Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> E. Conclusion: A Link Between Data Sets Is Indeed a New Form of Diversity <s> In various disciplines, information about the same phenomenon can be acquired from different types of detectors, at different conditions, different observations times, in multiple experiments or subjects, etc. We use the term "modality" to denote each such type of acquisition framework. Due to the rich characteristics of natural phenomena, as well as of the environments in which they occur, it is rare that a single modality can provide complete knowledge of the phenomenon of interest. The increasing availability of several modalities at once introduces new degrees of freedom, which raise questions beyond those related to exploiting each modality separately. It is the aim of this paper to evoke and promote various challenges in multimodal data fusion at the conceptual level, without focusing on any specific model, method or application. <s> BIB001 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> E. Conclusion: A Link Between Data Sets Is Indeed a New Form of Diversity <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> BIB002
The strength of IVA and coupled CPD over a set of unlinked factorizations lies in their ability to exploit commonalities among data sets. In IVA, it is the statistical dependence of sources across mixtures; in coupled CPD, it is the shared factors. In both scenarios, the links themselves are new types of information: the fact that data sets are linked, that elements in different data sets are related (or not), and the nature of these interactions, bring new types of constraints into the system that allow to reduce the number of degrees of freedom and thus enhance uniqueness, performance, interpretability, and robustness, among others. On top of that, the links among the data sets allow desired properties within one data set to propagate to the ensemble and enhance the properties of the whole decomposition BIB002 . This is a concrete mathematical manifestation of the raison d'être of data fusion that we have mentioned in Section II, implying that [11, Sec. 9],  , BIB001 An ensemble of data sets is ''more than the sum of its parts'' in the sense that it contains precious information that is lost if these relations are ignored. The models that we have just presented allow multiple data sets to inform each other and interact, as formulated in Definition I.2 and further elaborated in Section V-A. Therefore, in the same vein of the preceding discussion and Definition I.1, we conclude that BIB002 ,  , BIB001 Properly linking data sets can be regarded as introducing a new form of diversity, and this diversity is the basis and driving force of data fusion.