Nonnegative Matrix Factorization (NMF) is a popular data analysis tool for nonnegative data, able to extract meaningful features from a dataset, compress it and filter its noise. To do so, this method factorizes an input matrix Y as the product of two factors, the low-rank nonnegative matrices A and X. Recently, several interesting works have analyzed the possibility of imposing some additional structure to the NMF model, by requiring factors A and X to be based on nonnegative parametrizable functions, such as nonnegative splines or polynomials. When the input data is structured, for example for smooth continuous signals, this leads to factorizations that are less sensitive to noise and better able to find the characteristic elements of the dataset. This thesis is a continuation of that work. We first study NMF using polynomials and splines. We focus on the Hierarchical Alternating Least Squares (HALS) algorithm, which has the advantage of quickly finding a good solution, and converging to a stationary point under mild conditions. We generalize this algorithm and observe that it obtains good results and is competitive with existing approaches. We also seek to generalize the NMF model in a more formal way, so that any parameterizable function could be used in the factors. This leads to the definition of the H-NMF (NMF on Hilbert spaces), allowing to factorize a two-variable function (such as a matrix) as the sum of a small number of products of one-variable functions (the characteristic elements). This generalization is proved to have characteristics and theoretical guarantees similar to those of the standard NMF problem. Our generalization of HALS uses projections onto the functions under consideration (e.g. nonnegative polynomial or splines). As these projections can be costly and difficult to compute, we consider to approximate them. We are able to identify several cases where the algorithm keeps good convergence properties, and use these results to describe heuristic projections speeding up the algorithm without affecting its accuracy. Finally, we consider the case of rational functions which has no theoretical guarantees because rational functions of fixed degree do not form a convex set. In practice, we observe that our algorithms are very sensitive to their initialization in this case. Nevertheless, using rational functions in NMF can lead to very good results since rational functions are able to represent a wider range of shapes than polynomials and splines. (FSA - Sciences de l'ingénieur) -- UCL, 2022