machine learning, data analytics, and neural network optimization, emphasizing their scalability and distributed computation capabilities.

Contribution

It provides a detailed overview of tensor network models, their physical interpretations, and their application in scalable machine learning and data analysis tasks, extending prior foundational work.

Findings

Tensor networks enable distributed computations on large data sets.

Low-rank tensor approximations alleviate the curse of dimensionality.

Applications include regression, classification, eigenvalue problems, and neural network optimization.

Abstract

Part 2 of this monograph builds on the introduction to tensor networks and their operations presented in Part 1. It focuses on tensor network models for super-compressed higher-order representation of data/parameters and related cost functions, while providing an outline of their applications in machine learning and data analytics. A particular emphasis is on the tensor train (TT) and Hierarchical Tucker (HT) decompositions, and their physically meaningful interpretations which reflect the scalability of the tensor network approach. Through a graphical approach, we also elucidate how, by virtue of the underlying low-rank tensor approximations and sophisticated contractions of core tensors, tensor networks have the ability to perform distributed computations on otherwise prohibitively large volumes of data/parameters, thereby alleviating or even eliminating the curse of dimensionality. The usefulness of this concept is illustrated over a number of applied areas, including generalized regression and classification (support tensor machines, canonical correlation analysis, higher order partial least squares), generalized eigenvalue decomposition, Riemannian optimization, and in the optimization of deep neural networks. Part 1 and Part 2 of this work can be used either as stand-alone separate texts, or indeed as a conjoint comprehensive review of the exciting field of low-rank tensor networks and tensor decompositions.