We propose a novel rank-selection criterion for the Krylov-subspace-based filtering techniques such as the well-known multistage Wiener filter. We provide two necessary and sufficient conditions for the low-dimensional Krylov subspace to contain the optimal filter. The first condition is that the subspace is invariant under the transformation by the autocorrelation matrix associated with the subspace itself, and the second condition is its reverse inclusion. We derive two criteria based on the conditions; the criterion based on the first condition coincides with the conventional one, and the one based on the second is the proposed one. Simulation results indicate that the proposed criterion induces a natural relation between the threshold parameter and the average rank.