This thesis contributes to the emerging topic of data fitting algorithms that rely on the structure of the data. It tackles two problems: averaging a set of positive-definite (PD) matrices, and fitting a curve to a set of positive-semidefinite (PSD) matrices. Averaging a set of PD matrices is a subtask in several medical imaging and image processing algorithms. It is also used in classifiers for electroencephalogram signals in brain-computer interfaces. This last application motivates the development of fast and incremental averaging tools on PD matrices, to allow online learning and adaptation of the classifier. We propose and study accelerated versions of two algorithms for averaging PD matrices. The first is an incremental gradient descent algorithm on the manifold. The second computes the mean of the data points in a decentralized way, by relying on the expression for the mean of two data points. Finally, we apply our incremental algorithm to the EEG classification problem. For the second task, we restrict ourselves to the set of fixed-rank PSD matrices, to take advantage of the differentiable structure of this set. This restriction is applicable, for example, when working with low-rank estimations of large PSD matrices, or with large covariance matrices characterizing a dynamical process of lower dimension. We compute expressions for the main tools required for applying curve fitting algorithms to the manifold of fixed-rank PSD matrices. We finally apply these algorithms to two problems: parametric model order reduction and wind field estimation.