Chapter 4: Identifiability
Nicolas Gillis · Society for Industrial and Applied Mathematics eBooks · 2020
There are several issues when using NMF in practice, including for example the choice of the factorization rank r and of additional constraints that W and H should satisfy depending on the application at hand; see Chapter 5. However, the two main issues are arguably the NP-hardness of computing solutions and the nonuniqueness of the solutions. The NP-hardness of Exact NMF was discussed in Chapter 2, while NP-hardness of NMF will be discussed in Chapter 6. In this chapter, we discuss the nonuniqueness issue of the solutions of Exact NMF, also known as the identifiability issue. In other words, we discuss conditions under which the factors W and H in an Exact NMF decomposition X = WH are unique (up to scaling and permutation ambiguities) and hence correspond to the true factors that generated the data. This is crucial in many applications. For example, in blind HU (Section 1.3.2), it ensures that the recovered matrix W corresponds to the spectral signatures of the true endmembers and that the matrix H corresponds to the abundances of the endmembers within the pixels of the image. In audio source separation (Section 1.3.2), it ensures that the matrix W corresponds to the frequency response of the sources and that the matrix H corresponds to the activation of the sources over time. In Chapter 2, when studying the geometric interpretation of Exact NMF, we have encountered several matrices whose Exact NMF are not unique. As we will see, the geometric interpretation plays a crucial role in characterizing the solutions of Exact NMF.