CUR近似的盲误差估计

中文摘要 AI 辅助

低秩近似是进行可扩展矩阵计算的基础工具。虽然此类近似传统上通过截断奇异值分解（SVD）形成，但随机数值线性代数的最新进展已产生出在成本极低的情况下具有相当精度的方法。这些方法的一个关键优势是它们能够在仅能有限访问完整矩阵的情况下运行。一个突出的例子是CUR分解，它仅从矩阵的列和行的子集构建近似，使其特别适用于无法获得全局矩阵访问的场景，例如在光谱-显微实验的设计中，甚至无法计算矩阵-向量乘积。然而，这些方法固有的随机性引入了一个关键挑战：在仅能观测到矩阵条目子集的严格约束下，如何有效评估近似精度。我们推导出CUR分解的Frobenius误差的“盲”估计器。对于该估计器，我们推导了所需列查询数量的最坏情况下的下界，并提供了量化其不确定性的技术。最后，我们在合成问题与真实光谱-显微示例的混合上展示了该估计器及不确定性集合的性能。

英文摘要

Low-rank approximation is a fundamental tool for scalable matrix computations. While such approximations have classically been formed via the truncated SVD, recent advances in randomized numerical linear algebra have produced methods of comparable accuracy at a fraction of the cost. A key advantage of these approaches is their ability to operate with limited access to the full matrix. A prominent example is the CUR decomposition, which builds an approximation from only a subset of columns and rows, making it particularly well-suited to settings where global matrix access is unavailable, such as in the design of spectro-microscopy experiments where even matrix-vector products cannot be computed. However, the randomness inherent to these methods introduces a key challenge: efficiently assessing approximation accuracy under the stringent constraint that only a subset of matrix entries can be observed. We derive a "blind" estimator of the Frobenius error of a CUR decomposition. For this estimator, we derive a worst-case lower bound on the number of required column queries, and provide techniques for quantifying its uncertainty. Finally, we demonstrate the performance of the estimator and the uncertainty sets on a mix of synthetic problems and real spectro-microscopy examples.

发表机构

• Mathematical Institute, University of Oxford（牛津大学数学研究所）
• Department of Mathematics and Statistics, University of New Mexico（新墨西哥大学数学与统计系）

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