A Riemannian perspective on matrix recovery and constrained optimization - Goyens, F (2021)
Nonlinear matrix recovery is an emerging paradigm in which specific classes of high-rank matrices can be recovered from an underdetermined linear system of measurements. In particular, we consider matrices whose columns, seen as data points, belong to an algebraic variety, namely, a set defined by finitely many polynomial equations. The nonlinear structure is exploited using an appropriately chosen feature map, which induces a low-rank structure in the feature space, thus allowing recovery of the original (high-rank) matrix. The same framework is applicable to matrices whose columns are divided in clusters, so that we can address clustering tasks with missing entries.