  For the large-scale linear discrete ill-posed problem $\min\|Ax-b\|$ or
$Ax=b$ with $b$ contaminated by Gaussian white noise, there are four commonly
used Krylov solvers: LSQR and its mathematically equivalent CGLS, the Conjugate
Gradient (CG) method applied to $A^TAx=A^Tb$, CGME, the CG method applied to
$\min\|AA^Ty-b\|$ or $AA^Ty=b$ with $x=A^Ty$, and LSMR, the minimal residual
(MINRES) method applied to $A^TAx=A^Tb$. These methods have intrinsic
regularizing effects, where the number $k$ of iterations plays the role of the
regularization parameter. In this paper, we establish a number of
regularization properties of CGME and LSMR, including the filtered SVD
expansion of CGME iterates, and prove that the 2-norm filtering best
regularized solutions by CGME and LSMR are less accurate than and at least as
accurate as those by LSQR, respectively. We also prove that the
semi-convergence of CGME and LSMR always occurs no later and sooner than that
of LSQR, respectively. As a byproduct, using the analysis approach for CGME, we
improve a fundamental result on the accuracy of the truncated rank $k$
approximate SVD of $A$ generated by randomized algorithms, and reveal how the
truncation step damages the accuracy. Numerical experiments justify our results
on CGME and LSMR.
