  We adopt data structure in the form of cover trees and iteratively apply
approximate nearest neighbour (ANN) searches for fast compressed sensing
reconstruction of signals living on discrete smooth manifolds. Levering on the
recent stability results for the inexact Iterative Projected Gradient (IPG)
algorithm and by using the cover tree's ANN searches, we decrease the
projection cost of the IPG algorithm to be logarithmically growing with data
population for low dimensional smooth manifolds. We apply our results to
quantitative MRI compressed sensing and in particular within the Magnetic
Resonance Fingerprinting (MRF) framework. For a similar (or sometimes better)
reconstruction accuracy, we report 2-3 orders of magnitude reduction in
computations compared to the standard iterative method which uses brute-force
searches.
