  Approximate message passing (AMP) type algorithms have been widely used in
the signal reconstruction of certain large random linear systems. A key feature
of the AMP-type algorithms is that their dynamics can be correctly described by
state evolution. While state evolution is a useful analytic tool, its
convergence is not guaranteed. To solve the convergence problem of the state
evolution of AMP-type algorithms in principle, this paper proposes a
sufficient-statistic memory AMP (SS-MAMP) algorithm framework under the
conditions of right-unitarily invariant sensing matrices, Lipschitz-continuous
local processors and the sufficient-statistic constraint (i.e., the current
message of each local processor is a sufficient statistic of the signal vector
given the current and all preceding messages). We show that the covariance
matrices of SS-MAMP are L-banded and convergent, which is an optimal framework
(from the local MMSE/LMMSE perspective) for AMP-type algorithms given the
Lipschitz-continuous local processors. Given an arbitrary MAMP, we can
construct an SS-MAMP by damping, which not only ensures the convergence of the
state evolution, but also preserves the orthogonality, i.e., its dynamics can
be correctly described by state evolution. As a byproduct, we prove that the
Bayes-optimal orthogonal/vector AMP (BO-OAMP/VAMP) is an SS-MAMP. As an
example, we construct a sufficient-statistic Bayes-optimal MAMP (SS-BO-MAMP)
whose state evolution converges to the minimum (i.e., Bayes-optimal) mean
square error (MSE) predicted by replica methods when it has a unique fixed
point. In addition, the MSE of SS-BO-MAMP is not worse than the original
BO-MAMP. Finally, simulations are provided to support the theoretical results.
