Multiple Linked Tensor Factorization

Abstract: In biomedical research and other fields, it is now common to generate high content data that are both multi-source and multi-way. Multi-source data are collected from different high-throughput technologies while multi-way data are collected over multiple dimensions, yielding multiple tensor arrays. Integrative analysis of these data sets is needed, e.g., to capture and synthesize different facets of complex biological systems. However, despite growing interest in multi-source and multi-way factorization techniques, methods that can handle data that are both multi-source and multi-way are limited. In this work, we propose a Multiple Linked Tensors Factorization (MULTIFAC) method extending the CANDECOMP/PARAFAC (CP) decomposition to simultaneously reduce the dimension of multiple multi-way arrays and approximate underlying signal. We first introduce a version of the CP factorization with L2 penalties on the latent factors, leading to rank sparsity. When extended to multiple linked tensors, the method automatically reveals latent components that are shared across data sources or individual to each data source. We also extend the decomposition algorithm to its expectation-maximization (EM) version to handle incomplete data with imputation. Extensive simulation studies are conducted to demonstrate MULTIFAC's ability to (i) approximate underlying signal, (ii) identify shared and unshared structures, and (iii) impute missing data. The approach yields an interpretable decomposition on multi-way multi-omics data for a study on early-life iron deficiency.

Overview of "Multiple Linked Tensor Factorization"

The paper "Multiple Linked Tensor Factorization" by Kang et al. introduces a method dubbed MULTIFAC, designed to address the challenges associated with integrative analysis of multi-source and multi-way data, particularly in biomedical research contexts. The technique extends CANDECOMP/PARAFAC (CP) decomposition with $L_2$ penalties to simultaneously reduce the dimensionality of multiple multi-way arrays and estimate underlying signals. This work is particularly notable for its application in scenarios involving data collected from different high-throughput technologies, as well as its demonstration of efficacy in real-world applications such as the study of early-life iron deficiency.

The paper recognizes the complexity of modern datasets, which are often multi-dimensional (tensors) and sourced from varying technologies (multi-source). MULTIFAC aims to reveal both shared and individual structures within these datasets, a feature that distinguishes it from existing tensor decomposition approaches. To handle missing data, the authors further extend the method using an expectation-maximization (EM) version, thereby providing a comprehensive tool for imputation.

Numerical Results and Claims

The proposed MULTIFAC algorithm was subjected to extensive simulation and applied to real-world data. Some key results indicate its efficacy in approximating underlying signals and imputing missing data. For instance, the simulations show that MULTIFAC consistently achieves lower relative squared errors (RSE) compared to existing methods like tensor decompositions relying on nonlinear least squares (NLS), especially as noise ratios improve.

Moreover, the results also reveal that MULTIFAC can automatically select tensor rank and effectively distinguish shared from individual structures based on penalty terms influencing the singular values. This automatic rank determination is crucial as traditional tensor methods often require pre-specified ranks, which are not always feasible in practice.

Theoretical Implications

From a theoretical perspective, the integration of $L_2$ penalties results in sparse rank structures within the tensor decomposition context. The paper substantiates this claim through several theorems demonstrating that an $L_2$ penalty on factor matrices equates to a sparsity-inducing penalty on the component weights. This property is significant as it implies that the method can efficiently manage and simplify complex datasets by focusing computational resources on the most informative components.

These theoretical contributions not only provide a strong mathematical foundation for the method but also enhance its robustness in various applications. The capacity to handle linked tensors across different modes extends the applicability of tensor factorization to more diverse and interconnected data sets.

Practical Implications and Future Developments

Practically, MULTIFAC demonstrates significant potential for use in biomedical research, particularly in integrative analyses that require simultaneous handling of multiple, high-dimensional datasets. By offering a framework that can seamlessly integrate and analyze data from multiple sources, practitioners can glean more comprehensive insights into complex biological systems and diseases.

Looking forward, future research could explore more efficient parameter tuning techniques, potentially building on Bayesian approaches or theories from random tensor analysis. Additionally, extending the framework to accommodate tensors that share multiple modes would further increase its practicality and scope of applicability.

In conclusion, "Multiple Linked Tensor Factorization" offers an innovative solution for modern data complexities, marrying theoretical rigor with practical applicability. This positions the technique as a prominent tool in the quest to unlock meaningful patterns and insights from high-dimensional multi-source datasets.