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($\epsilon$). 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得该族中的每个矩阵均被近似对角化的任务。本文提出并分析了一种用于执行此任务的随机联合对角化（RJD）方法。RJD 通过对矩阵的随机线性组合应用标准特征值求解器来实现目标。与现有基于优化的方法不同，RJD 易于实现，并能充分利用现有高质量线性代数软件包。我们主要的创新贡献是证明了鲁棒恢复性：对于一族 $\epsilon$-接近于交换族的矩阵，RJD 能以高概率将这一族矩阵联合对角化，且误差范数为 O($\epsilon$)。此外，我们讨论了如何通过紧缩技术进一步改进该算法，并通过合成数据与真实数据的数值实验展示了其最先进的性能。