We derive a new adaptive leverage score sampling strategy for solving the Column Subset Selection Problem (CSSP). The resulting algorithm, called Adaptive Randomized Pivoting, can be viewed as a randomization of Osinsky's recently proposed deterministic algorithm for CSSP. It guarantees, in expectation, an approximation error that matches the optimal existence result in the Frobenius norm. Although the same guarantee can be achieved with volume sampling, our sampling strategy is much simpler and less expensive. To show the versatility of Adaptive Randomized Pivoting, we apply it to select indices in the Discrete Empirical Interpolation Method, in cross/skeleton approximation of general matrices, and in the Nystroem approximation of symmetric positive semi-definite matrices. In all these cases, the resulting randomized algorithms are new and they enjoy bounds on the expected error that match -- or improve -- the best known deterministic results. A derandomization of the algorithm for the Nystroem approximation results in a new deterministic algorithm with a rather favorable error bound.

翻译：本文针对列子集选择问题提出了一种新的自适应杠杆值采样策略。所得到的算法称为自适应随机主元选取法，可视为对Osinsky近期提出的确定性CSSP算法的随机化推广。该算法在期望意义上保证了与Frobenius范数下最优存在性结果相匹配的近似误差。尽管相同保证亦可通过体积采样实现，但本文所提采样策略更为简洁且计算代价更低。为展示自适应随机主元选取法的普适性，我们将其应用于离散经验插值法的指标选取、一般矩阵的交叉/骨架逼近，以及对称半正定矩阵的Nyström逼近。在所有案例中，所得随机算法均为首次提出，其期望误差界均达到或超越了当前最优确定性结果。通过对Nyström逼近算法进行去随机化处理，我们进一步得到了一种具有优越误差界的全新确定性算法。