Tensor Decomposition for Signal Processing and Machine Learning (1607.01668v2)

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.

• The paper presents tensor decomposition models, including CPD, Tucker, and HOSVD, that enable novel approaches in signal separation and feature extraction.
• It details efficient algorithms such as ALS, SGD, and Levenberg–Marquardt to address scalability and convergence challenges in multi-dimensional data analysis.
• The paper highlights practical applications in signal processing and machine learning, emphasizing improved blind source separation and recommender systems.

Tensor Decomposition for Signal Processing and Machine Learning

The paper by Sidiropoulos et al. presents an extensive overview of tensor decompositions and their applications in signal processing and machine learning. This essay summarizes the key aspects of the paper, emphasizing technical facets, applications, and implications for future research.

Background and Motivation

Tensors, generalizations of matrices to higher dimensions, have gained significant traction in various fields, including signal processing, statistics, data mining, and machine learning. The paper provides a comprehensive introduction to tensor factorization models, algorithms, and their applications, aiming to equip researchers with the foundational knowledge necessary to apply these techniques in practice.

Tensor Decomposition Models

The paper covers several core tensor decomposition models:

1. Canonical Polyadic Decomposition (CPD): Also known as PARAFAC, it represents a tensor as a sum of rank-1 tensors and is unique under mild conditions. This uniqueness makes it particularly useful for blind source separation and other applications requiring interpretable components.
2. Tucker Decomposition: This model generalizes CPD by decomposing a tensor into a core tensor multiplied by factor matrices along each mode. It is primarily used for dimensionality reduction and latent factor models.
3. Higher-Order Singular Value Decomposition (HOSVD): Similar to the SVD in matrices, HOSVD decomposes a tensor into orthogonal components, useful for tasks such as signal and image processing.

Algorithms for Tensor Decomposition

A variety of algorithms for computing tensor decompositions are discussed:

• Alternating Least Squares (ALS): A popular iterative method that alternates updates of each factor matrix while keeping the others fixed. Despite its simplicity, ALS can be computationally intensive.
• Gradient Descent and Stochastic Gradient Descent (SGD): These methods are particularly useful for large-scale tensors, offering a way to handle massive datasets efficiently.
• Nonlinear Least Squares (NLS) and Levenberg–Marquardt: These algorithms aim to improve convergence properties compared to ALS.

Statistical Performance Analysis

The paper explores the statistical performance of tensor decompositions, focusing on the Cramer-Rao bound (CRB) as a measure of estimator variance. The derivation of the Fisher Information Matrix (FIM) for CPD is detailed, demonstrating the complexities involved in estimating tensor components.

Applications in Signal Processing and Machine Learning

The applications described in the paper span numerous domains:

• Signal Processing: Tensor decompositions are applied to blind source separation (e.g., speech and communication signals), multi-dimensional harmonic retrieval, and radar emitter localization.
• Machine Learning: Applications include collaborative filtering for recommender systems, topic modeling, and classification. Tensor methods facilitate the extraction of latent factors and the handling of higher-order data such as user-item-time interactions.

Practical Implementations and Tools

Several software tools are highlighted for implementing tensor decompositions, such as Matlab's Tensor Toolbox and TensorLab. These tools provide a basis for researchers to experiment with tensor methods and apply them to real-world data sets.

Implications and Future Directions

The paper outlines several implications of tensor decomposition methods:

• Identifiability and Uniqueness: The uniqueness properties of CPD under mild conditions make it a powerful tool for blind source separation and latent factor extraction.
• Scalability: Methods such as SGD and parallel implementations are critical for handling large-scale data, a common scenario in modern applications.
• Cross-Disciplinary Applications: Tensor methods have broad applicability across fields, from cheminformatics to social network analysis, underscoring the versatility of these techniques.

Conclusion

The paper provides an in-depth overview of tensor decomposition models, algorithms, and applications, serving as a robust entry point for researchers interested in leveraging these methods for signal processing and machine learning tasks. The strong numerical results and theoretical insights presented indicate the potential of tensor decompositions to address complex, multi-dimensional problems in various domains. Future research is expected to focus on improving scalability, handling constraints effectively, and exploring new applications of tensor methods in emerging fields.