[Paper Review] A strong restricted isometry property, with an application to phaseless compressed sensing

Vladislav Voroninski, Zhiqiang Xu|arXiv (Cornell University)|Apr 15, 2014

Sparse and Compressive Sensing Techniques28 references3 citations

TL;DR

This paper introduces a Strong Restricted Isometry Property (SRIP) to enable exact recovery of $k$-sparse signals from phaseless compressed sensing measurements via $ε$-norm minimization. It proves that random Gaussian matrices with $O(k\log(n/k))$ measurements satisfy SRIP with high probability, ensuring that $\ell_1$ minimization over $|Ax| = |Ax_0|$ recovers $x_0$ up to global sign, extending standard compressed sensing to phaseless settings.

ABSTRACT

The many variants of the restricted isometry property (RIP) have proven to be crucial theoretical tools in the fields of compressed sensing and matrix completion. The study of extending compressed sensing to accommodate phaseless measurements naturally motivates a strong notion of restricted isometry property (SRIP), which we develop in this paper. We show that if $A \in \mathbb{R}^{m imes n}$ satisfies SRIP and phaseless measurements $|Ax_0| = b$ are observed about a $k$-sparse signal $x_0 \in \mathbb{R}^n$, then minimizing the $\ell_1$ norm subject to $ |Ax| = b $ recovers $x_0$ up to multiplication by a global sign. Moreover, we establish that the SRIP holds for the random Gaussian matrices typically used for standard compressed sensing, implying that phaseless compressed sensing is possible from $O(k \log (n/k))$ measurements with these matrices via $\ell_1$ minimization over $|Ax| = b$. Our analysis also yields an erasure robust version of the Johnson-Lindenstrauss Lemma.

Motivation & Objective

• To develop a strong restricted isometry property (SRIP) tailored for phaseless compressed sensing, where only magnitude measurements $|Ax_0|$ are available.
• To establish conditions under which minimizing the $\ell_1$-norm subject to $|Ax| = |Ax_0|$ recovers the original $k$-sparse signal $x_0$ up to global sign.
• To show that random Gaussian matrices with $m = O(k\log(n/k))$ measurements satisfy SRIP with high probability, enabling efficient and provably exact recovery.
• To derive an erasure-robust version of the Johnson-Lindenstrauss Lemma as a byproduct of the analysis.

Proposed method

• Introduce the Strong Restricted Isometry Property (SRIP) as a stronger variant of standard RIP, requiring that all sufficiently large submatrices of $A$ preserve $\ell_2$-norms of sparse vectors within a tight range.
• Prove that if $A$ satisfies SRIP of order $k$ with parameters $\theta_-, \theta_+$, then $\ell_1$ minimization over $|Ax| = |Ax_0|$ recovers $x_0$ up to global sign.
• Use concentration of measure inequalities and tail bounds to establish that $m \times n$ Gaussian matrices with $m = O(k\log(n/k))$ satisfy SRIP with high probability.
• Analyze the structure of sign patterns $\epsilon \in \{1,-1\}^m$ for which $Ax = b_\epsilon$ has solutions, and show that only $x = \pm x_0$ can achieve minimal $\ell_1$-norm under SRIP.
• Leverage known results on $\ell_1$-minimization under standard RIP to extend them to the phaseless setting via submatrix analysis and sign consistency.
• Derive an erasure-robust Johnson-Lindenstrauss Lemma, showing that dimensionality reduction remains approximately distance-preserving even after erasing a constant fraction of coordinates.

Experimental results

Research questions

• RQ1 Can $\ell_1$ minimization over the non-convex constraint set $|Ax| = |Ax_0|$ exactly recover a $k$-sparse signal $x_0$ from phaseless measurements?
• RQ2 What stronger isometry condition than standard RIP is required to guarantee exact recovery in phaseless compressed sensing?
• RQ3 Do random Gaussian matrices with $O(k\log(n/k))$ measurements satisfy this stronger condition with high probability?
• RQ4 Can the Johnson-Lindenstrauss Lemma be strengthened to remain valid under random coordinate erasures?
• RQ5 Is it possible to extend the phaseless recovery framework to complex-valued signals with the same measurement complexity?

Key findings

• The Strong Restricted Isometry Property (SRIP) ensures that $\ell_1$ minimization over $|Ax| = |Ax_0|$ recovers $x_0$ up to global sign, provided $A$ satisfies SRIP of order $k$ with appropriate parameters $\theta_-, \theta_+$.
• Random $m \times n$ Gaussian matrices with $m = O(k\log(n/k))$ satisfy SRIP of order $k$ with high probability, enabling provable recovery from $O(k\log(n/k))$ phaseless measurements.
• The analysis yields an erasure-robust version of the Johnson-Lindenstrauss Lemma, where dimensionality reduction preserves distances even after erasing a positive fraction of coordinates.
• For any $k$-sparse signal $x_0$, the solution to $\min \|x\|_1 \text{ s.t. } |Ax| = |Ax_0|$ is $x = \pm x_0$ with high probability when $A$ is a Gaussian matrix with $m = O(k\log(n/k))$.
• The proof relies on showing that for any sign pattern $\epsilon$, the solution $x_\epsilon$ to $Ax = b_\epsilon$ satisfies $\|x_\epsilon\|_1 \geq \|x_0\|_1$, with equality only if $x_\epsilon = \pm x_0$, under SRIP.
• The result establishes that phaseless compressed sensing is possible with the same $O(k\log(n/k))$ measurement complexity as standard compressed sensing, under the SRIP condition.