  Vector Approximate Message Passing (VAMP) provides the means of solving a
linear inverse problem in a Bayes-optimal way assuming the measurement operator
is sufficiently random. However, VAMP requires implementing the linear minimum
mean squared error (LMMSE) estimator at every iteration, which makes the
algorithm intractable for large-scale problems. In this work, we present a
class of warm-started (WS) methods that provides a scalable approximation of
LMMSE within VAMP. We show that a Message Passing (MP) algorithm equipped with
a method from this class can converge to the fixed point of VAMP while having a
per-iteration computational complexity proportional to that of AMP.
Additionally, we provide the Onsager correction and a multi-dimensional State
Evolution for MP utilizing one of the WS methods. Lastly, we show that the
approximation approach used in the recently proposed Memory AMP (MAMP)
algorithm is a special case of the developed class of WS methods.
