Multiple Linked Tensor Factorization

TL;DR

This paper introduces MULTIFAC, a novel tensor factorization method that extends CP decomposition to handle multiple linked multi-way data sources, enabling shared component discovery, missing data imputation, and interpretability in complex biological datasets.

Contribution

The paper proposes a new linked tensor factorization method, MULTIFAC, capable of analyzing multi-source multi-way data with shared and individual components, including an EM algorithm for incomplete data.

Findings

Successfully approximates underlying signals in simulated data

Identifies shared and source-specific latent components

Imputes missing data effectively

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.