Abstract

Combining various sources of information to discover hidden patterns is key in data analysis. These sources can often be represented as matrices and/or multiway arrays, or tensors, which can be factorized jointly, e.g., as sums of simple terms, to gain insight into the data. In this chapter, an overview of (the rationale behind) numerically well-founded optimization techniques based on a Gauss–Newton framework is given, which has superior convergence properties and allows all multilinear structure to be exploited. Prior knowledge in the form of parametric, box or soft constraints as well as regularization can be incorporated easily. We show how matrices and/or tensors can be coupled through (partially) shared factors or through common underlying variables. The framework is further extended to more general divergences allowing more suitable statistical assumptions. Finally, as tensor problems become large-scale quickly due to the curse of dimensionality, techniques used to alleviate or overcome this curse are discussed.

Code description

In the chapter, we discuss many examples for which the code is available in this package. Depending on the example, the code either reproduces the figures used in the chapter, or illustrates how the concept explained in the example can be implemented in Matlab.

Reference

N. Vervliet, L. De Lathauwer, "Numerical optimization-based algorithms for data fusion," in Chapter 4 of Data Fusion Methodology and Applications, (Cocchi M., ed.), vol. 33 of Data Handling in Science and Technology, Elsevier, pp. 81-128, Jan. 2019.

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