Regularized Computation of Approximate Pseudoinverse of Large Matrices Using Low-Rank Tensor Train Decompositions

We propose a new method for low-rank approximation of Moore-Penrose pseudoinverses (MPPs) of large-scale matrices using tensor networks. The computed pseudoinverses can be useful for solving or preconditioning large-scale overdetermined or underdetermined systems of linear equations. The computation is performed efficiently and stably based on the modified alternating least squares (MALS) scheme using lowrank tensor train (TT) decomposition. The large-scale optimization problem is reduced to sequential smaller-scale problems for which any standard and stable algorithms can be applied. Regularization technique is further introduced in order to alleviate illposedness and obtain low-rank solutions. Numerical simulation results illustrate that the pseudoinverses of a wide class of nonsquare or nonsymmetric matrices admit good approximate low-rank TT approximations. It is demonstrated that the computational cost of the proposed method is only logarithmic in the matrix size given that the TTranks of a data matrix and its approximate pseudoinverse are bounded. It is illustrated that a strongly nonsymmetric convection-diffusion problem can be efficiently solved by using the preconditioners computed by the proposed method.