Efficient computations in matrix factorization with missing components

The paper is devoted to the effective implementation of matrix factorization in the presence of missing components into a product of two lower rank matrices. The problem of estimating the parameters of the adopted data model is solved by multidimensional optimization. In practice, the large sizes of the matrices and vectors included in iterative algorithms give rise to the curse of dimensionality. It is proposed to drastically reduce the complexity of matrix operations by presenting them in block-diagonal form. The article substantiates the possibility of casting individual matrices to a block-diagonal form and describes the rules for block-by-block singular value decomposition of matrices. The results of block-by-block processing are illustrated by the example of data matrix factorization of different sizes and with different probabilities of missing components. The time for estimating parameters can be reduced by several orders of magnitude compared to the processing of matrices in the usual representation.