PARAFAC2-BASED COUPLED MATRIX AND TENSOR FACTORIZATIONS Carla Schenker12yXiulin Wang341yEvrim Acar1 1Simula Metropolitan Center for Digital Engineering Oslo Norway2Oslo Metropolitan University Oslo Norway

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PARAFAC2-BASED COUPLED MATRIX AND TENSOR FACTORIZATIONS

Carla Schenker1,2,†Xiulin Wang3,4,1†Evrim Acar1

1Simula Metropolitan Center for Digital Engineering, Oslo, Norway, 2Oslo Metropolitan University, Oslo, Norway

3Department of Radiology, Afﬁliated Zhongshan Hospital of Dalian University, Dalian, China

4School of Biomedical Engineering, Dalian University of Technology, Dalian, China

ABSTRACT

Coupled matrix and tensor factorizations (CMTF) have

emerged as an effective data fusion tool to jointly analyze

data sets in the form of matrices and higher-order tensors.

The PARAFAC2 model has shown to be a promising alter-

native to the CANDECOMP/PARAFAC (CP) tensor model

due to its ﬂexibility and capability to handle irregular/ragged

tensors. While fusion models based on a PARAFAC2 model

coupled with matrix/tensor decompositions have been re-

cently studied, they are limited in terms of possible regu-

larizations and/or types of coupling between data sets. In

this paper, we propose an algorithmic framework for ﬁtting

PARAFAC2-based CMTF models with the possibility of im-

posing various constraints on all modes and linear couplings,

using Alternating Optimization (AO) and the Alternating Di-

rection Method of Multipliers (ADMM). Through numerical

experiments, we demonstrate that the proposed algorithmic

approach accurately recovers the underlying patterns using

various constraints and linear couplings.

Index Terms—data fusion, PARAFAC2, coupled matrix

and tensor factorizations, AO-ADMM

1. INTRODUCTION

Joint analysis of heterogeneous data from multiple sources

has the potential to capture complementary information and

reveal underlying patterns of interest. Coupled matrix and

tensor factorizations (CMTF) are an effective approach to

jointly analyze such data in the form of matrices and tensors

in various ﬁelds, e.g., social network analysis [1, 2, 3], neu-

roscience [4, 5, 6], bioinformatics [7] and remote sensing [8].

CMTF models approximate each dataset using a low-rank

model, where some factors/patterns are shared between data

sets. Couplings with (linear) transformations have proven

useful in many applications, e.g., accounting for different

spatial, temporal or spectral relations between datasets [4, 8],

or modeling partially shared components [4, 5]. For analyz-

ing higher-order tensors, CMTF methods often rely on the

CANDECOMP/PARAFAC (CP) model [9, 10], which ap-

proximates the tensor as a sum of rank-one tensors. However,

†These authors contributed equally to the work.

the CP model has strict multilinearity assumptions and can-

not handle irregular tensors. The PARAFAC2 model [11, 12]

relaxes the CP model by allowing one factor matrix to vary

across tensor slices and enables the decomposition of irreg-

ular tensors. PARAFAC2 has shown to be advantageous in

chromatographic data analysis (with unaligned proﬁles) [13],

temporal phenotyping (with unaligned clinical visits) [14],

and tracing evolving patterns [15].

Recent CMTF studies have incorporated the PARAFAC2

model. For instance, Afshar et al. [16] use a non-negative

PARAFAC2 model coupled with a non-negative matrix fac-

torization to jointly analyze electronic health records and

patient demographic data. In [4], linearly coupled tensor

decompositions are used to jointly analyze neuroimaging

signals from different modalities, where PARAFAC2 is used

to cope with subject variability. However, previous studies

have been limited in terms of constraints on the factors and/or

different types of couplings between data sets. Usually, the

factor matrix of the varying mode in PARAFAC2 is estimated

implicitly [4, 12], which makes it challenging to impose con-

straints. The TASTE framework [16], therefore, adapts a

ﬂexible PARAFAC2 constraint [17], which allows for non-

negativity constraints on the varying mode. TASTE has also

been generalized to the coupling of a PARAFAC2 model with

a CP model together with different options for solving the

(non-negative) least-squares sub-problems [18]. Still, this

framework is limited to the unconstrained and non-negative

case, and does not support partial- or other linear couplings.

In this paper, we introduce an AO-ADMM-based al-

gorithmic approach for CMTF models incorporating the

PARAFAC2 model referred to as PARAFAC2-based CMTF.

The framework accommodates linear couplings with (multi-

ple) matrix- or CP-decompositions (Fig. 1 and 2), and a vari-

ety of possible constraints and regularizations on all modes.

Our algorithmic approach builds onto the AO-ADMM algo-

rithm [19] for constrained PARAFAC2, which allows for any

proximal constraint in any mode, and the ﬂexible framework

for CP-based CMTF [20]. Using numerical experiments,

we demonstrate the ﬂexibility and accuracy of the proposed

approach with different constraints and linear couplings. Fur-

thermore, we show the promise of PARAFAC2-based CMTF

models in terms of jointly analyzing dynamic and static data.

arXiv:2210.13054v1 [cs.LG] 24 Oct 2022

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PARAFAC2-BASEDCOUPLEDMATRIXANDTENSORFACTORIZATIONSCarlaSchenker1;2;yXiulinWang3;4;1yEvrimAcar11SimulaMetropolitanCenterforDigitalEngineering,Oslo,Norway,2OsloMetropolitanUniversity,Oslo,Norway3DepartmentofRadiology,AfliatedZhongshanHospitalofDalianUniversity,Dalian,China4SchoolofBiomedicalEngineeri...

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PARAFAC2-BASED COUPLED MATRIX AND TENSOR FACTORIZATIONS Carla Schenker12yXiulin Wang341yEvrim Acar1 1Simula Metropolitan Center for Digital Engineering Oslo Norway2Oslo Metropolitan University Oslo Norway

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