Tensor Decomposition for Signal Processing and Machine Learning

TLDR

Tensors are multi‑way arrays that generalize matrices and have become central to signal processing, statistics, data mining, and machine learning, building on a century of research and requiring only basic applied‑optimization knowledge. This overview article serves as a practical entry point for researchers and practitioners who want to learn about and apply tensor methods. It covers tensor rank, factorization models and identifiability, a range of algorithms from alternating optimization to stochastic gradient, statistical performance analysis, and diverse applications such as source separation, collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning, all presented with sufficient breadth and depth for graduate‑level readers.

Abstract

Tensors or {\em multi-way arrays} are functions of three or more indices $(i,j,k,\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.