Adaptive filter with Riemannian manifold constraint

The adaptive filtering theory has been extensively developed, and most of the proposed algorithms work under the assumption of Euclidean space. However, in many applications, the data to be processed comes from a non-linear manifold. In this article, we propose an alternative adaptive filter that works on a manifold, thus generalizing the filtering task to non-Euclidean spaces. To this end, we generalized the least-mean-squared algorithm to work on a manifold using an exponential map. Our experiments showed that the proposed method outperforms other state-of-the-art algorithms in several filtering tasks.