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Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors

Wei Zhang, Zhenni Wang

Abstract

The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices.

Restricted Isometry Property of Rank-One Measurements with Random Unit-Modulus Vectors

Abstract

The restricted isometry property (RIP) is essential for the linear map to guarantee the successful recovery of low-rank matrices. The existing works show that the linear map generated by the measurement matrices with independent and identically distributed (i.i.d.) entries satisfies RIP with high probability. However, when dealing with non-i.i.d. measurement matrices, such as the rank-one measurements, the RIP compliance may not be guaranteed. In this paper, we show that the RIP can still be achieved with high probability, when the rank-one measurement matrix is constructed by the random unit-modulus vectors. Compared to the existing works, we first address the challenge of establishing RIP for the linear map in non-i.i.d. scenarios. As validated in the experiments, this linear map is memory-efficient, and not only satisfies the RIP but also exhibits similar recovery performance of the low-rank matrices to that of conventional i.i.d. measurement matrices.
Paper Structure (16 sections, 11 theorems, 66 equations, 5 figures)

This paper contains 16 sections, 11 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. RIP Analysis of Linear Map
  3. Sufficient Condition of RIP
  4. Inequalities of Tail Bounds
  5. Connection with the All-One Matrix
  6. RIP of Rank-One Unit-Modulus Measurements
  7. Numerical Experiments
  8. Conclusion
  9. Proof of \ref{['lemma:all one ineq']}
  10. Proof of Preliminaries
  11. Proof of \ref{['lem:poly max']}
  12. Proof of \ref{['lem: vector case']}
  13. Proof of \ref{['lem: vector case c']}
  14. Proof of \ref{['lem:holder N2']}
  15. Proof of \ref{['lem:holder N general']}
  16. ...and 1 more sections

Key Result

Theorem 1

Let ${\mathcal{A}}(\cdot):{\mathbb{C}}^{M \times N} \rightarrow {\mathbb{C}}^{K \times 1}$ be a linear map with random measurement matrices obeying the following condition: for any given $\mathbf{X} \in {\mathbb{C}}^{M \times N}$ and any fixed $0<\alpha<1$ holds for fixed constants $C,c>0$ (which may depend on $\alpha$). Then if $K\ge D\max\{M,N \}r$, the linear map ${\mathcal{A}}(\cdot)$ satisfie

Figures (5)

  • Figure 1: Probability Density Function of $\| {\mathcal{A}}(\mathbf{X})\|_2^2$ with Different Number of Measurements
  • Figure 2: Comparison of Two Expectations: All-One Matrix Scenario $\frac{1}{M^tN^t} {\mathbb{E}}[ |\mathbf{u}^H \bm{1} \bm{1}^T\mathbf{v}|^{2t}]$ and Arbitrary Matrix Scenario ${\mathbb{E}}[ |\mathbf{u}^H \mathbf{X}\mathbf{v}|^{2t}]$
  • Figure 3: Recovery Error of Low-Rank Matrix by Using Rank-One Unit-Modulus and i.i.d. Gaussian Measurements with Different Number of Measurements
  • Figure 4: Recovery Error of Alternating Minimization Method by Using Rank-One Unit-Modulus and i.i.d. Gaussian Measurements with Different Number of Measurements
  • Figure 5: Recovery Error of Gradient-Based Method by Using Rank-One Unit-Modulus and i.i.d. Gaussian Measurements with Different Number of Measurements

Theorems & Definitions (13)

  • Definition 1: Standard RIP over Low-Rank Matrices candes2011tight
  • Theorem 1: candes2011tight, Theorem 2.3
  • Remark 1: recht2010guaranteedcandes2011tight
  • Theorem 2
  • Lemma 1
  • Corollary 1
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 3 more