Low-rank approximations with applications to principal singular component learning systems

In this paper, we present several dynamical systems for efficient and accurate computation of optimal low rank approximation of a real matrix. The proposed dynamical systems are gradient flows or weighted gradient flows derived from unconstrained optimization of certain objective functions. These systems are then modified to obtain power-like methods for computing a few dominant singular triplets of very large matrices simultaneously rather than just one at a time, by incorporating upper-triangular and diagonal matrices. The validity of the proposed algorithms was demonstrated through numerical experiments.