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Unsupervised hyperspectral unmixing methods aim to extract endmember spectra and infer the proportion of each of these spectra in each observed pixel when considering linear mixtures. However, the interaction between sunlight and the Earth’s surface is often very complex, so that observed spectra are then composed of nonlinear mixing terms. This nonlinearity is generally bilinear or linear quadratic. In this work, unsupervised hyperspectral unmixing methods, designed for the bilinear and linear-quadratic mixing models, are proposed. These methods are based on bilinear or linear-quadratic matrix factorization with non-negativity constraints. Two types of algorithms are considered. The first ones only use the projection of the gradient, and are therefore linked to an optimal manual choice of their learning rates, which remains the limitation of these algorithms. The second developed algorithms, which overcome the above drawback, employ multiplicative projective update rules with automatically chosen learning rates. In addition, the endmember proportions estimation, with three alternative approaches, constitutes another contribution of this work. Besides, the reduction of the number of manipulated variables in the optimization processes is also an originality of the proposed methods. Experiments based on realistic synthetic hyperspectral data, generated according to the two considered nonlinear mixing models, and also on two real hyperspectral images, are carried out to evaluate the performance of the proposed approaches. The obtained results show that the best proposed approaches yield a much better performance than various tested literature methods. Full article