  In this work we propose a new paradigm for designing fast plug-and-play (PnP)
algorithms using dimensionality reduction techniques. Unlike existing
approaches which utilize stochastic gradient iterations for acceleration, we
propose novel multi-stage sketched gradient iterations which first perform
downsampling dimensionality reduction in the image space, and then efficiently
approximate the true gradient using the sketched gradient in the
low-dimensional space. This sketched gradient scheme can also be naturally
combined with PnP-SGD methods for further improvement on computational
complexity. As a generic acceleration scheme, it can be applied to accelerate
any existing PnP/RED algorithm. Our numerical experiments on X-ray fan-beam CT
demonstrate the remarkable effectiveness of our scheme, that a computational
free-lunch can be obtained using this dimensionality reduction in the image
space.
