Recent zbMATH articles in MSC 41A52https://www.zbmath.org/atom/cc/41A522022-05-16T20:40:13.078697ZWerkzeugComplex best \(r\)-term approximations almost always exist in finite dimensionshttps://www.zbmath.org/1483.410102022-05-16T20:40:13.078697Z"Qi, Yang"https://www.zbmath.org/authors/?q=ai:qi.yang"MichaĆek, Mateusz"https://www.zbmath.org/authors/?q=ai:michalek.mateusz"Lim, Lek-Heng"https://www.zbmath.org/authors/?q=ai:lim.lek-hengSummary: We show that in finite-dimensional nonlinear approximations, the best \(r\)-term approximant of a function \(f\) almost always exists over \(\mathbb{C}\) but that the same is not true over \(\mathbb{R}\), i.e., the infimum \(\inf_{f_1, \dots, f_r \in D} \| f - f_1 - \dots - f_r \|\) is almost always attainable by complex-valued functions \(f_1, \ldots, f_r\) in \(D\), a set (dictionary) of functions (atoms) with some desired structures. Our result extends to functions that possess properties like symmetry or skew-symmetry under permutations of arguments. When \(D\) is the set of separable functions, this is the best rank-\(r\) tensor approximation problem. We show that over \(\mathbb{C}\), any tensor almost always has a unique best rank-\(r\) approximation. This extends to other notions of ranks such as symmetric and alternating ranks, to best \(r\)-block-terms approximations, and to best approximations by tensor networks. Applied to sparse-plus-low-rank approximations, we obtain that for any given \(r\) and \(k\), a general tensor has a unique best approximation by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with a fixed sparsity pattern; a problem arising in covariance estimation of Gaussian model with \(k\) observed variables conditionally independent given \(r\) hidden variables. The existential (but not uniqueness) part of our result also applies to best approximations by a sum of a rank-\(r\) tensor and a \(k\)-sparse tensor with no fixed sparsity pattern, and to tensor completion problems.