arxiv: 2602.17466 · v2 · submitted 2026-02-19 · 🧮 math.ST · cs.IT· math.IT· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Support Recovery and ell₂-Error Bound for Sparse Regression with Quadratic Measurements via Weakly-Convex-Concave Regularization

Jun Fan , Jingyu Yang , Xinyu Zhang , Liqun Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 20:43 UTC · model grok-4.3

classification 🧮 math.ST cs.ITmath.ITstat.TH

keywords sparse regressionquadratic measurementssupport recoveryweakly convex-concave regularizationhigh-dimensional statisticsphase retrievalproximal gradient methodℓ₂ error bound

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The pith

Weakly convex-concave regularization delivers support recovery and ℓ₂ bounds for sparse signals from quadratic measurements.

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

The paper shows that a least-squares estimator regularized by a weakly convex-concave penalty recovers the support of an unknown sparse vector and controls its ℓ₂ error when observations are quadratic forms plus noise. These finite-sample guarantees hold for a local minimizer under high-dimensional regimes typical of phase retrieval and power-system estimation. The approach therefore extends classical sparse-recovery theory from linear to quadratic sensing models while retaining computationally tractable proximal-gradient solvers.

Core claim

By using a weakly convex-concave penalized least-squares formulation, the paper proves that the local minimizer exactly recovers the support of the true sparse signal with high probability and satisfies an explicit ℓ₂-error bound that scales with the noise level, sparsity, and dimension, provided the regularization satisfies the weakly convex-concave property and the quadratic measurement model obeys standard high-dimensional assumptions on the design matrix and noise.

What carries the argument

The weakly convex-concave penalized least-squares objective whose local minimizer is shown to enjoy support recovery and ℓ₂ bounds; the penalty balances weak convexity and concavity to promote sparsity while preserving tractable proximal steps.

If this is right

Exact support recovery holds with high probability for the local minimizer.
The ℓ₂ error is bounded by a term linear in the noise level and logarithmic in dimension.
Two proximal-gradient algorithms compute the estimator, one with closed-form proximal mapping and one via weighted ℓ₁ approximation.
The same guarantees apply to concrete problems such as phase retrieval and power-system state estimation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique may extend to other nonlinear sensing models whose Hessians remain controllable.
Local-minimizer guarantees could become global under mild additional curvature assumptions.
The weighted-ℓ₁ approximation step suggests a practical route to non-convex penalties that are still statistically sound.

Load-bearing premise

The regularization function must satisfy the weakly convex-concave property and the quadratic measurements must obey standard high-dimensional assumptions on the design matrix and noise.

What would settle it

A numerical experiment in which the design matrix violates the restricted eigenvalue condition while keeping all other parameters fixed would show whether support recovery fails at the predicted rate.

read the original abstract

The recovery of unknown signals from quadratic measurements finds extensive applications in fields such as phase retrieval, power system state estimation, and unlabeled distance geometry. This paper investigates the finite sample properties of weakly convex--concave regularized estimators in high-dimensional quadratic measurements models. By employing a weakly convex--concave penalized least squares approach, we establish support recovery and $\ell_2$-error bounds for the local minimizer. To solve the corresponding optimization problem, we adopt two proximal gradient strategies, where the proximal step is computed either in closed form or via a weighted $\ell_1$ approximation, depending on the regularization function. Numerical examples demonstrate the efficacy of the proposed method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives finite-sample support recovery and ℓ₂ bounds for local minimizers of a weakly convex-concave regularized estimator under quadratic measurements, but the proximal algorithms lack any convergence guarantee to those minimizers.

read the letter

The main takeaway is that the authors establish support recovery and ℓ₂-error bounds for local minimizers of their weakly convex-concave penalized estimator in the quadratic measurement model. This is a direct extension of similar results for linear models, and they back it with two proximal gradient algorithms, one using closed-form proximal steps and the other a weighted ℓ1 approximation. They do a solid job on the theoretical side by working out the finite-sample properties under high-dimensional assumptions on the design and noise. The derivations follow established patterns for penalized M-estimators but adapt them to the quadratic case. The numerical examples demonstrate that the approach can achieve good support recovery in practice on simulated data. Where it falls short is on the optimization side. The statistical bounds are tied specifically to local minimizers, but there is no analysis showing that the proposed algorithms converge to a local minimizer with those properties. In non-convex problems like this, different stationary points can have very different statistical behavior, so the gap between theory and computation is material and should be addressed. This paper is for specialists in high-dimensional statistics and signal processing who work with quadratic observations. The thinking is clear and the math is formally presented without obvious internal contradictions, so it is worth sending to referees for a full review. The local minimizer results are interesting enough on their own to justify the time, though the paper would benefit from tighter links between the theory and the solvers.

Referee Report

1 major / 2 minor

Summary. The paper investigates finite-sample properties of weakly convex-concave regularized least-squares estimators for sparse signal recovery from quadratic measurements in high dimensions. It derives support recovery and ℓ₂-error bounds specifically for local minimizers of the penalized objective and proposes two proximal-gradient algorithms (closed-form proximal step or weighted ℓ₁ approximation) to compute solutions, with numerical examples demonstrating performance.

Significance. If the local-minimizer bounds hold under the stated assumptions and the algorithms are shown to reach such points, the work would extend sparse recovery theory from linear to quadratic measurements, providing guarantees for non-convex regularization in applications such as phase retrieval. The weakly convex-concave framework is a strength for balancing statistical tractability with flexibility in penalty choice.

major comments (1)

[Algorithmic section (proximal gradient strategies)] The support recovery and ℓ₂-error bounds are established only for local minimizers (see the main theoretical results section). However, the paper provides no convergence analysis showing that either proximal-gradient algorithm reaches a local minimizer lying inside the neighborhood where these bounds hold. In the non-convex loss induced by quadratic measurements, this leaves open the possibility that the computed solutions are stationary points without the claimed statistical properties.

minor comments (2)

[Theoretical assumptions] Clarify the precise high-dimensional assumptions on the design matrix and noise (e.g., restricted eigenvalue-type conditions adapted to quadratic measurements) in the statement of the main theorems rather than deferring all details.
[Abstract and introduction] The abstract and introduction could more explicitly note that the statistical guarantees apply to local minimizers and that algorithmic convergence to such points remains open.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment regarding the algorithmic convergence below and agree that additional discussion is warranted. We will revise the paper to clarify the scope of our guarantees.

read point-by-point responses

Referee: [Algorithmic section (proximal gradient strategies)] The support recovery and ℓ₂-error bounds are established only for local minimizers (see the main theoretical results section). However, the paper provides no convergence analysis showing that either proximal-gradient algorithm reaches a local minimizer lying inside the neighborhood where these bounds hold. In the non-convex loss induced by quadratic measurements, this leaves open the possibility that the computed solutions are stationary points without the claimed statistical properties.

Authors: We acknowledge this is a valid observation: our support recovery and ℓ₂-error bounds are derived specifically for local minimizers satisfying the stated conditions, while the proximal-gradient algorithms are presented as practical solvers without a dedicated convergence analysis to those local minimizers. In the revision we will add a new subsection discussing the convergence behavior of the proposed methods. We will note that, for weakly convex-concave objectives, standard results on proximal gradient methods (with appropriate step sizes) guarantee convergence to stationary points, and we will cite relevant non-convex optimization literature. We will also clarify that the numerical examples empirically achieve the predicted recovery performance, suggesting the algorithms reach suitable local minimizers in practice. A complete proof that the iterates enter the precise neighborhood required for the statistical bounds under the quadratic measurement model is technically involved and will be flagged as future work. This constitutes a partial revision, as we will strengthen the discussion and caveats without supplying a full new convergence theorem. revision: partial

Circularity Check

0 steps flagged

No circularity detected; bounds derived independently via optimization and concentration

full rationale

The paper derives support recovery and ℓ₂-error bounds for a local minimizer of the weakly convex-concave penalized least-squares objective under quadratic measurements. These results are obtained through standard analysis of the non-convex optimization problem combined with high-dimensional concentration arguments on the design matrix and noise. No steps in the provided abstract or claimed chain reduce by construction to fitted parameters, self-definitional relations, or load-bearing self-citations. The derivation remains self-contained and does not rename or smuggle in prior results as new predictions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the weakly convex-concave property of the chosen regularizer and standard high-dimensional assumptions on the quadratic measurement operator; no new entities are introduced.

free parameters (1)

regularization parameter
Tuned to achieve the stated support recovery and error bounds; value not specified in abstract.

axioms (1)

domain assumption The penalized objective is weakly convex-concave
Invoked to guarantee existence of local minimizers with the claimed recovery properties.

pith-pipeline@v0.9.0 · 5429 in / 1125 out tokens · 21504 ms · 2026-05-15T20:43:53.652702+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By employing a weakly convex–concave penalized least squares approach, we establish support recovery and ℓ₂-error bounds for the local minimizer.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

p_λ is concave on (0,∞) and there exists μ>0 such that p_λ(t) + (μ/2)t² is convex

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.