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Living near the edge: A lower-bound on the phase transition of total variation minimization

Daei, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1109/TIT.2019.2940673
  3. Publisher: Institute of Electrical and Electronics Engineers Inc , 2020
  4. Abstract:
  5. This work is about the total variation (TV) minimization which is used for recovering gradient-sparse signals from compressed measurements. Recent studies indicate that TV minimization exhibits a phase transition behavior from failure to success as the number of measurements increases. In fact, in large dimensions, TV minimization succeeds in recovering the gradient-sparse signal with high probability when the number of measurements exceeds a certain threshold; otherwise, it fails almost certainly. Obtaining a closed-form expression that approximates this threshold is a major challenge in this field and has not been appropriately addressed yet. In this work, we derive a tight lower-bound on this threshold in case of any random measurement matrix whose null space is distributed uniformly with respect to the Haar measure. In contrast to the conventional TV phase transition results that depend on the simple gradient-sparsity level, our bound is highly affected by generalized notions of gradient-sparsity. Our proposed bound is very close to the true phase transition of TV minimization confirmed by simulation results. © 1963-2012 IEEE
  6. Keywords:
  7. Sample complexity ; Statistical dimension ; Image denoising ; Closed-form expression ; High probability ; Large dimensions ; Random measurement ; Sparse signals ; Total variation ; Total variation minimization ; Transition behavior ; Signal reconstruction
  8. Source: IEEE Transactions on Information Theory ; Volume 66, Issue 5 , 2020 , Pages 3261-3267
  9. URL: https://ieeexplore.ieee.org/document/8832269