Tensor factorizations have been widely used for the task of uncovering patterns in various domains. Often, the input is time-evolving, shifting the goal to tracking the evolution of underlying patterns instead. To adapt to this more complex setting, existing methods incorporate temporal regularization but they either have overly constrained structural requirements or lack uniqueness which is crucial for interpretation. In this paper, in order to capture the underlying evolving patterns, we introduce t(emporal)PARAFAC2 which utilizes temporal smoothness regularization on the evolving factors. We propose an algorithmic framework that employs Alternating Optimization (AO) and the Alternating Direction Method of Multipliers (ADMM) to fit the model. Furthermore, we extend the algorithmic framework to the case of partially observed data. Our numerical experiments on both simulated and real datasets demonstrate the effectiveness of the temporal smoothness regularization, in particular, in the case of data with missing entries. We also provide an extensive comparison of different approaches for handling missing data within the proposed framework.

The PARAFAC2 model provides a flexible alternative to the popular CANDECOMP/PARAFAC (CP) model for tensor decompositions. Unlike CP, PARAFAC2 allows factor matrices in one mode (i.e., evolving mode) to change across tensor slices, which has proven useful for applications in different domains such as chemometrics, and neuroscience. However, the evolving mode of the PARAFAC2 model is traditionally modelled implicitly, which makes it challenging to regularise it. Currently, the only way to apply regularisation on that mode is with a flexible coupling approach, which finds the solution through regularised least-squares subproblems. In this work, we instead propose an alternating direction method of multipliers (ADMM)-based algorithm for fitting PARAFAC2 and widen the possible regularisation penalties to any proximable function. Our numerical experiments demonstrate that the proposed ADMM-based approach for PARAFAC2 can accurately recover the underlying components from simulated data while being both computationally efficient and flexible in terms of imposing constraints.

In many areas of science, various sensing technologies are used to obtain information about a single system of interest. Often, none of the datasets alone contains a complete view of the system, but the data measured from different modalities can complement each other. For instance, brain activity patterns can be captured using both electroencephalography (EEG) and functional magnetic resonance imaging (fMRI) signals, which have complementary temporal and spatial resolutions. Similarly, in metabolomics, multiple analytical techniques such as LCMS (Liquid Chromatography - Mass Spectrometry) and NMR (Nuclear Magnetic Resonance) spectroscopy are used to measure chemical compounds in biological samples, providing a more complete picture of underlying biological processes. Joint analysis of datasets from multiple sources, also referred to as data fusion (or multi-modal data mining), exploits these complementary measurements, and allows for better interpretability and, potentially, more accurate recovery of patterns characterizing the underlying phenomena. Nevertheless, data fusion poses many challenges, and there is an emerging need for data fusion methods that can take into account different characteristics of data from multiple sources in many disciplines [1-4]. Data from multiple sources can often be represented in the form of matrices and higher-order tensors. Coupled matrix and tensor factorizations (CMTF) are an effective approach for joint analysis of such datasets in many domains including social network analysis [5-8], neuroscience [9-13], and chemometrics [2, 14]. In such coupled factorizations, each dataset is modelled by a low-rank approximation.