Given a family of nearly commuting symmetric matrices, we consider the task of computing an orthogonal matrix that nearly diagonalizes every matrix in the family. In this paper, we propose and analyze randomized joint diagonalization (RJD) for performing this task. RJD applies a standard eigenvalue solver to random linear combinations of the matrices. Unlike existing optimization-based methods, RJD is simple to implement and leverages existing high-quality linear algebra software packages. Our main novel contribution is to prove robust recovery: Given a family that is $\epsilon$-close to a commuting family, RJD jointly diagonalizes this family, with high probability, up to an error of norm O($\epsilon$). No other existing method is known to enjoy such a universal robust recovery guarantee. We also discuss how the algorithm can be further improved by deflation techniques and demonstrate its state-of-the-art performance by numerical experiments with synthetic and real-world data.

翻译：考虑到一个几乎通勤对称矩阵的大家庭,我们考虑计算一个几乎对每个家庭矩阵进行对等的正方矩阵的任务。在本文件中,我们提议和分析随机联合对等化(RJD)来完成这项任务。RJD将标准电子价值求解器应用于该矩阵的随机线性组合。与现有的优化方法不同,RJD易于实施和利用现有的高质量线性代数软件包。我们的主要新贡献是证明恢复力强:鉴于一个家庭接近一个通勤家庭,RJD共同对这个家庭进行对等化,极有可能出现O($/eplon$)标准错误。没有其他现有方法可以享受这种普遍强力恢复保证。我们还讨论如何通过通缩技术进一步改进算法,并通过以合成和现实世界数据的数字实验来展示其最新表现。