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Robust Instance Optimal Phase-Only Compressed Sensing

T0 review · 1 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Phase-only compressed sensing achieves uniform instance optimality over the unit sphere with error at most C s^{-1/2} times the l1 approximation error to the nearest s-sparse signal.

desk verdict This paper upgrades the phase-only CS recovery from nonuniform to uniform instance optimality over the whole sphere and adds near-optimal robustness bounds for noise and corruption. read the letter →

arxiv 2408.06275 v4 submitted 2024-08-12 cs.IT eess.SPmath.IT

Junren Chen , Michael K. Ng , Jonathan Scarlett This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords phase-onlycompressedsensinginstanceoptimalityrestrictedisometrypropertyrobustrecoverysparsesignalreconstructionbasispursuituniformguarantees
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that recovering an approximately sparse signal from the phases of its complex Gaussian measurements, via a linearization step followed by quadratically constrained basis pursuit, satisfies a uniform bound on the Euclidean recovery error that holds simultaneously for every signal on the unit sphere. The bound takes the form of a constant times s to the power -1/2 multiplied by the signal's l1 distance to its best s-sparse approximation. This uniform guarantee is obtained by proving that the family of linearized sensing matrices arising from all approximately sparse signals obeys the restricted isometry property at once. The same estimator is additionally shown to degrade gracefully under small dense noise and under adversarial replacement of a fraction of the phase measurements.

What carries the argument

The family of linearized sensing matrices arising from phase-only observations of all approximately sparse signals on the sphere, shown to satisfy the restricted isometry property uniformly.

What would settle it

An explicit construction or numerical instance, for sufficiently large dimension n and sparsity s, of an approximately sparse unit-norm vector x together with its phase-only measurements such that the recovery error ||x^sharp - x||_2 exceeds any fixed multiple of s^{-1/2} σ_{ℓ1}(x, Σ^n_s).

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Extended reading notes

Core claim

There exists a universal constant C such that the estimator x^sharp obtained from phase-only measurements satisfies ||x^sharp - x||_2 ≤ C s^{-1/2} σ_{ℓ1}(x, Σ^n_s) for every x in the unit Euclidean sphere, where σ_{ℓ1}(x, Σ^n_s) denotes the ℓ1 distance from x to its closest s-sparse vector. The proof proceeds by establishing that the new sensing matrices corresponding to all approximately sparse signals satisfy the restricted isometry property simultaneously. The estimator remains robust when dense noise of amplitude at most τ0 is added either before or after phase retention, increasing the error by O(τ0), and when an arbitrary ζ0-fraction of measurements is adversarially corrupted, which in

Load-bearing premise

The linearized sensing matrices for all approximately sparse signals simultaneously satisfy the restricted isometry property.

Editorial extensions

If this is right

  • The recovery procedure yields a uniform instance-optimal guarantee rather than one that holds only for individual fixed signals.
  • Bounded dense noise, whether introduced before or after phase extraction, increases the Euclidean error by a term linear in the noise amplitude.
  • Adversarial corruption of a ζ0 fraction of the phase measurements increases the error by a term of order sqrt(ζ0 log(1/ζ0)).
  • Combining the uniform bound with the two robustness results produces an overall robust instance-optimal guarantee analogous to the standard one for linear compressed sensing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The uniform RIP property over the derived matrices may allow the same analysis technique to be applied to other nonlinear measurement models that admit a linearization step.
  • Because the bound is uniform, it directly supports the use of the estimator inside iterative algorithms that refine sparsity level or support estimates.
  • The near-optimality result for dense noise suggests that phase-only sensing cannot improve upon linear sensing in the presence of additive noise beyond logarithmic factors.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper claims to strengthen nonuniform instance optimality results for phase-only compressed sensing (PO-CS) to a uniform guarantee over the entire unit sphere. Using a linearization procedure that recasts PO-CS as linear compressed sensing followed by quadratically constrained basis pursuit, it proves existence of a universal constant C such that ||x^sharp - x||_2 ≤ C s^{-1/2} σ_ℓ1(x, Σ^n_s) for all x, achieved via simultaneous RIP of the linearized sensing matrices for all approximately sparse signals. It further derives robustness to dense bounded noise (additive O(τ0) error) and adversarial phase corruptions (additive O(√(ζ0 log(1/ζ0))) error), yielding a combined robust instance-optimal bound.

Significance. If the simultaneous RIP holds with constants yielding a dimension-independent C, the uniform instance optimality is a meaningful strengthening of prior nonuniform results in PO-CS and aligns the theory more closely with standard linear CS. The near-optimal robustness bounds (up to log factors) are a further strength, as is the reduction to existing CS tools via linearization. These results would be of interest to the compressed sensing community for both theoretical and applied phase-retrieval settings.

major comments (1)
  1. [Abstract] Abstract and main result: The uniform bound over all x ∈ S^{n-1} rests on the claim that the family of linearized sensing matrices (one per approximately sparse signal) simultaneously satisfy the RIP with constants sufficient for a universal C. This is load-bearing; the manuscript should explicitly state the achieved RIP parameters (δ_s, s) and the argument establishing uniformity over the family (e.g., via a covering or union-bound argument), including how they produce the stated C independent of n.
minor comments (2)
  1. Notation: Define Σ^n_s and σ_ℓ1(x, Σ^n_s) explicitly on first use; the current description in the abstract is clear but the main text should include the precise definition to avoid ambiguity with other sparsity measures.
  2. The robustness statements distinguish noise appearing prior vs. posterior to phase retention; a brief remark on whether the analysis treats these cases identically or requires separate arguments would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation of minor revision, and the constructive comment on clarifying the RIP parameters and uniformity argument. We address the point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main result: The uniform bound over all x ∈ S^{n-1} rests on the claim that the family of linearized sensing matrices (one per approximately sparse signal) simultaneously satisfy the RIP with constants sufficient for a universal C. This is load-bearing; the manuscript should explicitly state the achieved RIP parameters (δ_s, s) and the argument establishing uniformity over the family (e.g., via a covering or union-bound argument), including how they produce the stated C independent of n.

    Authors: We agree that making the RIP parameters and the uniformity argument explicit will improve the presentation. In the revised manuscript we will add the following statement to the abstract and the main theorem: the family of linearized sensing matrices satisfies the RIP of order s with constant δ_s ≤ 1/3 simultaneously for all approximately s-sparse signals on the unit sphere, provided m ≥ C s log(n/s) for a sufficiently large universal C. Uniformity over the family is obtained by a standard ε-net argument on the set of s-sparse vectors (covering number at most (3/ε)^s) together with a union bound over the net; each fixed linearized matrix obeys the required concentration, and the union-bound failure probability remains exponentially small in m, independent of n. The resulting δ_s < 1/3 then feeds into the standard basis-pursuit error bound, producing a universal constant C (independent of n) in the instance-optimality statement. We will also include a short paragraph in Section 3 summarizing the covering and union-bound steps. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on independent RIP proof

full rationale

The paper's central claim is a uniform instance-optimality bound obtained by proving that the linearized sensing matrices (one per approximately sparse signal) simultaneously satisfy the RIP with constants independent of dimension. This RIP verification is presented as a new technical step that reduces PO-CS to standard linear CS instance optimality; it does not invoke self-citations for uniqueness theorems, does not rename fitted parameters as predictions, and does not define the target bound in terms of itself. The argument therefore remains self-contained against external CS theory and does not reduce by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard compressed sensing assumptions plus the paper-specific claim that linearized matrices satisfy simultaneous RIP for all approximately sparse signals.

assumptions (2)
  • domain assumption Complex Gaussian phases permit a linearization that recasts PO-CS as standard linear compressed sensing
    Invoked in the abstract to apply basis pursuit after linearization.
  • ad hoc to paper The new sensing matrices for all approximately sparse signals simultaneously satisfy the RIP
    This simultaneous RIP property is the key step used to obtain the uniform bound.

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Cite this review

Pith. "Pith review of Robust Instance Optimal Phase-Only Compressed Sensing." pith.science (2026). https://pith.science/paper/2408.06275

@misc{pith2026240806275,
  author       = {Pith},
  title        = {Pith review of: Robust Instance Optimal Phase-Only Compressed Sensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2408.06275}},
  note         = {Machine review of arXiv:2408.06275}
}
abstract

Phase-only compressed sensing (PO-CS) concerns the recovery of sparse signals from the phases of complex measurements. Recent results show that sparse signals in the standard sphere $\mathbb{S}^{n-1}$ can be exactly recovered from complex Gaussian phases by a linearization procedure, which recasts PO-CS as linear compressed sensing and then applies (quadratically constrained) basis pursuit to obtain $\mathbf{x}^\sharp$. This paper focuses on the instance optimality and robustness of $\mathbf{x}^{\sharp}$. First, we strengthen the nonuniform instance optimality of Jacques and Feuillen (2021) to a uniform one over the entire signal space. We show the existence of some universal constant $C$ such that $\|\mathbf{x}^\sharp-\mathbf{x}\|_2\le Cs^{-1/2}\sigma_{\ell_1}(\mathbf{x},\Sigma^n_s)$ holds for all $\mathbf{x}$ in the unit Euclidean sphere, where $\sigma_{\ell_1}(\mathbf{x},\Sigma^n_s)$ is the $\ell_1$ distance of $\mathbf{x}$ to its closest $s$-sparse signal. This is achieved by showing the new sensing matrices corresponding to all approximately sparse signals simultaneously satisfy RIP. Second, we investigate the estimator's robustness to noise and corruption. We show that dense noise with entries bounded by some small $\tau_0$, appearing either prior or posterior to retaining the phases, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(\tau_0)$. This is near-optimal (up to log factors) for any algorithm. On the other hand, adversarial corruption, which changes an arbitrary $\zeta_0$-fraction of the measurements to any phase-only values, increments $\|\mathbf{x}^\sharp-\mathbf{x}\|_2$ by $O(\sqrt{\zeta_0\log(1/\zeta_0)})$. The developments are then combined to yield a robust instance optimal guarantee that resembles the standard one in linear compressed sensing.

Figures

Figures reproduced from arXiv: 2408.06275 by the authors.

Figure 1
Figure 1. Reconstruction errors under post-sign bounded by [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Reconstruction errors under pre-sign bounded by [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. The impact of sparse phase corruption on [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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