Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> III. MULTIMODALITY AS A FORM OF DIVERSITY <s> The choosing of a set of factors likely to correspond to the real psychological unitary traits in a situation usually reduces to finding a satisfactory rotation in a Thurstone centroid analysis. Seven principles, three of which are new, are described whereby rotation may be determined and/or judged. It is argued that the most fundamental is the principle of “parallel proportional profiles” or “simultaneous simple structure.” A mathematical proof of the uniqueness of determination by this means is attempted and equations are suggested for discovering the unique position. <s> BIB001 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> III. MULTIMODALITY AS A FORM OF DIVERSITY <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> BIB002 </s> Multimodal Data Fusion: An Overview of Methods, Challenges, and Prospects <s> III. MULTIMODALITY AS A FORM OF DIVERSITY <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> BIB003
In this section, we discuss data fusion from a theoretical perspective. In order to contain our discussion, we focus on data-driven methods. Within these, we restrict our examples to a class of problems known as blind separation, and within these, to data and observations that can be represented by (multi)linear relationships. Reasons are as follows. First, by definition, data-driven models may be useful to numerous applications, as will be explained in Section III-B. Second, there exist much established theory and numerous models that fit into this framework. Third, it is impossible to cover all types of models. Still, the ideas that these examples illustrate go far beyond these specific models. A key property in any analytical model is uniqueness. Uniqueness is necessary in order to achieve interpretability, i.e., attach physical meaning to the output BIB001 , BIB002 . In order to establish uniqueness, all blind separation problems invariably rely on one or more types of diversity BIB003 : concrete mathematical examples will be given in Section III-C1. In this section, we show how the concept of ''diversity'' plays part, under different guises, in data fusion. In particular, we show that multimodality can provide a new form of diversity that can achieve uniqueness even in cases that are not unique otherwise. The rest of this section is as follows. Section III-A presents some basic mathematical preliminaries that will serve us to provide a more concrete meaning to the ideas that we lay out in the rest of this work. Section III-B explains the model-driven approach versus the data-driven approach, and motivates the latter. Section III-C discusses diversity and data fusion in data sets that are stacked in a single matrix or a higher order array, also known as a tensor. In Section III-D, we go beyond single-array data analysis, and establish the idea of ''a link between data sets as a new form of diversity'' as the key to advanced data fusion. We conclude our claims and summarize these ideas in Section III-E.