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Bisparse Blind Deconvolution through Hierarchical Sparse Recovery

Axel Flinth, Ingo Roth, Gerhard Wunder

TL;DR

The HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied and the approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the hierarchical sparsity framework.

Abstract

The hierarchical sparsity framework, and in particular the HiHTP algorithm, has been successfully applied to many relevant communication engineering problems recently, particularly when the signal space is hierarchically structured. In this paper, the applicability of the HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied. The bi-sparse blind deconvolution setting here consists of recovering $h$ and $b$ from the knowledge of $h*(Qb)$, where $Q$ is some linear operator, and both $b$ and $h$ are both assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the \emph{hierarchical sparsity framework}. %In particular, the efficient HiHTP algorithm is proposed for performing the recovery. Then, for a Gaussian draw of the random matrix $Q$, it is theoretically shown that an $s$-sparse $h \in \mathbb{K}^μ$ and $σ$-sparse $b \in \mathbb{K}^n$ with high probability can be recovered when $μ\succcurlyeq s\log(s)^2\log(μ)\log(μn) + sσ\log(n)$.

Bisparse Blind Deconvolution through Hierarchical Sparse Recovery

TL;DR

The HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied and the approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the hierarchical sparsity framework.

Abstract

The hierarchical sparsity framework, and in particular the HiHTP algorithm, has been successfully applied to many relevant communication engineering problems recently, particularly when the signal space is hierarchically structured. In this paper, the applicability of the HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied. The bi-sparse blind deconvolution setting here consists of recovering and from the knowledge of , where is some linear operator, and both and are both assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the \emph{hierarchical sparsity framework}. %In particular, the efficient HiHTP algorithm is proposed for performing the recovery. Then, for a Gaussian draw of the random matrix , it is theoretically shown that an -sparse and -sparse with high probability can be recovered when .
Paper Structure (26 sections, 17 theorems, 122 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 26 sections, 17 theorems, 122 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. HiHTP
  3. Blind deconvolution
  4. The main result
  5. Extensions
  6. Outline of the paper
  7. Notation
  8. Literature Review
  9. Lifted approaches
  10. Direct approaches
  11. Conclusion
  12. Bisparse blind deconvolution with HiHTP
  13. Deconvolution and demixing.
  14. Motivation
  15. Multi-level hierarchical sparsity
  16. ...and 11 more sections

Key Result

Theorem 1

hiHTP If $\delta_{3s,2\sigma}<1/\sqrt{3}$, the sequence $w^k$ defined by HiHTP for a $y = \mathcal{A} w_0 + e$ for a $(s,\sigma)$-sparse ground truth $w_0$ and error vector $e\in \mathbb{K}^m$ satisfies with $\rho = (2\delta_{3s,2\sigma}/(1-\delta_{2s,2\sigma}^2))^{1/2}<1$, $\tau = 5.15/(1-\rho)$. In particular, when the measurements $y$ are noise-free, $w^k$ converges towards $w_0$ at a linear r

Figures (3)

  • Figure 1: Two tensors which are both $9$-sparse. However, only the left one is hierarchically sparse -- it is $(3,3)$-sparse.
  • Figure 2: Recovery probability plots over $s=1,2,\dots,7$ and $\mu=10,20,\dots,250$ for different values of $n$ and $\sigma$.
  • Figure 3: The empirical recovery probabilities and fitted logistic curves plotted against the optimal predictor $\lambda_{(0,1,0)}$(left) and the one suggested by Theorem \ref{['th:main_result']}$\lambda_{(1,1,2)}$ (right). The vertical axes have been cropped to highlight the phase transition region.

Theorems & Definitions (43)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • Definition 3
  • Corollary 3: Blind Deconvolution and demixing guarantee
  • ...and 33 more