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：给定一族近交换的对称矩阵，我们考虑计算一个能近对角化族中每个矩阵的正交矩阵的任务。本文提出并分析了用于执行该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。与现有基于优化的方法不同，RJD实现简单，并可利用现有优质线性代数软件包。我们主要的创新贡献在于证明了鲁棒恢复：给定一个与交换族$\epsilon$-接近的矩阵族，RJD能以高概率联合对角化该族，误差范数为O($\epsilon$)。目前已知无其他方法具备这种通用鲁棒恢复保证。我们还讨论了如何通过收缩技术进一步改进算法，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。