  The Recently proposed Vector Approximate Message Passing (VAMP) algorithm
demonstrates a great reconstruction potential at solving compressed sensing
related linear inverse problems. VAMP provides high per-iteration improvement,
can utilize powerful denoisers like BM3D, has rigorously defined dynamics and
is able to recover signals measured by highly undersampled and ill-conditioned
linear operators. Yet, its applicability is limited to relatively small problem
sizes due to the necessity to compute the expensive LMMSE estimator at each
iteration. In this work we consider the problem of upscaling VAMP by utilizing
Conjugate Gradient (CG) to approximate the intractable LMMSE estimator. We
propose a rigorous method for correcting and tuning CG withing CG-VAMP to
achieve a stable and efficient reconstruction. To further improve the
performance of CG-VAMP, we design a warm-starting scheme for CG and develop
theoretical models for the Onsager correction and the State Evolution of
Warm-Started CG-VAMP (WS-CG-VAMP). Additionally, we develop robust and accurate
methods for implementing the WS-CG-VAMP algorithm. The numerical experiments on
large-scale image reconstruction problems demonstrate that WS-CG-VAMP requires
much fewer CG iterations compared to CG-VAMP to achieve the same or superior
level of reconstruction.
