Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges

Arnab Auddy, Dong Xia, Ming Yuan

TL;DR

This survey outlines how tensor methods unlock analysis of high-dimensional, multiway data across fields, highlighting the interpretability, identifiability, and robust inference benefits of preserving multilinear structure. It surveys core tools—CP and Tucker decompositions, plus algorithms like alternating minimization, HOOI, tensor power iterations, and gradient methods—and discusses strategies to address nonconvexity via spectral initialization and convex relaxations. The eight statistical settings (tensor SVD, multiway PCA, ICA, mixtures, tensor completion, tensor regression, higher-order networks, and tensor time series) reveal a pervasive theme: a tradeoff between statistical optimality and computational feasibility, with clear computational-statistical gaps in several problems. The paper emphasizes a cross-disciplinary mix of statistics, optimization, and numerical linear algebra, and highlights practical implications for applications in science and engineering where tensor-structured data are prevalent.

Abstract

Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization and numerical linear algebra among other fields. Despite these hurdles, significant progress has been made in the last decade. This review seeks to examine some of the key advancements and identify common threads among them, under eight different statistical settings.

Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges

TL;DR

This survey outlines how tensor methods unlock analysis of high-dimensional, multiway data across fields, highlighting the interpretability, identifiability, and robust inference benefits of preserving multilinear structure. It surveys core tools—CP and Tucker decompositions, plus algorithms like alternating minimization, HOOI, tensor power iterations, and gradient methods—and discusses strategies to address nonconvexity via spectral initialization and convex relaxations. The eight statistical settings (tensor SVD, multiway PCA, ICA, mixtures, tensor completion, tensor regression, higher-order networks, and tensor time series) reveal a pervasive theme: a tradeoff between statistical optimality and computational feasibility, with clear computational-statistical gaps in several problems. The paper emphasizes a cross-disciplinary mix of statistics, optimization, and numerical linear algebra, and highlights practical implications for applications in science and engineering where tensor-structured data are prevalent.

Abstract

Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity of such data provides vast new opportunities for modeling and analysis, but efficiently extracting information content from them, both statistically and computationally, presents unique and fundamental challenges. Addressing these challenges requires an interdisciplinary approach that brings together tools and insights from statistics, optimization and numerical linear algebra among other fields. Despite these hurdles, significant progress has been made in the last decade. This review seeks to examine some of the key advancements and identify common threads among them, under eight different statistical settings.
Paper Structure (45 sections, 61 equations, 4 figures, 2 algorithms)

This paper contains 45 sections, 61 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Why Tensor Methods
  3. Conceptual Benefit -- Interpretability
  4. Benefit for Modeling -- Identifiability
  5. Benefit for Inferences -- Restricted Effect of Perturbation
  6. Challenges of Using Tensor Methods
  7. Organization
  8. Background and Tools
  9. Notation
  10. Matricization
  11. Outer Product
  12. Multiplication
  13. Norms
  14. Tensor Decompositions
  15. CP Decomposition
  16. ...and 30 more sections

Figures (4)

  • Figure 1: Loadings of the first principal components for different brain regions: the darkness of the dots represent the magnitude of the loadings. Reproduced from liu2022characterizing.
  • Figure 2: ICA vs PCA: $\kappa_4(\mathbf{u}^\top \mathbf{X})=\mathbb{E}(\mathbf{u}^\top \mathbf{X})^4-3$ and $\texttt{cov}(\mathbf{u}^\top \mathbf{X})$ as functions of $\mathbf{u}$ over the unit circle.
  • Figure 3: Effect of eigengap on PCA and ICA.
  • Figure 4: Statistical and computational tradeoff for tensor SVD: horizontal axis represents the moment condition for noise, vertical axis corresponds to the signal strength. Reproduced and modified from auddy2021estimating.