  We consider the problem of localizing change points in high-dimensional
linear regression. We propose an Approximate Message Passing (AMP) algorithm
for estimating both the signals and the change point locations. Assuming
Gaussian covariates, we give an exact asymptotic characterization of its
estimation performance in the limit where the number of samples grows
proportionally to the signal dimension. Our algorithm can be tailored to
exploit any prior information on the signal, noise, and change points. It also
enables uncertainty quantification in the form of an efficiently computable
approximate posterior distribution, whose asymptotic form we characterize
exactly. We validate our theory via numerical experiments, and demonstrate the
favorable performance of our estimators on both synthetic data and images.
