In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems.

翻译：本文提出了一种新的随机算法，用于在矩阵自由设置下估计二到无穷范数与一到二范数，仅需使用矩阵向量乘法。我们的方法基于对Hutchinson对角估计器及其Hutch++版本的适当改进。我们为两种改进提供了预言机复杂度界。我们进一步展示了所提算法在图像分类任务中用于深度神经网络训练时基于雅可比矩阵正则化的实际效用。我们还证明了该方法可应用于缓解推荐系统领域中对抗性攻击的影响。