Tensor Decompositions.pdf

Tensor decompositions enable a fundamental paradigm shift in data analysis, moving from standard flat-view matrix models to more versatile and polynomial models with guaranteed uniqueness under mild conditions. This is achieved through the power of multilinear algebra, allowing for flexible choice of constraints and discovery of more general latent components in data.

Key aspects of tensor decompositions include:
- Canonical Polyadic Decomposition (CPD): A decomposition method that represents a tensor as a sum of rank-1 tensors.
- Tucker Decomposition: A method that represents a tensor as a core tensor multiplied by factor matrices.
- Multiway Component Analysis: An approach that extends traditional two-way component analysis to multiway data, enabling the capture of multiple interactions and couplings.

Tensor decompositions have various applications, including:
- Signal Processing: Enabling natural generalizations of signal processing paradigms such as canonical correlation and subspace techniques.
- Data Analysis: Providing efficient and promising tools for modern data analysis and machine learning applications.
- Machine Learning: Extending benefits to vector/matrix data through tensorization.

Historically, the roots of multiway analysis date back to the 19th century, with contributors including Gauss, Kronecker, and Hilbert. The Tucker decomposition and CPD were introduced in psychometrics and linguistics, respectively, and have since been adopted in diverse branches of data analysis.

Notations and conventions in tensor analysis include:
- Tensor Order: The number of modes or dimensions in a tensor.
- Tensor Entries: Denoted by ai1,i2,...,iN for a real-valued tensor of order N.
- Subtensors: Parts of the original data tensor, created by fixing a subset of indices.
- Fibers and Slices: Vector-valued and matrix-valued subtensors, respectively.

Interpretable components in two-way data analysis aim to decompose a data matrix into factor matrices, representing unknown source signals or latent variables. This is achieved through methods such as blind source separation, factor analysis, and latent variable analysis.