The Geometric Matrix Mean: an Adaptation for Structured Matrices
Ben Jeuris, Raf Vandebril, Dario A. Bini, Bruno Iannazzo · Lirias · 2014
Positive definite matrices can be encountered in a widespread collection of applications, such as signal processing, bio-informatics, and radar technology. As a consequence, these matrices and their geometry has been well researched, resulting in a natural and optimal geometry. In many applications, not one, but multiple positive definite matrices are provided through measurements. It is desired to find an average representation of the matrices, which best describes the corresponding measurement. To this end, the geometric matrix mean is often used, because it has interesting properties with regard to positive definite matrices, such as invariance under inversion, invariance under congruence, etc. Of the various instances for the geometric mean, the Karcher mean appears the most natural through its definition as the barycenter of the matrices under the positive definite geometry. Computationally, the barycenter is located using matrix manifold optimization techniques. Often positive definiteness is not the only structural property present in the measured matrices. Additional structures can be, e.\,g., Toeplitz or Hankel structure, low displament rank, etc. These structures are usually linked to some physical interpretation in the application, so it is desirable for the result to preserve the structure and hence the interpretation. Unfortunately, the Karcher mean is typically not structure preserving. We present an adaptation of the Karcher mean which preserves desired structures by computing an optimizer of the barycenter problem over a restricted search space. The interesting properties of the geometric mean are generalized to take the additional structure into account. To compute the new barycenter, two preconditioned gradient descent methods are investigated for linear submanifolds of the positive definite matrix manifold. The preconditioners are derived using differential geometry, where the natural positive definite geometry results in a more involved, but also more efficient preconditioner. In a second approach, we focus on the set of positive definite Toeplitz matrices. An application-inspired transformation maps such a matrix to the product space of the positive scalars and a multiple of the complex unit circle. Separately, these sets are naturally endowed with the geometry of positive scalars (the one-dimensional equivalent of the positive definite geometry) and the hyperbolic geometry, respectively. Combined, this allows the definition of a new barycenter. Finally, we test a generalization of the previous barycenter to the set of Block-Toeplitz Toeplitz-block (positive definite) matrices. The new blockwise transformation generalises the positive scalars to positive definite matrices and provides a natural extension of the hyperbolic unit circle to the matrix setting. The final algorithms are again designed to maintain the original structure of the matrices. The performance of all mentioned algorithms is tested in numerical experiments, both in a theoretical environment and in real-world applications. The unstructured Karcher mean is applied in an experiment in bio-informatics where correlation between human genes and physical traits is combined over several independent measurements. For the structured versions, an application is found in radar detection, where positive definite Toeplitz matrices need to be averaged.