Adaptive filtering for estimation of a low-rank positive semidefinite matrix

In this paper, we adopt a geometric viewpoint to tackle the problem of estimating a linear model whose parameter is a fixed-rank positive semidefinite matrix. We consider two gradient descent flows associated to two distinct Riemannian quotient geometries that underlie this set of matri- ces. The resulting algorithms are non-linear and can be viewed as a generalization of Least Mean Squares that instrically constrain the parameter within the manifold search space. Such algorithms designed for low-rank matrices find applications in high-dimensional distance learning problems for classification or clustering.