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Instance optimality in phase retrieval

Yu Xia, Zhiqiang Xu

TL;DR

This work extends instance optimality from linear compressive sensing to phase retrieval with phaseless measurements. By introducing a phaseless bi-Lipschitz condition and studying $\ell_p$-minimization decoders, the authors show that, for real and complex signals, one can achieve $(2,1)$ and $(1,1)$ instance optimality with $m=O(k\log(n/k))$ measurements using $|\boldsymbol{A}\boldsymbol{x}|$, and provide a non-uniform $(2,2)$-instance optimality guarantee for a fixed vector in probability. They prove that uniform $(2,2)$-instance optimality is impossible when $m\ll n$, but establish non-uniform results for fixed vectors and high-probability recovery with Gaussian measurements. Overall, the results reveal strong parallels between phase retrieval and classical compressed sensing, via a phaseless bi-Lipschitz framework and rigid bounds for sparse recovery under phaseless observations.

Abstract

Compressed sensing has demonstrated that a general signal $\boldsymbol{x} \in \mathbb{F}^n$ ($\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$) can be estimated from few linear measurements with an error {proportional to} the best $k$-term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the $\ell_p$-minimization decoder, where $p \in (0, 1]$, for both real and complex cases. More specifically, we prove that $(2,1)$ and $(1,1)$-instance optimality of order $k$ can be achieved with $m =O(k \log(n/k))$ phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately $k$-sparse signals from $m = O(k \log(n/k))$ phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of $(2,2)$-instance optimality result in probability applicable to any fixed vector $\boldsymbol{x} \in \mathbb{F}^n$. These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.

Instance optimality in phase retrieval

TL;DR

This work extends instance optimality from linear compressive sensing to phase retrieval with phaseless measurements. By introducing a phaseless bi-Lipschitz condition and studying -minimization decoders, the authors show that, for real and complex signals, one can achieve and instance optimality with measurements using , and provide a non-uniform -instance optimality guarantee for a fixed vector in probability. They prove that uniform -instance optimality is impossible when , but establish non-uniform results for fixed vectors and high-probability recovery with Gaussian measurements. Overall, the results reveal strong parallels between phase retrieval and classical compressed sensing, via a phaseless bi-Lipschitz framework and rigid bounds for sparse recovery under phaseless observations.

Abstract

Compressed sensing has demonstrated that a general signal () can be estimated from few linear measurements with an error {proportional to} the best -term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the -minimization decoder, where , for both real and complex cases. More specifically, we prove that and -instance optimality of order can be achieved with phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately -sparse signals from phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of -instance optimality result in probability applicable to any fixed vector . These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.
Paper Structure (10 sections, 11 theorems, 94 equations)

This paper contains 10 sections, 11 theorems, 94 equations.

Table of Contents

  1. introduction
  2. Linear measurements
  3. Phaseless measurements
  4. Our contribution
  5. Organization
  6. Main Results
  7. Proof of Theorem \ref{['th:main11']}
  8. Proof of Theorem \ref{['th:lip']}
  9. Proof of Proposition \ref{['thm:bad_22']}
  10. Proof of Theorem \ref{['th:22']}

Key Result

Theorem 2.3

Let $k \in \{1,\ldots,n\}$ be an integer and ${\boldsymbol A} \in \mathbb{F}^{m\times n}$ satisfy the phaseless bi-Lipschitz condition on the set $\mathcal{X}=\{\boldsymbol{x}\in \mathbb{F}^n\ :\ \|\boldsymbol{x}\|_0\leq (r+4)k\}$ with positive constants $L$ and $U$. The constant $r$ can be any posi Then, for $p\in (0,1]$, the following holds for all $\boldsymbol x\in \mathbb F^n$: where $\Delta_

Theorems & Definitions (28)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • Remark 2.8
  • ...and 18 more