Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> V. CHALLENGES AT THE MODEL DESIGN LEVEL <s> This paper is concerned with the study of similarities and differences in factor structures between different groups. A common situation occurs when a battery of tests has been administered to samples of examinees from several populations.A very general model is presented, in which any parameter in the factor analysis models (factor loadings, factor variances, factor covariances, and unique variances) for the different groups may be assigned an arbitrary value or constrained to be equal to some other parameter. Given such a specification, the model is estimated by the maximum likelihood method yielding a large samplex2 of goodness of fit. By computing several solutions under different specifications one can test various hypotheses.The method is capable of dealing with any degree of invariance, from the one extreme, where nothing is invariant, to the other extreme, where everything is invariant. Neither the number of tests nor the number of common factors need to be the same for all groups, but to be at all interesting, it is assumed that there is a common core of tests in each battery that is the same or at least content-wise comparable. <s> BIB001 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> V. CHALLENGES AT THE MODEL DESIGN LEVEL <s> We present a unified view of matrix factorization that frames the differences among popular methods, such as NMF, Weighted SVD, E-PCA, MMMF, pLSI, pLSI-pHITS, Bregman co-clustering, and many others, in terms of a small number of modeling choices. Many of these approaches can be viewed as minimizing a generalized Bregman divergence, and we show that (i) a straightforward alternating projection algorithm can be applied to almost any model in our unified view; (ii) the Hessian for each projection has special structure that makes a Newton projection feasible, even when there are equality constraints on the factors, which allows for matrix co-clustering; and (iii) alternating projections can be generalized to simultaneously factor a set of matrices that share dimensions. These observations immediately yield new optimization algorithms for the above factorization methods, and suggest novel generalizations of these methods such as incorporating row and column biases, and adding or relaxing clustering constraints. <s> BIB002 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> V. CHALLENGES AT THE MODEL DESIGN LEVEL <s> Relational learning is concerned with predicting unknown values of a relation, given a database of entities and observed relations among entities. An example of relational learning is movie rating prediction, where entities could include users, movies, genres, and actors. Relations encode users' ratings of movies, movies' genres, and actors' roles in movies. A common prediction technique given one pairwise relation, for example a #users x #movies ratings matrix, is low-rank matrix factorization. In domains with multiple relations, represented as multiple matrices, we may improve predictive accuracy by exploiting information from one relation while predicting another. To this end, we propose a collective matrix factorization model: we simultaneously factor several matrices, sharing parameters among factors when an entity participates in multiple relations. Each relation can have a different value type and error distribution; so, we allow nonlinear relationships between the parameters and outputs, using Bregman divergences to measure error. We extend standard alternating projection algorithms to our model, and derive an efficient Newton update for the projection. Furthermore, we propose stochastic optimization methods to deal with large, sparse matrices. Our model generalizes several existing matrix factorization methods, and therefore yields new large-scale optimization algorithms for these problems. Our model can handle any pairwise relational schema and a wide variety of error models. We demonstrate its efficiency, as well as the benefit of sharing parameters among relations. <s> BIB003 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> V. CHALLENGES AT THE MODEL DESIGN LEVEL <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> BIB004 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> V. CHALLENGES AT THE MODEL DESIGN LEVEL <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> BIB005
In this section, we confront the unavoidable ''how'' question, presenting some guidelines that might be helpful in the actual design of data fusion solutions, from a model design perspective. This question has already been raised by numerous authors, e.g., in BIB004 -  , BIB005 , BIB001 , BIB002 , BIB003 , among others, and the following discussion builds upon these foundations. In a sense, this section concludes our paper. It complements Section III by proposing theoretical model design principles that allow diversity to manifest itself. It complements Sections II and IV by presenting model design principles that can accommodate the practical data-level challenges presented in Section IV and the numerous tasks given in Section II. It provides examples of approaches that allow data sets to interact and inform each other, in the sense of Definition I.2. As in previous sections, due to the vastness of the field, the discussion in this section is far from being exhaustive: we only touch at certain topics, and leave others, such as computation, algorithms and fusion of large-scale data, outside the scope of this overview. The rest of this section is organized as follows. In Section V-A, we discuss different strategies to data fusion, and address, in particular, at which level of abstraction, reduction and simplification the data should be fused. Section V-B discusses mathematical models for links between data sets that maximally exploit diversity and enhance interpretability and performance. Section V-C discusses some theoretical approaches to the analysis of the ensemble of linked data sets. Section V-D brings together the numerous model design steps and considerations in a unified framework of ''structured data fusion.'' We conclude our discussion with validation issues in Section V-E.