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 algorithmic 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, showing that it is both necessary and sufficient. This insight elevates Gu and Eisenstat's rank-revealing QR as an archetypal rank-revealer, and devise a version that is less than $2\times$ 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：本文研究一类称为秩揭示算法的矩阵秩结构揭示方法。此类算法构成矩阵压缩、奇异值估计及列子集选择问题的基础组件。尽管基于列主元QR分解因其实用性被广泛采用，但并非总能作为秩揭示器。相反，采用全局最大体积主元策略的高斯消元法虽能保证估计矩阵奇异值，但其指数级算法复杂度使其仅具理论价值。研究表明，基于局部最大体积主元的概念是高斯消元法和QR分解两类秩揭示器中关键且实用的主元策略，并证明该策略既是必要条件也是充分条件。这一发现将Gu与Eisenstat的秩揭示QR分解提升为秩揭示器的典型范例，并设计出计算复杂度比CPQR低两倍以内的改进版本。通过同时研究高斯消元法与QR分解，我们统一了秩揭示器领域的研究框架，并证明任一主元策略的成功性均可通过以局部最大体积主元为基准进行评判。