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