Geodesically-convex optimization for averaging partially observed covariance matrices
Geodesically-convex optimization for averaging partially observed covariance matrices
复制标题
DOI：
--
复制
发表时间：
2020-09
期刊：
影响因子：
--
通讯作者：
F. Yger;S. Chevallier;Quentin Barthélemy;S. Sra
中科院分区：
文献类型：
--
作者：
F. Yger;S. Chevallier;Quentin Barthélemy;S. Sra
对称正定 (SPD) 矩阵渗透到许多科学学科，包括机器学习、优化和信号处理。配备了黎曼几何，SPD 矩阵的空间受益于令人信服的特性，其导出的黎曼均值现在是某些应用中的黄金标准，例如脑机接口（BCI）。本文解决了缺失变量的协方差矩阵的平均问题。这种情况经常发生在廉价或不可靠的传感器上，或者当伪影抑制技术删除损坏的传感器时，导致矩阵排名不足，从而阻碍黎曼几何在基于协方差的方法中的使用。另一种但有问题的方法是删除缺少变量的矩阵，从而减少训练集的大小。我们解决了这些限制，并提出了一种基于测地线凸性的新公式。我们的方法在具有受控数量的缺失变量和已知基线的生成数据集上进行评估，证明了所提出的估计器的稳健性。这种方法的实际意义是在真实的 BCI 数据集上进行评估的。我们的结果表明，所提出的平均值比经典数据插补方法更稳健，更适合分类。
Symmetric positive definite (SPD) matrices permeates numerous scientific disciplines, including machine learning, optimization, and signal processing. Equipped with a Riemannian geometry, the space of SPD matrices benefits from compelling properties and its derived Riemannian mean is now the gold standard in some applications, e.g. brain-computer interfaces (BCI). This paper addresses the problem of averaging covariance matrices with missing variables. This situation often occurs with inexpensive or unreliable sensors, or when artifact-suppression techniques remove corrupted sensors leading to rank deficient matrices, hindering the use of the Riemannian geometry in covariance-based approaches. An alternate but questionable method consists in removing the matrices with missing variables, thus reducing the training set size. We address those limitations and propose a new formulation grounded in geodesic convexity. Our approach is evaluated on generated datasets with a controlled number of missing variables and a known baseline, demonstrating the robustness of the proposed estimator. The practical interest of this approach is assessed on real BCI datasets. Our results show that the proposed average is more robust and better suited for classification than classical data imputation methods.