Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces

We propose and justify a matrix reduction method for calculating the optimal approximation of an observed matrix A ∈ Cm × n by a sum ∑ᵢ₌₁ᵖ ∑ⱼ₌₁q BᵢXᵢⱼCⱼ of matrix products where each Bᵢ ∈ Cm × gᵢ and Cⱼ ∈ Chⱼ × n is known and where the unknown matrix kernels Xᵢⱼ are determined by minimizing the Frobenius norm of the error. The sum can be represented as a bounded linear mapping $BXC$ with unknown kernel X from a prescribed subspace T ⊆ Cⁿ onto a prescribed subspace S ⊆ Cᵐ defined, respectively, by the collective domains and ranges of the given matrices C₁,…,Cq and B₁,…,Bₚ. We show that the optimal kernel is X = B^AC^ and that the optimal approximation BB^AC^C is the projection of the observed mapping A onto a mapping from T to S. If A is large, B and C may also be large and direct calculation of B^ and C^ becomes unwieldy and inefficient. The proposed method avoids this difficulty by reducing the solution process to finding the pseudo-inverses of a collection of much smaller matrices. This significantly reduces the computational burden.