We propose a continuous optimization algorithm for the Column Subset Selection Problem (CSSP) and Nystr\"om approximation. The CSSP and Nystr\"om method construct low-rank approximations of matrices based on a predetermined subset of columns. It is well known that choosing the best column subset of size $k$ is a difficult combinatorial problem. In this work, we show how one can approximate the optimal solution by defining a penalized continuous loss function which is minimized via stochastic gradient descent. We show that the gradients of this loss function can be estimated efficiently using matrix-vector products with a data matrix $X$ in the case of the CSSP or a kernel matrix $K$ in the case of the Nystr\"om approximation. We provide numerical results for a number of real datasets showing that this continuous optimization is competitive against existing methods.

翻译：本文提出一种用于列子集选择问题（CSSP）和Nyström近似的连续优化算法。CSSP和Nyström方法基于预先确定的列子集构造矩阵的低秩近似。众所周知，选择大小为$k$的最优列子集是一个困难的组合问题。在本工作中，我们展示了如何通过定义一个经随机梯度下降最小化的带惩罚连续损失函数来逼近最优解。我们证明，该损失函数的梯度可通过与数据矩阵$X$（针对CSSP情况）或核矩阵$K$（针对Nyström近似情况）的矩阵-向量乘积进行高效估计。我们在多个真实数据集上给出数值结果，表明该连续优化方法与现有方法相比具有竞争力。