# Inexact Gradient Projection and Fast Data Driven Compressed Sensing

Research output: Contribution to journalArticlepeer-review

## Abstract

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 language English 6707 - 6721 IEEE Transactions on Information Theory 64 10 28 May 2018 https://doi.org/10.1109/TIT.2018.2841379 Published - Oct 2018

## Keywords

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

## Fingerprint

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• ### Exploiting low dimensional models in sensing, computation and signal processing

Davies, M.

EU government bodies

1/09/1631/08/22

Project: Research

• ### Next Generation Compressive and Computational Sensing and Signal Processing

Davies, M.

UK-based charities

1/10/1630/09/21

Project: Research

• ### CQ-MRI: Compressed Quantitative MRI

Marshall, I. & Davies, M.

EPSRC

1/07/1531/12/18

Project: Research

• ### Multi-shot Echo Planar Imaging for accelerated Cartesian MR Fingerprinting: an alternative to conventional spiral MR Fingerprinting

Benjamin, A. J. V., Gómez, P. A., Golbabaee, M., Mahbub, Z., Sprenger, T., Menzel, M. I., Davies, M. & Marshall, I., 10 May 2019, (E-pub ahead of print) 21 p.

Research output: Contribution to journalArticlepeer-review

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