Refining uniform approximation algorithm for low-rank Chebyshev embeddings

Nowadays, low-rank approximations are a critical component of many numerical procedures. Traditionally the problem of low-rank approximation of matrices is solved in unitary invariant norms such as Frobenius or spectral norm due to the existence of efficient methods for constructing approximations. However, recent results discover the potential of low-rank approximations in the Chebyshev norm, which naturally arises in many applications. In this paper, we investigate the problem of uniform approximation of vectors, which is the main component in the low-rank approximations of matrices in the Chebyshev norm. The principal novelty of this paper is the accelerated algorithm for solving uniform approximation problems. We also analyze the iterative procedure of the proposed algorithm and demonstrate that it has a geometric convergence rate. Finally, we provide an extensive numerical evaluation, which demonstrates the effectiveness of the proposed procedures.