We study algorithms called rank-revealers that reveal a matrix's rank structure. Such algorithms form a fundamental component in matrix compression, singular value estimation, and column subset selection problems. While column-pivoted QR has been widely adopted due to its practicality, it is not always a rank-revealer. Conversely, Gaussian elimination (GE) with a pivoting strategy known as global maximum volume pivoting is guaranteed to estimate a matrix's singular values but its exponential complexity limits its interest to theory. We show that the concept of local maximum volume pivoting is a crucial and practical pivoting strategy for rank-revealers based on GE and QR. In particular, we prove that it is both necessary and sufficient; highlighting that all local solutions are nearly as good as the global one. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and we implement a version that is observed to be at most $2\times$ more computationally expensive than CPQR. We unify the landscape of rank-revealers by considering GE and QR together and prove that the success of any pivoting strategy can be assessed by benchmarking it against a local maximum volume pivot.

翻译：我们研究一类称为秩揭示算法的算法，这类算法能够揭示矩阵的秩结构。此类算法构成了矩阵压缩、奇异值估计以及列子集选择问题中的基础组件。尽管列主元QR分解因其实用性而被广泛采用，但它并非总是秩揭示算法。相反，采用全局最大体积主元策略的高斯消元法（GE）能够保证估计矩阵的奇异值，但其指数级的计算复杂度使其主要局限于理论兴趣。我们证明了局部最大体积主元策略是基于GE和QR的秩揭示算法中关键且实用的主元选取策略。具体而言，我们证明了该策略既是必要的也是充分的；并强调所有局部解几乎与全局解一样好。这一见解将Gu和Eisenstat的秩揭示QR算法提升为典型的秩揭示算法，并且我们实现了一个版本，其计算开销最多比CPQR高$2\times$。通过将GE和QR一同考虑，我们统一了秩揭示算法的研究图景，并证明任何主元选取策略的成功与否，都可以通过将其与局部最大体积主元策略进行基准测试来评估。