On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices
This work provides theoretical guarantees for low-rank approximation in structured matrices, benefiting numerical linear algebra and approximation theory communities.
The paper shows that for symmetric semidefinite or diagonally dominant matrices, a maximum volume submatrix can always be chosen as a principal submatrix. It provides new error bounds for cross approximation with complete pivoting, proving that for doubly diagonally dominant matrices the error remains within a modest factor of the best approximation error.
The problem of finding a $k \times k$ submatrix of maximum volume of a matrix $A$ is of interest in a variety of applications. For example, it yields a quasi-best low-rank approximation constructed from the rows and columns of $A$. We show that such a submatrix can always be chosen to be a principal submatrix if $A$ is symmetric semidefinite or diagonally dominant. Then we analyze the low-rank approximation error returned by a greedy method for volume maximization, cross approximation with complete pivoting. Our bound for general matrices extends an existing result for symmetric semidefinite matrices and yields new error estimates for diagonally dominant matrices. In particular, for doubly diagonally dominant matrices the error is shown to remain within a modest factor of the best approximation error. We also illustrate how the application of our results to cross approximation for functions leads to new and better convergence results.