  We consider the problem of signal estimation in a generalized linear model
(GLM). GLMs include many canonical problems in statistical estimation, such as
linear regression, phase retrieval, and 1-bit compressed sensing. Recent work
has precisely characterized the asymptotic minimum mean-squared error (MMSE)
for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there
is a significant gap between the MMSE and the performance of the best known
feasible estimators. In this work, we address this issue by considering GLMs
defined via spatially coupled sensing matrices. We propose an efficient
approximate message passing (AMP) algorithm for estimation and prove that with
a simple choice of spatially coupled design, the MSE of a carefully tuned AMP
estimator approaches the asymptotic MMSE in the high-dimensional limit. To
prove the result, we first rigorously characterize the asymptotic performance
of AMP for a GLM with a generic spatially coupled design. This characterization
is in terms of a deterministic recursion (`state evolution') that depends on
the parameters defining the spatial coupling. Then, using a simple spatially
coupled design and a judicious choice of functions for the AMP algorithm, we
analyze the fixed points of the resulting state evolution and show that it
achieves the asymptotic MMSE. Numerical results for phase retrieval and
rectified linear regression show that spatially coupled designs can yield
substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when
used with AMP algorithms.
