The problem of low rank approximation is ubiquitous in science. Traditionally this problem is solved in unitary invariant norms such as Frobenius or spectral norm due to existence of efficient methods for building approximations. However, recent results reveal the potential of low rank approximations in Chebyshev norm, which naturally arises in many applications. In this paper we tackle the problem of building optimal rank-1 approximations in the Chebyshev norm. We investigate the properties of alternating minimization algorithm for building the low rank approximations and demonstrate how to use it to construct optimal rank-1 approximation. As a result we propose an algorithm that is capable of building optimal rank-1 approximations in Chebyshev norm for small matrices.

翻译：低秩逼近问题在科学领域普遍存在。传统上，由于存在高效构建近似的方法，这一问题通常在酉不变范数（如Frobenius范数或谱范数）下求解。然而，最新研究揭示了切比雪夫范数下低秩逼近的潜力，该范数自然出现在许多应用中。本文探讨了在切比雪夫范数下构建最优秩1逼近的问题。我们研究了用于构建低秩逼近的交替最小化算法的性质，并展示了如何利用该算法构建最优秩1逼近。最终，我们提出了一种能够对小规模矩阵在切比雪夫范数下构建最优秩1逼近的算法。