The Nystr\"om method offers an effective way to obtain low-rank approximation of SPD matrices, and has been recently extended and analyzed to nonsymmetric matrices (leading to the generalized Nystr\"om method). It is a randomized, single-pass, streamable, cost-effective, and accurate alternative to the randomized SVD, and it facilitates the computation of several matrix low-rank factorizations. In this paper, we take these advancements a step further by introducing a higher-order variant of Nystr\"om's methodology tailored to approximating low-rank tensors in the Tucker format: the multilinear Nystr\"om technique. We show that, by introducing appropriate small modifications in the formulation of the higher-order method, strong stability properties can be obtained. This algorithm retains the key attributes of the generalized Nystr\"om method, positioning it as a viable substitute for the randomized higher-order SVD algorithm.

翻译：Nyström方法为对称正定矩阵的低秩逼近提供了一种有效途径，并最近被推广至非对称矩阵（从而形成广义Nyström方法）。该方法是随机化、单次遍历、可流式处理、高效且准确的随机SVD替代方案，并便于计算多种矩阵低秩分解。本文通过引入面向Tucker格式低秩张量逼近的Nyström方法论高阶变体——多线性Nyström技术，进一步推进了上述进展。我们证明，在高阶方法的公式中引入适当的微小修改，即可获得强稳定性。该算法保留了广义Nyström方法的关键特性，使其成为随机化高阶SVD算法的可行替代方案。