Congedo Marco, Afsari Bijan, Barachant Alexandre, Moakher Maher
GIPSA-lab, CNRS and Grenoble University, Grenoble, France.
Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland, United States of America.
PLoS One. 2015 Apr 28;10(4):e0121423. doi: 10.1371/journal.pone.0121423. eCollection 2014.
We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.
一组对称正定（SPD）矩阵的几何均值估计及其近似联合对角化（AJD）。如今，在赋予了费希尔信息度量的SPD矩阵流形中估计SPD矩阵集的几何均值受到了广泛关注。所得均值具有若干重要的不变性属性，并且在生物医学和图像数据处理等各种工程应用中已证明非常有用。虽然对于两个SPD矩阵，均值有代数闭式解，但对于多于两个的SPD矩阵集，只能通过迭代算法进行估计。然而，现有的迭代算法都没有同时具备快速收敛、每次迭代计算复杂度低以及收敛保证的特点。因此，最近人们考虑了基于对称散度度量（如巴氏散度）的几何均值的其他定义。所得均值虽然在实际中可能有用，但并不满足所有理想的不变性属性。在本文中，我们考虑在高维时间序列上估计的协方差矩阵的几何均值，假设数据是根据瞬时混合模型生成的，这在信号处理中非常常见。我们表明，在这些情况下，我们可以通过采用一种高效的AJD算法来近似费希尔信息几何均值。与其他方法相比，我们的近似通常更接近费希尔信息几何均值，并验证了许多不变性属性。此外，保证了收敛性，计算复杂度低且收敛速度是二次的。通过仿真证明了这种新的几何均值近似的准确性。
