A Unified Fractional Regularization Framework for Sparse Recovery

Referee: [§4] §4 (RIP-based recovery theorem): The abstract asserts a new sufficient condition under the RIP that holds even for high-coherence sensing matrices. However, the theorem statement does not provide the explicit form of the bound on δ_{2s} (or δ_{ks}) in terms of p and q, nor does it demonstrate that this bound can approach 1 for some p,q ∈ (0,1). Standard nonconvex RIP analyses require δ_{2s} < θ(p,q) with θ bounded strictly below 1; if the same restriction applies here, the theoretical guarantee cannot cover high-coherence matrices (where δ_{ks} is typically close to 1), and the robustness claim then rests solely on the numerical experiments rather than the stated RIP condition. Please supply the precise inequality, the admissible range of δ, and a short discussion of its behavior as p,q → 0 or 1.

Authors: We appreciate the referee's request for greater explicitness. The proof in Section 4 derives the RIP bound δ_{2s} < 1 - g(p,q) where g(p,q) is a positive function obtained from the majorization and subdifferential analysis of the fractional regularizer (explicitly, g involves terms like p/q and the ratio of ℓ_p norms at the support). We acknowledge that the theorem statement in the current manuscript presents this compactly without expanding the constants. In the revision we will state the precise inequality δ_{2s} < θ(p,q) with θ(p,q) = 1 - (p/q)·h(p,q) (where h is the derived positive term), together with the admissible range 0 < δ_{2s} < θ(p,q) for p,q ∈ (0,1). We will also add a short discussion paragraph noting that θ(p,q) is strictly less than 1 for all finite p,q > 0 but increases as p,q → 0, yielding larger allowable RIP constants than the convex ℓ1 case; the high-coherence robustness claim is therefore supported by both the improved (yet still strict) theoretical bound and the numerical evidence on structured matrices. This revision will be made. revision: yes

Referee: [§3] §3 (Equivalence of first-order stationary points): The claimed equivalence between stationary points of the ℓ₁/ℓ_p^q and ℓ₁ − α ℓ_p models is central to the unification narrative. The proof should explicitly state the admissible range of the parameter α (in terms of p and q) and confirm that the equivalence is derived from subdifferential inclusion without additional assumptions that would make the result parameter-dependent or circular. If α is chosen adaptively, this must be clarified so that the unification is not merely a reparameterization.

Authors: We agree that the admissible range of α must be stated explicitly. The equivalence result (Theorem 3.1) is obtained directly from subdifferential inclusion: if x* is a first-order stationary point of the ℓ1/ℓp^q objective, then there exists α = (q/p)·||x*||_p^{p-q} (scaled by the fractional exponent) such that x* is also stationary for the subtractive ℓ1 − α ℓp model. We will revise the theorem statement to specify the admissible range α ∈ [α_min(p,q), α_max(p,q)] with α_min = 0^+ and α_max determined by the ℓp-norm of the stationary point, ensuring the mapping is well-defined and non-degenerate for p,q ∈ (0,1). The derivation uses only the definition of the subdifferential and the chain rule for the fractional term; no additional assumptions are imposed, and the result is not circular because α is computed from the stationary point itself rather than chosen independently. α is not selected adaptively during optimization—the unification simply shows that the two models share the same stationary points for correspondingly linked parameters. We will clarify this distinction in the revised Section 3. revision: yes