Inexact Gradient Projection and Fast Data Driven Compressed Sensing

Mohammad Golbabaee, Michael Davies

Research output: Contribution to journalArticlepeer-review


We study convergence of the iterative projected gradient (IPG) algorithm for arbitrary (possibly nonconvex) sets and when both the gradient and projection oracles are computed approximately. We consider different notions of approximation of which we show that the Progressive Fixed Precision (PFP) and the $(1+\epsilon)$-optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We show that the former scheme is also able to maintain the (linear) rate of convergence of the exact algorithm, under the same embedding assumption. In contrast, the $(1+\epsilon)$-approximate oracle requires a stronger embedding condition, moderate compression ratios and it typically slows down the convergence. We apply our results to accelerate solving a class of data driven compressed sensing problems, where we replace iterative exhaustive searches over large datasets by fast approximate nearest neighbour search strategies based on the cover tree data structure. For datasets with low intrinsic dimensions our proposed algorithm achieves a complexity logarithmic in terms of the dataset population as opposed to the linear complexity of a brute force search. By running several numerical experiments we conclude similar observations as predicted by our theoretical analysis.
Original languageEnglish
Pages (from-to) 6707 - 6721
JournalIEEE Transactions on Information Theory
Issue number10
Early online date28 May 2018
Publication statusPublished - Oct 2018


  • Convergence
  • iterative projected gradient
  • approximate updates
  • Linear convergence
  • compressed sensing
  • constrained least squares
  • data driven models
  • cover trees
  • approximate nearest neighbour search


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