The partial pivoted Cholesky approximation accurately represents matrices that are close to being low-rank. Meanwhile, the Vecchia approximation accurately represents matrices with inverse Cholesky factors that are close to sparse. What happens if a partial Cholesky approximation is combined with a Vecchia approximation of the residual? The sum can be rewritten as a Vecchia approximation of the original matrix with an amplified sparsity pattern. Thus, the Vecchia approximation is a superset for other factored matrix approximations and it has broad applicability.