Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis (1403.4462v1)

Abstract: The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under verymild and natural conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker decomposition, HOSVD, tensor networks, Tensor Train.

• The paper demonstrates how tensor decompositions overcome matrix limitations to capture intricate multi-dimensional patterns in signal processing.
• It details tensor models like CPD and Tucker, offering robust methods for unique and interpretable component analysis.
• It discusses computational strategies, including ALS and compressed sensing, to efficiently manage large-scale multiway data.

Tensor Decompositions for Signal Processing Applications: From Two-Way to Multiway Component Analysis

The paper authored by A. Cichocki et al. explores the pivotal role of tensor decompositions in the field of signal processing. With the growing complexity of data driven by multi-sensor technology and extensive datasets, conventional matrix models fall short in adequately capturing the inherent multi-dimensional patterns present in such data. This paper argues for the transition to tensor-based approaches, highlighting their versatility and robustness in representing complex data structures.

Key Concepts and Historical Background

The manuscript revisits the historical evolution of tensor decompositions, starting from the 19th-century studies on homogeneous polynomials by mathematicians like Gauss and Hilbert, to the application of tensors in modern scientific computing and signal processing. Noteworthy milestones include the introduction of the Tucker decomposition in psychometrics and the Canonical Polyadic Decomposition (CPD) in various fields like chemometrics and linguistics. These foundational works underscore the mathematical depth and practical applications of multiway arrays, setting the stage for current tensor-based methodologies.

From Matrix to Tensor Representations

Traditional matrix-based techniques, such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), and Nonnegative Matrix Factorization (NMF), offer powerful tools for two-way data analysis. However, when applied to multi-dimensional datasets, these methods often require an initial flattening of the data, which can obscure essential multiway interactions. Tensor decompositions, in contrast, retain the inherent multi-dimensional nature of data, allowing for more sophisticated modeling and analysis.

Tensor Decomposition Models

Canonical Polyadic Decomposition (CPD)

CPD represents multi-dimensional data as a sum of rank-1 tensors, ensuring a minimum rank through its canonical form. This enables unique decompositions under mild conditions, thus enhancing interpretability and robustness. The CPD can be computed via algorithms like Alternating Least Squares (ALS), which, despite their simplicity, may require improvements in convergence and handling ill-conditioned cases.

Tucker Decomposition

Tucker decomposition generalizes PCA to multi-dimensional data, modeling it as a multilinear transformation of a core tensor with factor matrices. This approach is particularly valuable for identifying subspace structures and performing dimensionality reduction. Unlike CPD, Tucker decompositions are not inherently unique but can be constrained by properties such as orthogonality, statistical independence, or sparsity to derive meaningful components.

Applications and Computational Aspects

Applications in Signal Processing

Tensor decompositions have been successfully applied in various signal processing domains, including sensor array processing, machine learning, and biomedical engineering. Case studies in the paper illustrate the efficacy of tensor methods in resolving complex signal interferences, enhancing feature extraction, and facilitating blind source separation.

Computational Efficiency

Given their higher-dimensional nature, tensor decompositions face significant computational challenges. The paper discusses strategies like tensor factorization via block processing, compressed sensing, and hierarchical tensor formats (e.g., Tensor Train (TT) decomposition) to efficiently handle large-scale datasets. These methods leverage the structure of tensor data to significantly reduce computational and storage requirements.

Future Directions

The paper sets the stage for several future research directions:

• Coupled Tensor Decompositions: Integrating multiple datasets to identify shared and individual components requires robust theoretical frameworks and efficient algorithms.
• Advanced Algorithms: Developing iterative algorithms beyond ALS to ensure more reliable and faster convergence.
• Component Estimation: Automating the determination of the number of components and their dimensions, especially in the presence of noise.
• Handling Large-Scale Data: Innovating methods to process and store ultra-large-scale tensors, potentially through quantized tensor networks.
• Probabilistic Tensor Models: Incorporating prior knowledge and probabilistic frameworks to enhance the modeling accuracy of complex data interactions.

Conclusion

The paper by Cichocki et al. underscores the transformative potential of tensor decompositions in addressing the limitations of traditional matrix-based methods for multiway data analysis. By leveraging the structural richness of tensors, the proposed techniques offer a robust and versatile framework for modern signal processing applications, paving the way for more accurate and insightful analysis of high-dimensional datasets. As research in this domain progresses, it promises to unlock new capabilities and applications in various scientific and engineering fields.