An Overview of Linked Component Analysis from Matrices to High Order Tensors
This paper presents a detailed exploration of linked component analysis methods, transitioning from matrix-based approaches to high order tensor decompositions, with a particular emphasis on applications within biomedical data analysis. As the availability of diverse sensor technologies increases, researchers face the challenge of processing and interpreting vast amounts of multi-block data, which often comes in multidimensional forms necessitating advanced methods of analysis.
The authors begin by reviewing existing two-way component analysis methods, including well-known techniques such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Sparse Component Analysis (SCA), Nonnegative Matrix Factorization (NMF), and Smooth Component Analysis (SmCA). Each of these methods provides a framework for extracting latent components under specific constraints, such as statistical independence, nonnegativity, or smoothness. These foundational methods offer significant insights but often fall short in the context of multi-block and higher-order data.
Building on the limitations of matrix methods, the paper transitions into the field of tensors, emphasizing the flexibility and power of Tensor Decomposition techniques. The paper discusses both the Tucker and Canonical Polyadic (CP) decomposition models, offering a nuanced examination of how these models provide a basis for linked component analysis in multi-block data scenarios. The CP decomposition's unique capability to function as a multilinear blind source separation model is particularly noted for its applicability to data blocks of uniform size, sharing common components across dimensions.
The paper further extends this exploration into Multiway Blind Source Separation (MBSS), where tensor decompositions allow for constraints on components, resulting in models that better capture the intrinsic properties of the data. Importantly, the uniqueness conditions for tensor decompositions such as CP are elaborated, with implications for applications in EEG, fMRI, and other domains where physiological data are abundant.
In practical terms, the paper highlights several methods for linked analysis of multi-block data, such as joint ICA models and their extensions into tensor space with Independent Vector Analysis (IVA). These models exploit statistical correlations across datasets, providing a more flexible and robust framework for data fusion and analysis than traditional methods that assume identical components across all data blocks.
A core contribution of the paper is the development of Common and Individual Feature Analysis (CIFA), which innovatively separates shared features from unique ones within multi-block datasets. CIFA offers practical insights for extracting both shared and individual components, enhancing the interpretability and utility of models in understanding complex biomedical data structures.
Dimensionality reduction techniques are also examined, with robust methods like RPCA (Robust PCA) and tensor completion strategies discussed as essential tools for handling noisy data and estimating missing values. These methods underscore the importance of low-rank tensor approximations in modern data analysis, particularly in biomedical applications with inherent uncertainty and variability.
Overall, the paper provides both a theoretical framework and practical guidance for researchers interested in employing advanced linked component analysis techniques on multi-dimensional datasets, particularly within the biomedical field. The frameworks discussed are expected to play a crucial role in future developments in machine learning and AI, particularly in their application to large-scale, complex data environments.