Low-rank Approximation of Linear Maps

This work provides closed-form solutions and minimal achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of linear bounded operators and p-th Schatten norms previous results obtained in the finite dimensional case for the Frobenius norm. The theorem is illustrated in various settings, including low-rank approximation problems with respect to the trace norm, the 2-induced norm or the Hilbert-Schmidt norm. The theorem provides also the basics for the design of tractable algorithms for kernel-based or continuous DM