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。由于存在构建近似的高效方法，该问题通常在西不变范数（如Frobenius范数或谱范数）下求解。然而，最新研究揭示了Chebyshev范数在低秩逼近中的潜力，该范数自然出现在众多应用中。本文致力解决Chebyshev范数下的最优秩1逼近问题。我们探究交替最小化算法构建低秩逼近的特性，并展示如何利用该算法构造最优秩1逼近。最终提出一种能够针对小规模矩阵构建Chebyshev范数最优秩1逼近的算法。