Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> The model for three-mode factor analysis is discussed in terms of newer applications of mathematical processes including a type of matrix process termed the Kronecker product and the definition of combination variables. Three methods of analysis to a type of extension of principal components analysis are discussed. Methods II and III are applicable to analysis of data collected for a large sample of individuals. An extension of the model is described in which allowance is made for unique variance for each combination variable when the data are collected for a large sample of individuals. <s> BIB001 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Given several Gramian matrices, a least-square fit to all the matrices by one factor matrix, with a predetermined number of factors, is shown to be the principal axes solution of the average of the matrices. <s> BIB002 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Simple structure and other common principles of factor rotation do not in general provide strong grounds for attributing explanatory significance to the factors which they select. In contrast, it is shown that an extension of Cattell's principle of rotation to Proportional Profiles (PP) offers a basis for determining explanatory factors for three-way or higher order multi-mode data. Conceptual models are developed for two basic patterns of multi-mode data variation, systemand object-variation, and PP analysis is found to apply in the system-variation case. Although PP was originally formulated as a principle of rotation to be used with classic two-way factor analysis, it is shown to embody a latent three-mode factor model, which is here made explicit and generalized frown two to N "parallel occasions". As originally formulated, PP rotation was restricted to orthogonal factors. The generalized PP model is demonstrated to give unique "correct" solutions with oblique, non-simple structure, and even non-linear factor structures. A series of tests, conducted with synthetic data of known factor composition, demonstrate the capabilities of linear and non-linear versions of the model, provide data on the minimal necessary conditions of uniqueness, and reveal the properties of the analysis procedures when these minimal conditions are not fulfilled. In addition, a mathematical proof is presented for the uniqueness of the solution given certain conditions on the data. Three-mode PP factor analysis is applied to a three-way set of real data consisting of the fundamental and first three formant frequencies of 11 persons saying 8 vowels. A unique solution is extracted, consisting of three factors which are highly meaningful and consistent with prior knowledge and theory concerning vowel quality. The relationships between the three-mode PP model and Tucker's multi-modal model, McDonald's non-linear model and Carroll and Chang's multi-dimensional scaling model are explored. <s> BIB003 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations. <s> BIB004 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the “triple product” [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars—under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotatability characteristic can hold even when the number of factors extracted is greater thanevery dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments. <s> BIB005 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Abstract A three-way array X (or three-dimensional matrix) is an array of numbers xijk subscripted by three indices. A triad is a multiplicative array, xijk = aibjck. Analogous to the rank and the row rank of a matrix, we define rank (X) to be the minimum number of triads whose sum is X, and dim1(X) to be the dimensionality of the space of matrices generated by the 1-slabs of X. (Rank and dim1 may not be equal.) We prove several lower bounds on rank. For example, a special case of Theorem 1 is that rank(X)⩾dim 1 (UX) + rank(XW) − dim 1 (UXW) , where U and W are matrices; this generalizes a matrix theorem of Frobenius. We define the triple product [A, B, C] of three matrices to be the three-way array whose (i, j, k) element is given by ⩞rairbjrckr; in other words, the triple product is the sum of triads formed from the columns of A, B, and C. We prove several sufficient conditions for the factors of a triple product to be essentially unique. For example (see Theorem 4a), suppose [A, B, C] = [ A , B , C ] , and each of the matrices has R columns. Suppose every set of rank (A) columns of A are independent, and similar conditions hold for B and C. Suppose rank (A) + rank (B) + rank (C) ⩾ 2R + 2. Then there exist diagonal matrices Λ, M, N and a permutation matrix P such that A = APΛ, B = BP M , C = CP N . Our results have applications to arithmetic complexity theory and to statistical models used in three-way multidimensional scaling. <s> BIB006 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints. <s> BIB007 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Abstract The independent component analysis (ICA) of a random vector consists of searching for a linear transformation that minimizes the statistical dependence between its components. In order to define suitable search criteria, the expansion of mutual information is utilized as a function of cumulants of increasing orders. An efficient algorithm is proposed, which allows the computation of the ICA of a data matrix within a polynomial time. The concept of ICA may actually be seen as an extension of the principal component analysis (PCA), which can only impose independence up to the second order and, consequently, defines directions that are orthogonal. Potential applications of ICA include data analysis and compression, Bayesian detection, localization of sources, and blind identification and deconvolution. <s> BIB008 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> The paper considers an application of blind identification to beamforming. The key point is to use estimates of directional vectors rather than resort to their hypothesised value. By using estimates of the directional vectors obtained via blind identification, i.e. without knowing the array manifold, beamforming is made robust with respect to array deformations, distortion of the wave front, pointing errors etc., so that neither array calibration nor physical modelling is necessary. Rather suprisingly, ‘blind beamformers’ may outperform ‘informed beamformers’ in a plausible range of parameters, even when the array is perfectly known to the informed beamformer. The key assumption on which blind identification relies is the statistical independence of the sources, which is exploited using fourth-order cumulants. A computationally efficient technique is presented for the blind estimation of directional vectors, based on joint diagonalisation of fourth-order cumulant matrices; its implementation is described, and its performance is investigated by numerical experiments. <s> BIB009 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis (xij = ΣRr = 1airbjr) to a trilinear model (xijk = ΣRr = 1airbjrckr). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the method's robustness against nonstationary factor structures and qualitative (nonproportional) factor change. <s> BIB010 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Blind separation of communication signals invariably relies on some form(s) of diversity to overdetermine the problem and thereby recover the signals of interest. More often than not, linear (e.g., spreading) diversity is employed, i.e., each diversity branch provides a linear combination of the unknown signals, albeit with possibly unknown weights. If multiple forms of linear diversity are simultaneously available, then the resulting data exhibit multilinear structure, and the blind recovery problem can be shown to be tantamount to low-rank decomposition of the multi-dimensional received data array. This paper generalizes Kruskal's fundamental result on the uniqueness of low-rank decomposition of 3-way arrays to the case of multilinear decomposition of 4- and higher-way arrays. The result characterizes diversity combining for blind identifiability when N forms of linear diversity are available; that is the balance between different forms of diversity that guarantees blind recovery of all signals involved. <s> BIB011 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> This paper links the direct-sequence code-division multiple access (DS-CDMA) multiuser separation-equalization-detection problem to the parallel factor (PARAFAC) model, which is an analysis tool rooted in psychometrics and chemometrics. Exploiting this link, it derives a deterministic blind PARAFAC DS-CDMA receiver with performance close to non-blind minimum mean-squared error (MMSE). The proposed PARAFAC receiver capitalizes on code, spatial, and temporal diversity-combining, thereby supporting small sample sizes, more users than sensors, and/or less spreading than users. Interestingly, PARAFAC does not require knowledge of spreading codes, the specifics of multipath (interchip interference), DOA-calibration information, finite alphabet/constant modulus, or statistical independence/whiteness to recover the information-bearing signals. Instead, PARAFAC relies on a fundamental result regarding the uniqueness of low-rank three-way array decomposition due to Kruskal (1977, 1988) (and generalized herein to the complex-valued case) that guarantees identifiability of all relevant signals and propagation parameters. These and other issues are also demonstrated in pertinent simulation experiments. <s> BIB012 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> We generalize Kruskal's fundamental result on the uniqueness of trilinear decomposition of three-way arrays to the case of multilinear decomposition of four- and higher-way arrays. The result is surprisingly general and simple and has several interesting ramifications. Copyright © 2000 John Wiley & Sons, Ltd. <s> BIB013 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Blind signal separation (BSS) and independent component analysis (ICA) are emerging techniques of array processing and data analysis that aim to recover unobserved signals or "sources" from observed mixtures (typically, the output of an array of sensors), exploiting only the assumption of mutual independence between the signals. The weakness of the assumptions makes it a powerful approach, but it requires us to venture beyond familiar second order statistics, The objectives of this paper are to review some of the approaches that have been developed to address this problem, to illustrate how they stem from basic principles, and to show how they relate to each other. <s> BIB014 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Abstract Finding the means to efficiently summarize electroencephalographic data has been a long-standing problem in electrophysiology. A popular approach is identification of component modes on the basis of the time-varying spectrum of multichannel EEG recordings—in other words, a space/frequency/time atomic decomposition of the time-varying EEG spectrum. Previous work has been limited to only two of these dimensions. Principal Component Analysis (PCA) and Independent Component Analysis (ICA) have been used to create space/time decompositions; suffering an inherent lack of uniqueness that is overcome only by imposing constraints of orthogonality or independence of atoms. Conventional frequency/time decompositions ignore the spatial aspects of the EEG. Framing of the data being as a three-way array indexed by channel, frequency, and time allows the application of a unique decomposition that is known as Parallel Factor Analysis (PARAFAC). Each atom is the tri-linear decomposition into a spatial, spectral, and temporal signature. We applied this decomposition to the EEG recordings of five subjects during the resting state and during mental arithmetic. Common to all subjects were two atoms with spectral signatures whose peaks were in the theta and alpha range. These signatures were modulated by physiological state, increasing during the resting stage for alpha and during mental arithmetic for theta. Furthermore, we describe a new method (Source Spectra Imaging or SSI) to estimate the location of electric current sources from the EEG spectrum. The topography of the theta atom is frontal and the maximum of the corresponding SSI solution is in the anterior frontal cortex. The topography of the alpha atom is occipital with maximum of the SSI solution in the visual cortex. We show that the proposed decomposition can be used to search for activity with a given spectral and topographic profile in new recordings, and that the method may be useful for artifact recognition and removal. <s> BIB015 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Let X be a real-valued three-way array. The Candecomp/Parafac (CP) decomposition is written as (X) under bar = (Y) under bar ((1)) + ... + (Y) under bar ((R)) + E, where Y-(r) are rank-1 arrays and (E) under bar is a rest term. Each rank-1 array is defined by the outer product of three vectors a((r)). b((r)) and c((r)), i.e. y(ijk)((r)) =a(i)((r)) b(j)((r)) c(k)((t)). These vectors make up the R columns of the component matrices A, B and C. If 2R + 2 is less than or equal to the sum of the k-ranks of A, B and C, then the fitted part of the decomposition is unique up to a change in the order of the rank-1 arrays and rescaling/counterscaling of each triplet of vectors (a((r)), b((r)), c((r))) forming a rank-1 array. This classical result was shown by Kruskal. His proof is, however, rather inaccessible and does not seem intuitive. In order to contribute to a better understanding of CP uniqueness, this paper provides an accessible and intuitive proof of Kruskal's condition. The proof is both self-contained and compact and can easily be adapted for the complex-valued CP decomposition. (c) 2006 Elsevier Inc. All rights reserved. <s> BIB016 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Abstract We discuss model-free analysis of multisubject or multisession FMRI data by extending the single-session probabilistic independent component analysis model (PICA; Beckmann and Smith, 2004. IEEE Trans. on Medical Imaging, 23 (2) 137–152) to higher dimensions. This results in a three-way decomposition that represents the different signals and artefacts present in the data in terms of their temporal, spatial, and subject-dependent variations. The technique is derived from and compared with parallel factor analysis (PARAFAC; Harshman and Lundy, 1984. In Research methods for multimode data analysis, chapter 5, pages 122–215. Praeger, New York). Using simulated data as well as data from multisession and multisubject FMRI studies we demonstrate that the tensor PICA approach is able to efficiently and accurately extract signals of interest in the spatial, temporal, and subject/session domain. The final decompositions improve upon PARAFAC results in terms of greater accuracy, reduced interference between the different estimated sources (reduced cross-talk), robustness (against deviations of the data from modeling assumptions and against overfitting), and computational speed. On real FMRI ‘activation’ data, the tensor PICA approach is able to extract plausible activation maps, time courses, and session/subject modes as well as provide a rich description of additional processes of interest such as image artefacts or secondary activation patterns. The resulting data decomposition gives simple and useful representations of multisubject/multisession FMRI data that can aid the interpretation and optimization of group FMRI studies beyond what can be achieved using model-based analysis techniques. <s> BIB017 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> In this paper, we present a powerful technique for the blind extraction of direct-sequence code-division multiple access (DS-CDMA) signals from convolutive mixtures received by an antenna array. The technique is based on a generalization of the canonical or parallel factor decomposition (CANDECOMP/PARAFAC) in multilinear algebra. We present a bound on the number of users under which blind separation and deconvolution is guaranteed. The solution is computed by means of an alternating least squares (ALS) algorithm. The excellent performance is illustrated by means of a number of simulations. We include an explicit expression of the Cramer-Rao bound (CRB) of the transmitted symbols. <s> BIB018 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> We propose a nonparametric extension to the factor analysis problem using a beta process prior. This beta process factor analysis (BP-FA) model allows for a dataset to be decomposed into a linear combination of a sparse set of factors, providing information on the underlying structure of the observations. As with the Dirichlet process, the beta process is a fully Bayesian conjugate prior, which allows for analytical posterior calculation and straightforward inference. We derive a varia-tional Bayes inference algorithm and demonstrate the model on the MNIST digits and HGDP-CEPH cell line panel datasets. <s> BIB019 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> A nonparametric Bayesian extension of Factor Analysis (FA) isproposed where observed data Y is modeled as a linear superposi-tion, G, of a potentially in nite number of hidden factors, X. TheIndian Bu et Process (IBP) is used as a prior on G to incorporatesparsity and to allow the number of latent features to be inferred.The model’s utility for modeling gene expression data is investigatedusing randomly generated datasets based on a known sparse connec-tivity matrix for E. Coli, and on three biological datasets of increasingcomplexity. <s> BIB020 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> As a consequence of our information society, not only more and larger data sets become available, but also data sets that include multiple sorts of information regarding the same system. Such data sets can be denoted by the terms coupled, linked, or multiset data, and the associated data analysis can be denoted by the term data fusion. In this paper, we first give a formal description of coupled data, which allows the data-analyst to typify the structure of a coupled data set at hand. Second, we list two meta-questions and a series of complicating factors that may be useful to focus the initial content-driven research questions that go with coupled data, and to choose a suitable data-analytic method. Third, we propose a generic framework for a family of decomposition-based models pertaining to an important subset of data fusion problems. This framework is intended to constitute both a means to arrive at a better understanding of the features and the interrelations of the specific models subsumed by it, and as a powerful device for the development of novel, custom-made data fusion models. We conclude the paper by showing how the proposed formal data description, meta-questions, and generic model may assist the data-analyst in choosing and developing suitable strategies for the treatment of coupled data in practice. Throughout the paper we illustrate with examples from the domain of systems biology. <s> BIB021 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Polarization analysis in Raman spectroscopy provides a powerful tool in chemical physics to identify the symmetry of the vibration modes of molecules, macromolecules and crystals, according to their crystallographic structure and local spatial orientation. In this paper we propose two new approaches in polarized Raman spectroscopy for data with rotational and spatial diversity, respectively. We show that the joint use of parallel and crossed polarization data yields more accurate source separation results and improves the uniqueness properties of the solution in blind nonnegative source separation algorithms. The proposed approaches are validated on two real polarized Raman data sets. <s> BIB022 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Several problems in signal processing have been formulated in terms of the Canonical Polyadic Decomposition of a higher-order tensor with one or more Vandermonde constrained factor matrices. We first propose new, relaxed uniqueness conditions. We explain that, under these conditions, the number of components may simply be estimated as the rank of a matrix. We propose an efficient algorithm for the computation of the factors that only resorts to basic linear algebra. We demonstrate the use of the results for various applications in wireless communication and array processing. <s> BIB023 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Given multivariate multiblock data (e.g., subjects nested in groups are measured on multiple variables), one may be interested in the nature and number of dimensions that underlie the variables, and in differences in dimensional structure across data blocks. To this end, clusterwise simultaneous component analysis (SCA) was proposed which simultaneously clusters blocks with a similar structure and performs an SCA per cluster. However, the number of components was restricted to be the same across clusters, which is often unrealistic. In this paper, this restriction is removed. The resulting challenges with respect to model estimation and selection are resolved. <s> BIB024 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Abstract Recent technological advances enable us to collect huge amounts of data from multiple sources. Jointly analyzing such multi-relational data from different sources, i.e., data fusion (also called multi-block, multi-view or multi-set data analysis), often enhances knowledge discovery. For instance, in metabolomics, biological fluids are measured using a variety of analytical techniques such as Liquid Chromatography–Mass Spectrometry and Nuclear Magnetic Resonance Spectroscopy. Data measured using different analytical methods may be complementary and their fusion may help in the identification of chemicals related to certain diseases. Data fusion has proved useful in many fields including social network analysis, collaborative filtering, neuroscience and bioinformatics. In this paper, unlike many studies demonstrating the success of data fusion, we explore the limitations as well as the advantages of data fusion. We formulate data fusion as a coupled matrix and tensor factorization (CMTF) problem, which jointly factorizes multiple data sets in the form of higher-order tensors and matrices by extracting a common latent structure from the shared mode. Using numerical experiments on simulated and real data sets, we assess the performance of coupled analysis compared to the analysis of a single data set in terms of missing data estimation and demonstrate cases where coupled analysis outperforms analysis of a single data set and vice versa. <s> BIB025 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> The widespread use of multisensor technology and the emergence of big data sets have highlighted the limitations of standard flat-view matrix models and the necessity to move toward more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift toward models that are essentially polynomial, the uniqueness of which, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints which match data properties and extract more general latent components in the data than matrix-based methods. <s> BIB026 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> Canonical Polyadic Decomposition (CPD) of a third-order tensor is a minimal decomposition into a sum of rank-$1$ tensors. We find new mild deterministic conditions for the uniqueness of individual rank-$1$ tensors in CPD and present an algorithm to recover them. We call the algorithm"algebraic"because it relies only on standard linear algebra. It does not involve more advanced procedures than the computation of the null space of a matrix and eigen/singular value decomposition. Simulations indicate that the new conditions for uniqueness and the working assumptions for the algorithm hold for a randomly generated $I\times J\times K$ tensor of rank $R\geq K\geq J\geq I\geq 2$ if $R$ is bounded as $R\leq (I+J+K-2)/2 + (K-\sqrt{(I-J)^2+4K})/2$ at least for the dimensions that we have tested. This improves upon the famous Kruskal bound for uniqueness $R\leq (I+J+K-2)/2$ as soon as $I\geq 3$. In the particular case $R=K$, the new bound above is equivalent to the bound $R\leq(I-1)(J-1)$ which is known to be necessary and sufficient for the generic uniqueness of the CPD. An existing algebraic algorithm (based on simultaneous diagonalization of a set of matrices) computes the CPD under the more restrictive constraint $R(R-1)\leq I(I-1)J(J-1)/2$ (implying that $R<(J-\frac{1}{2})(I-\frac{1}{2})/\sqrt{2}+1$). On the other hand, optimization-based algorithms fail to compute the CPD in a reasonable amount of time even in the low-dimensional case $I=3$, $J=7$, $K=R=12$. By comparison, in our approach the computation takes less than $1$ sec. We demonstrate that, at least for $R\leq 24$, our algorithm can recover the rank-$1$ tensors in the CPD up to $R\leq(I-1)(J-1)$. <s> BIB027 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> 1) Diversity in Matrix Decomposition Models: <s> We present structured data fusion (SDF) as a framework for the rapid prototyping of knowledge discovery in one or more possibly incomplete data sets. In SDF, each data set—stored as a dense, sparse, or incomplete tensor—is factorized with a matrix or tensor decomposition. Factorizations can be coupled, or fused, with each other by indicating which factors should be shared between data sets. At the same time, factors may be imposed to have any type of structure that can be constructed as an explicit function of some underlying variables. With the right choice of decomposition type and factor structure, even well-known matrix factorizations such as the eigenvalue decomposition, singular value decomposition and QR factorization can be computed with SDF. A domain specific language (DSL) for SDF is implemented as part of the software package Tensorlab, with which we offer a library of tensor decompositions and factor structures to choose from. The versatility of the SDF framework is demonstrated by means of four diverse applications, which are all solved entirely within Tensorlab’s DSL. <s> BIB028
Perhaps the most simple yet useful implementation of (1) is In many applications, model (2) is generalized as where i ¼ 1; . . . ; I, j ¼ 1; . . . ; J. An often-used interpretation is that x ij is a linear combination of R signals b j1 ; . . . ; b jR impinging on sensor i at sample index j, with weights a i1 ; . . . ; a iR . Equation (3) can be rewritten in a matrix form as such that x ij is the ði; jÞth entry of X 2 K IÂJ , K 2 fR; Cg, and similarly for A 2 K IÂR and B 2 K JÂR . The rth column vectors of A and B are a r ¼ ½a 1r ; . . . ; a Ir > and b r ¼ ½b 1r ; . . . ; b Jr > , respectively. The model in (4) provides I linear combinations of the columns of B and J linear combinations of the columns of A BIB011 . In the terminology of BIB011 , X provides I-fold diversity for B and J-fold diversity for A. Unfortunately, these types of diversity are generally insufficient to retrieve the underlying factor matrices A and B. For any R Â R invertible matrix T, it always holds that Hence, the pairs ðAT À1 ; TB > Þ and ðA; B > Þ have the same contribution to the observations X and thus cannot be distinguished. Consequently, one cannot uniquely identify the rank-1 terms a r b > r unless R 1  Lemma 4i] . We refer to this matter as the indeterminacy problem. A prevalent approach is to reduce T to a unitary matrix using a simplifying assumption that the columns of B are decorrelated. In such cases, the indeterminacy (5)  JÞ. The latter amounts to T ¼ Pm, where P is a permutation matrix and m is diagonal and invertible. The presence of P implies that the indexing 1; . . . ; R is arbitrary. This indeterminacy is inherent to the problem and thus inevitable. If all decompositions yield the same rank-1 terms then we say that the model is unique. The fact that the factorization of a matrix into a product of several matrices is generally not unique for R > 1 unless additional constraints are imposed is well known  Sec. 3] . We now discuss approaches to fix the indeterminacy in (5). In a general algebraic context, matrix factorizations such as singular value decomposition (SVD) and eigenvalue decomposition (EVD) are made unique by imposing orthogonality on the underlying matrices and inequality on the singular or eigenvalues  Sec. 3] , BIB007 . Such constraints are convenient mathematically but usually physically implausible since they yield noninterpretable results BIB010 . It is thus desirable to find other types of constraints that allow for better representation of the natural properties of the data. Depending on the application, the matrix factorization model in (4) may be interpreted in different ways that give rise to different types of constraints. When the model in (4) is used to analyze data, it is sometimes termed factor analysis (FA)  . In the signal processing community, when the columns of B represent signal samples and the goal is to recover these signals given only the observations X, model (4) is commonly associated with the blind source separation (BSS) problem  , BIB014 . The goal of FA and BSS is to represent X as a sum of low-rank terms with interpretable factors BIB026 , where the difference lies in the type of assumptions being used. In FA, one approach to fixing the indeterminacy (5) is by imposing external constraints BIB003 Sec. I] . This is not a data-driven approach and is thus excluded from our discussion. A data-driven approach to FA is to use physically meaningful constraints on the factor matrices that reduce the number of degrees of freedom. For example, a specific arrangement of a receive antenna array or other properties of a communication system may be imposed via a Vandermonde  - BIB023 or Toeplitz BIB018 structure. Alternatively, a factor may reflect a specific signal type such as constant modulus or finite alphabet BIB011 , BIB012 . Another approach is to use sparsity BIB019 - BIB020 . Probably the most well-known BSS approach to fix the indeterminacy in (5) is independent component analysis (ICA). ICA is more commonly formulated as where sðtÞ ¼ ½s 1 ðtÞ; . . . ; s R ðtÞ > 2 K RÂ1 is a vector of R statistically independent random processes known as ''sources,'' and xðtÞ 2 K IÂ1 their observations. A is full column rank. The link with (4) is established via X ¼ ½xð1Þ; . . . ; xðTÞ, J ¼ T, and B > ¼ ½sð1Þ; . . . ; sðTÞ such that the R columns of B represent samples from the R statistically independent random processes. ICA uses the ''spatial diversity'' provided by an array of sensors, which amounts to the I-fold diversity for B mentioned before, together with an assumption of statistical independence on the sources, in order to obtain estimates of sðtÞ whose entries are as statistically independent as possible. This amounts to fixing the indeterminacy BIB003 . Under these assumptions, separation can be achieved if the statistically independent sources are nonstationary, nonwhite, or nonGaussian BIB014 , BIB008 -  . The first two can be interpreted as diversity across time or diversity in the spectral domain: the sources must have different nonstationarity profiles or power spectra  Sec. 6] . Non-Gaussianity is associated with diversity in higher order statistics (HOS). A plethora of methods has been devised to exploit this diversity  , BIB008 , BIB009 -  , and the matter is far from being exhausted. Both FA and ICA have been used for decades and with much success to analyze a very broad range of data, their success being much due to the simplicity of their basic idea and the fact that very robust algorithms exist that yield satisfying results. Therefore, they are at the focus of our discussion. It should be kept in mind, however, that in practice, many observations can be better explained by other types of underlying models that are not limited to decomposition into a sum of rank-1 terms, statistical independence, linear relationships, or even matrix factorizations. Other properties that are often used to achieve uniqueness, improve numerical robustness and enhance interpretability are, for example, nonnegativity, sparsity, and smoothness  . Proving uniqueness for these types of factorizations is a matter of ongoing research. Any type of constraint or assumption on the underlying variables that helps achieve essential uniqueness can be regarded as a ''diversity.'' 2) Going Up to Higher Order Arrays: In Section III-C1, we have seen that the two linear types of diversity that are present in the rows and columns of X are not sufficient in order to obtain a unique matrix factorization. We saw that uniqueness can be established by imposing sufficiently strong constraints on the factor matrices A and B in (4). An alternative approach is to enrich the observational domain, without constraining the factor matrices. For example, if the two linear diversities given by the 2-D array X are interpreted as spatial and temporal, it is possible to obtain uniqueness by adding a third diversity in the frequency domain, without imposing constraints on the factor matrices. We now explain how this can be done. The two-way model (4) can be generalized by extending (3) to a ir b jr c kr BIB001 with i ¼ 1; . . . ; I, j ¼ 1; . . . ; J, k ¼ 1; . . . ; K. These observations can be collected into a three-way array (third-order tensor) with dimensions whose ði; j; kÞth entry is are matrices whose column vectors are a r , b r , and c r ¼ ½c 1r ; . . . ; c Kr > , respectively. Here, a r b r c r 2 K IÂJÂK is an outer product of three vectors and thus is a rank-1 term. Its ði; j; kÞth entry is a ir b jr c kr . When (8) holds and is irreducible in the sense that R is minimal, it is sometimes referred to as the canonical polyadic decomposition (CPD) BIB004 ,  . Note that (4) can be rewritten as X ¼ P R r¼1 a r b r . In striking difference to (5), the pair fðA; B; CÞ; ðA; B; CÞg has the same triple product (8) if and only if there exists an R Â R permutation matrix P and three diagonal matrices m A ; m B ; m C such that even for R > 1, under very mild constraints on A, B, C  , BIB006 ,  . Equation (9) can be reformulated as X ¼ P R r¼1 ð r a r Þ ð r b r Þ ð r c r Þ 8 r r r ¼ 1. If a threeway array is subject only to these trivial indeterminacies (alternatively: if all CPDs yield the same rank-1 terms) then we say that it is (essentially) unique. The key difference between matrix and tensor factorizations is that CPD is inherently ''essentially unique'' up to a scaled permutation matrix, whereas in the bilinear case the indeterminacy is an arbitrary nonsingular matrix. The uniqueness of the CPD becomes even more pronounced when it is joined with the fact that it holds also for R > maxðI; J; KÞ BIB006 , BIB005 . This is in contrast to FA, where it holds only for R minðI; JÞ. The immediate outcome is that underdetermined cases of ''more sources than sensors'' can be handled straightforwardly. In addition, the factor matrices A; B; C need not be full column rank BIB006 , BIB005  and BIB005 . These results have later been extended to higher order arrays, where ''order'' indicates the number N of indices x ijkÁÁÁ and N ! 3 BIB011 ,  , BIB013 . Recently, more relaxed bounds that guarantee uniqueness for larger R have been derived; see, e.g., BIB016 - BIB027 and references therein. In analogy to (4), the three-way array X provides three modes of linear diversity. It contains JK linear combinations of the columns of A, IK of B and IJ of C BIB011 . The fact that there exist multiple linear relationships within the model gives it the name ''multilinear.'' As argued in BIB011 , in many real-life scenarios, often there exist N ! 3 linear types of diversity that admit the multilinear decomposition BIB021 and thus guarantee uniqueness without any further assumptions. For example, in direct-sequence code-division multiple-access (DS-CDMA) communication systems, one may exploit (spatial Â temporal Â spreading code) BIB011 or (sensor Â polarization Â source signal) types of diversity; in psychometrics, (occasions Â persons Â tests)  or (observations Â scores Â variables) BIB024 ; in chemometrics and metabolomics, (sample Â frequency Â emission profile Â excitation profile) BIB021 ,  , BIB025 ; in polarized Raman spectroscopy, (polarization Â spatial diversity Â wavenumber) BIB022 ; in EEG, (time Â frequency Â electrode)  - BIB015 ; and in fMRI, (voxels Â scans Â subjects) BIB017 . Each type of constraint, structural (i.e., on the factor matrices) or observational (i.e., any of the nondegenerate modes of a matrix or a higher order array), that contributes to the unique decomposition and thus to the identifiability of the model, and cannot be deduced from the other constraints, i.e., is ''disjoint'' BIB028 , can be regarded as a ''diversity.'' In particular, each observational mode in the Nth order tensor (8) is a ''diversity.'' Hence, a tensor order corresponds to the number of types of (observational) diversity BIB011 , BIB012 . The explicit link between tensor order as a diversity and data fusion has been made in BIB028 . The fact that we can now associate ''diversity'' with well-defined mathematical properties of an analytical model implies that we can now link results on uniqueness, identifiability, and performance with the number of types of diversity that this model involves. Hence, the contribution of each ''diversity'' to the model can now be characterized and quantified BIB011 ,  . An application of this idea is the question raised in BIB011 as to how the number of types of observational diversity, i.e., tensor order N ! 3, contributes to the identifiability. To answer this question, it is shown that as N increases, indeed the bound on the number of rank-1 terms that can be uniquely identified becomes more relaxed. In other words, more observational modes allow to identify more sources in the same setup. Hence, this is a proof that increasing observational diversity improves identifiability. This is an example how questions regarding multimodality and diversity are a stimulus for new mathematical and theoretical insights. Until now, we have looked at N-way arrays as a way to represent simultaneously N (multi)linear types of diversity. An interesting link with the matrix factorization problem in Section III-C1 is achieved if we look at an N-way array as a structure that stores ðN À 1Þ-way arrays by stacking them along the Nth dimension. As noted, e.g., in BIB004 , BIB003 ,  , BIB005 , and BIB002 , the CPD can be thought of as a generalization of FA, as follows. Let denote K instances of the FA problem (4) where the diagonal R Â R matrix m k ¼ diagfc k1 ; . . . ; c kR g can be regarded as a scaling of the rows of B. It can be readily verified that stacking the K matrices X k in parallel along the third dimension results in (8). As we already know from (9), the rotation problem is eliminated BIB005 . It is thus no surprise that the tensor decomposition (8) is also known as parallel factor analysis (PARAFAC) BIB003 . Combining this observation with the perspective of data fusion, it has been noted that a tensor decomposition can be regarded as a way to fuse and jointly analyze data of multiple observations when all the data sets have the same size and share the same type of decomposition BIB028 . Note that this notion applies also to two-way arrays. For example, if we associate a BSS interpretation to the model in (4), the ith row can be regarded as the contribution of the ith sensor, and stacking all I observations yields the I Â J observation matrix X BIB028 . Model (10) can be linked not only to FA but also to BSS, as follows. In Section III-C1, we mentioned that uniqueness of BSS can be achieved if the sources are nonGaussian, nonstationary, or not spectrally flat (i.e., colored). These properties can be reformulated algebraically as a symmetric joint diagonalization (JD) of several matrices  , BIB009 , i.e., a special case of (10) when A ¼ B. As we have just explained, JD can be interpreted as a simple data fusion problem in which several data sets share the same mixing matrix. A key point is that diagonalization of a single matrix has an infinite number of solutions, and each of these ''nonproperties''  provides a set of at least two matrices that can be jointly diagonalized, thus fixing the indeterminacies. The discussion in this section implies that if we can represent our observations in terms of N ! 3 linear types of diversity or stack multiple data sets in an Nth-order tensor then we may benefit from the following powerful properties.