Best low-rank approximations and Kolmogorov n-widths

We relate the problem of best low-rank approximation in the spectral norm for a matrix A to Kolmogorov n-widths and corresponding optimal spaces. We characterize all the optimal spaces for the image of the Euclidean unit ball under A, and we show that any orthonormal basis in an n-dimensional optimal space generates a best rank-n approximation to A. We also present a simple and explicit construction to obtain a sequence of optimal n-dimensional spaces once an initial optimal space is known. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.