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 $\epsilon$-near 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：给定一族近似交换的对称矩阵，我们考虑计算一个正交矩阵，使其能近似对角化该族中每个矩阵的任务。本文提出并分析了随机联合对角化（RJD）方法以完成该任务。RJD将标准特征值求解器应用于矩阵的随机线性组合。与现有基于优化的方法不同，RJD实现简单，并能利用现有高质量线性代数软件包。我们的主要创新贡献在于证明了鲁棒恢复性：给定一族与交换族ε-近似的矩阵，RJD能以高概率实现该族的联合对角化，误差范数为O(ε)。目前尚无其他已知方法具备如此通用的鲁棒恢复保证。我们还讨论了如何通过紧缩技术进一步改进该算法，并通过合成数据与实际数据的数值实验展示了其最先进的性能。