Many real-world unmixing problems involve inherent non-negativity constraints. Most physical quantities are non-negative: lengths, weights, amounts of radiation, and so on. For example, in the field of air quality, the amount of a particulate from a given source in a particular sample must be non-negative; and in musical audio signal processing, each musical note contributes a nonnegative amount to the signal power spectrum. This type of non-negativity constraint also arises in, e.g.

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... al image analysis for remote sensing, positron emission tomography (PET) image sequences in medical applications, or semantic analysis of text documents. Often we lose this non-negativity constraint when, for example, we subtract the mean from the data, such as when we perform the usual pre-whitening process for independent component analysis (ICA). However, we need to be aware that doing this may lose us important information that could help find the solution to our unmixing problem. Even where the non-negativity constraint is not inherently part of the problem, analogies with biological information processing systems suggest that this is an interesting direction to investigate, since information in neural systems is typically communicated using spikes, and the spike rate is a non-negative quantity. In this chapter we discuss some algorithms for the use of non-negativity constraints in unmixing problems, including positive matrix factorization (PMF) [71], non-negative matrix factorization (NMF), and their combination with other unmixing methods such as non-negative ICA and sparse non-negative matrix factorization. The 2-D models can be naturally extended to multiway array (tensor) decompositions, especially Non-negative Tensor Factorization (NTF) and Non-negative Tucker Decomposition (NTD). Rather than using a gradient descent direction to reduce the Euclidean cost function E in (1.3), we can use a Newton-like method to find alternately the S and A that directly minimizes E .