Phase Transitions in Phase-Only Compressed Sensing

TL;DR

This work analyzes phase-only compressed sensing (PO-CS), where phase measurements $\mathbf{z}=\mathrm{sign}(\mathbf{\Phi x})$ enable recovery of structured signals via a linearized constrained norm-minimization. It establishes an asymptotically exact phase-transition threshold $\zeta_{\mathrm{PO}}(\mathbf{x};f)$ governing successful recovery and proves, using the Gaussian min-max theorem and a near-Gaussian reformulation of the sensing matrix, that the PO transition occurs at or below the corresponding linear CS threshold $\zeta_{\mathrm{LN}}$. The authors derive explicit asymptotically exact formulas for PO-CS in two canonical settings—sparse vectors with $\ell_1$ regularization and low-rank matrices with the nuclear norm—showing that $\zeta_{\mathrm{PO}}$ typically differs from $\zeta_{\mathrm{LN}}$, often significantly, especially for highly structured signals. They also provide simulations illustrating that PO-CS can attain exact recovery with substantially fewer phase measurements than linear measurements, and quantify the ratio of PO to LN thresholds across regimes. Overall, the results offer precise phase-transition characterizations and practical guidance for designing phase-only sensing systems for structured signals.

Abstract

The goal of phase-only compressed sensing is to recover a structured signal $\mathbf{x}$ from the phases $\mathbf{z} = {\rm sign}(\mathbfΦ\mathbf{x})$ under some complex-valued sensing matrix $\mathbfΦ$. Exact reconstruction of the signal's direction is possible: we can reformulate it as a linear compressed sensing problem and use basis pursuit (i.e., constrained norm minimization). For $\mathbfΦ$ with i.i.d. complex-valued Gaussian entries, this paper shows that the phase transition is approximately located at the statistical dimension of the descent cone of a signal-dependent norm. Leveraging this insight, we derive asymptotically precise formulas for the phase transition locations in phase-only sensing of both sparse signals and low-rank matrices. Our results prove that the minimum number of measurements required for exact recovery is smaller for phase-only measurements than for traditional linear compressed sensing. For instance, in recovering a 1-sparse signal with sufficiently large dimension, phase-only compressed sensing requires approximately 68% of the measurements needed for linear compressed sensing. This result disproves earlier conjecture suggesting that the two phase transitions coincide. Our proof hinges on the Gaussian min-max theorem and the key observation that, up to a signal-dependent orthogonal transformation, the sensing matrix in the reformulated problem behaves as a nearly Gaussian matrix.

Figures (4)

• Figure 1: We fix $n=1000$ and plot the (approximate) curves of $\zeta_{\rm PO}/\zeta_{\rm LN}$ v.s. $s=1:1000$ under $\|\mathbf{x}\|_1=\sqrt{s}$, $0.7\sqrt{s}+0.3,~0.3\sqrt{s}+0.7$. These curves are monotonically increasing.
• Figure 2: The left figure plots $R_{\rm sp}(u,1)$ and $R_{\rm sp}(u,0.6)$, showing $\lim_{u\to 0^+}R_{\rm sp}(u,1)\approx 0.678$ and $\lim_{u\to 0^+}R_{\rm sp}(u,0.6)\approx 0.808$; the former indicates that for recovering $s$-sparse $\mathbf{x}\in\mathbb{S}^{n-1}$ with nonzero entries being $\pm 1/\sqrt{s}$, if $s$ is fixed and $n\to \infty$, then PO-CS requires no more than $0.68 \zeta_{\rm LN}(\mathbf{x};\|\cdot\|_1)$ phases to succeed, as we highlighted in the abstract. Similarly, the right figure plots $R_{\rm lr}(u,1,1)$ and $R_{\rm lr}(u,1,0.6)$, and we further report $\lim_{u\to 0^+} R_{\rm lr}(u,1,1)\approx 0.758$ and $\lim_{u\to 0^+} R_{\rm lr}(u,1,0.6)\approx 0.856$.
• Figure 3: The left figure shows that the empirical phase transitions of recovering equal amplitude sparse vectors in $\mathbb{R}^{100}$ are consistent with $\hat{\zeta}_{\rm PO}(\mathbf{x};\|\cdot\|_1)$. The right figure shows the empirical success rates of recovering $9$-sparse signals in $\mathbb{R}^{300}$ with $\ell_1$-norm varying between $[1.1,3]$, confirming earlier phase transitions under larger $\|\mathbf{x}\|_1$.
• Figure 4: The left figure shows that the empirical phase transitions of recovering low-rank $30\times 30$ matrices can be precisely predicted by $pq\Psi(\frac{r}{p},\frac{p}{q},\frac{\|\mathbf{X}\|_{nu}^2}{r})$. The right figure shows the empirical success rates of recovering rank-$2$$30\times 30$ matrices with nuclear norm varying between $[1.05,\sqrt{2}]$, confirming earlier phase transitions under larger $\|\mathbf{X}\|_{nu}$.

Theorems & Definitions (12)

• Lemma 1: Conditions for success and failure
• Lemma 2
• Lemma 3: Nearly-Gaussianity of $\mathbf{A}_{\mathbf{z}}$
• Theorem 1
• proof : Proof Sketch
• Proposition 1
• Remark 1
• Theorem 2
• Theorem 3
• Remark 2