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arxiv: 2603.09722 · v2 · submitted 2026-03-10 · 🧮 math.FA · cs.IT· math.CA· math.IT

Transformed ell_p Minimization Model and Sparse Signal Recovery

Ziwei Li , Wengu Chen , Huanmin Ge , Dachun Yang This is my paper

Pith reviewed 2026-05-15 13:09 UTC · model grok-4.3

classification 🧮 math.FA cs.ITmath.CAmath.IT
keywords transformed lp minimizationsparse signal recoveryrestricted isometry propertynon-convex penaltyiteratively reweighted least squaresexact recoverystable recovery
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The pith

A two-parameter transformed ℓ_p penalty guarantees exact sparse recovery under a tunable RIP bound that sharpens to known thresholds.

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

The paper introduces a non-convex transformed ℓ_p penalty with adjustable parameters a and p to promote sparsity in signal recovery problems. Using the sparse convex-combination technique, it proves that this TLp minimization model achieves exact recovery of s-sparse vectors and stable recovery of nearly sparse vectors as long as the measurement matrix obeys a restricted isometry property whose constant is bounded by an expression in a and p. The authors also present the IRLSTLp algorithm based on modified iteratively re-weighted least squares together with convergence guarantees. A key feature is that the derived RIP upper bound reduces to the sharp Zhang-Li bound as a tends to infinity and recovers the classic δ_{2s} < √2/2 result when p equals 1.

Core claim

The central claim is that the TLp minimization model with the two-parameter transformed ℓ_p penalty ensures exact and stable recovery of s-sparse signals whenever the sensing matrix satisfies a restricted isometry property condition whose explicit upper bound on δ_{2s} depends on a and p, with the bound becoming the sharp known threshold in the limit a to infinity and recovering δ_{2s} < √2/2 exactly when p=1.

What carries the argument

The transformed ℓ_p (TLp) penalty function with parameters a > 0 and p ∈ (0,1], which is minimized subject to linear measurements and whose recovery guarantees are obtained by translating the RIP condition through the sparse convex-combination technique.

If this is right

  • Exact recovery holds for every s-sparse signal when the RIP constant lies below the parameter-dependent threshold.
  • Stable recovery holds for signals that are close to s-sparse in the presence of noise.
  • The recovery condition approaches the sharpest previously known bound as a tends to infinity for any fixed p in (0,1].
  • When p is set to 1 the model reproduces the classical sharp bound δ_{2s} < √2/2.

Where Pith is reading between the lines

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

  • Choosing different pairs (a,p) may allow practitioners to trade off recovery sharpness against numerical stability in applications with varying noise levels.
  • The introduced notion of relaxation degree could be used to compare other multi-parameter penalties and guide selection of parameters that keep the model close to ℓ_0 behavior.
  • Because the bound recovers known sharp results in limiting cases, the TLp model offers a single framework that interpolates between several existing non-convex penalties.

Load-bearing premise

The measurement matrix must satisfy the restricted isometry property with a constant small enough for the chosen values of a and p.

What would settle it

Construct or identify a matrix whose RIP constant δ_{2s} equals the paper's derived upper bound for a given a and p, then check whether every s-sparse signal is exactly recovered by the TLp minimizer or whether a counterexample signal fails to be recovered.

Figures

Figures reproduced from arXiv: 2603.09722 by Dachun Yang, Huanmin Ge, Wengu Chen, Ziwei Li.

Figure 1
Figure 1. Figure 1: Exact recovery for sparse e Let us further clarify how RDP measures the approximation degree for the given penalty function P to approach ℓ0 through an example. Example 2.3. Let p ∈ (0, 1] and Pp(x) := kxk p ℓp (2.3) for any x ∈ R N, where k · kℓp is as in (1.3). It is easy to prove RDPp = N 1 2 − 1 p . We find that RDPp decreases to 0 as p decreases to 0, which coincides with the known fact that ℓp penalt… view at source ↗
Figure 2
Figure 2. Figure 2: Level lines of P0.1,p with p ∈ {0.5, 0.7, 0.9, 1} P0.5,0.7 P1,0.7 P10,0.7 P0.7 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Level lines of Pa,0.7 with a ∈ {0.5, 1, 10} and P0.7 P5,0.7 ℓ 0.7 5 P1,0.7 ℓ 0.7 1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Level lines of P5,0.7, ℓ 0.7 5 , P1,0.7, and ℓ 0.7 1 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical tests on p and a by 64 × 256 Gaussian matrix RDPa,p ≈ 2.4×10−4 . We find that the success rates keep almost the same with RDPa,p being fixed although both a and p change. (a, p) (0.18, 0.9) (0.3, 0.85) (0.54, 0.8) (1.1, 0.75) (4, 0.7) Success Rate 76% 75% 84% 79% 79% [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Numerical tests by 64 × 256 Gaussian matrix when the sparsity is 24; (b) graphs of a as a function of p with N = 256 and various RDPa,p [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Numerical tests on different RDPa,p by 64 × 256 Gaussian matrix 4.2 Comparison with DCATL1 Zhang and Xin [63] investigated the DC algorithm for the TL1 function and test the performance. Although the TLp function reduces to the TL1 function when p = 1, the IRLSTLp implementation for p = 1 differs somewhat from the DCATL1. A performance comparison between these two methods is therefore necessary. We test wi… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison tests between the IRLSTLp and the DCATL1 4.3.1 Tests by Gaussian Random Matrices We use the class of Gaussian matrices generated by the multi-variable normal dis￾tribution N(0, Σ) to test the four algorithms above, where the covariance matrix Σ := {(1 − r)1{i=j} + r}i, j with r ∈ [0, 1). Generally speaking, it will be more difficult to recover the true sparse signal as r gets larger. We test by … view at source ↗
Figure 9
Figure 9. Figure 9: Numerical tests by 64 × 256 Gaussian matrix with different r 4.3.2 Tests by Over-Sampled DCT Matrices We use the class of over sampled DCT matrices A = (a1, . . . , aN) ∈ R M×N to compare the performance of these four algorithms under varying degrees of matrix coherence, where, for any i ∈ {1, . . . , N}, ai := 1 √ M cos 2π[i − 1]ω F ! , [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Numerical tests by 100 × 1500 over-sampled DCT matrix with different F [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗
read the original abstract

In this article, we introduce a minimization model via a non-convex transformed $\ell_p$ (TLp) penalty function with two parameters $a\in(0,\infty)$ and $p\in(0,1]$, where the case $p=1$ is known and was established by S. Zhang and J. Xin. Using the sparse convex-combination technique, we establish the exact and the stable sparse signal recovery based on the restricted isometry property (RIP). We apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to give the IRLSTLp algorithm for unconstrained TLp minimization and prove some convergence results. Finally, we conduct some numerical experiments to show the robustness of the IRLSTLp and the flexibility of the TLp minimization model. The novelty of these results lies in three aspects: (i) We introduce the concept of the relaxation degree RD$_P$ of a separable penalty function $P$ to quantitatively measure how closely $P$ approaches $\ell_0$, whose significance also lies in revealing the functional relationship of the parameters involved to keep a high performance of a multi-parameter minimization model. (ii) We introduce the TLp penalty, which includes two aforementioned adjustable parameters, offering more flexibility and stronger sparsity-promotion capability of the TLp minimization model, compared with the $\ell_p$ and the TL1 minimization models. (iii) The obtained RIP upper bound for signal recovery via TLp minimization can reduce, when $p\in(0,1]$ and as $a\to \infty$, to the sharp RIP bound obtained by R. Zhang and S. Li and, especially, can recover, when $p=1$, the well-known sharp bound $\delta_{2s}<\frac{\sqrt{2}}{2}$.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces a two-parameter transformed ℓ_p (TLp) penalty function (a>0, p∈(0,1]) that generalizes the TL1 case of Zhang-Xin, establishes exact and stable sparse recovery guarantees from the RIP via the sparse convex-combination technique, proposes the IRLSTLp algorithm (modified IRLS + DCA) with convergence results, and reports numerical experiments. It further defines a relaxation degree RD_P for separable penalties and claims that the derived RIP threshold for TLp recovery reduces exactly to the sharp Zhang-Li bound as a→∞ (including δ_{2s}<√2/2 when p=1).

Significance. If the RIP threshold derivation is shown to recover the sharp constants without residual looseness in the a→∞ limit, the work supplies a tunable non-convex model with an explicit quantitative measure (RD_P) of proximity to ℓ_0, potentially improving flexibility over single-parameter ℓ_p or TL1 models while inheriting known sharp recovery thresholds.

major comments (2)
  1. [§3] §3 (Recovery guarantees): The sparse convex-combination argument must be expanded to display the explicit functional form of the RIP threshold δ_{2s} < f(a,p) and to verify, term-by-term, that lim_{a→∞} f(a,1) equals exactly √2/2 with no a-dependent slack remaining from the weighting coefficients; the current reduction claim in the abstract is not yet load-bearing without this limit calculation.
  2. [§3.1] §3.1 (Null-space property or convex-combination step): The translation from the TLp penalty to the RIP condition appears to introduce an auxiliary factor that depends on a; an explicit computation showing this factor tends to 1 (or to the precise Zhang-Li multiplier) as a→∞ is required to substantiate the sharp-bound recovery.
minor comments (2)
  1. [§2] Notation for the transformed penalty (definition of TLp) should be stated once with both parameters a and p made explicit before any recovery statements.
  2. [§5] The numerical section would benefit from a table comparing recovery success rates and CPU times against ℓ_p and TL1 baselines for the same RIP constants.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The points raised concern the explicit form of the RIP threshold and the verification of the sharp limit as a→∞. We will revise the manuscript to include these calculations, thereby strengthening the recovery guarantees section.

read point-by-point responses

  1. Referee: [§3] §3 (Recovery guarantees): The sparse convex-combination argument must be expanded to display the explicit functional form of the RIP threshold δ_{2s} < f(a,p) and to verify, term-by-term, that lim_{a→∞} f(a,1) equals exactly √2/2 with no a-dependent slack remaining from the weighting coefficients; the current reduction claim in the abstract is not yet load-bearing without this limit calculation.

    Authors: We agree that an explicit expression for the RIP threshold and a detailed term-by-term limit verification are necessary to make the reduction claim fully rigorous. In the revised manuscript we will expand the sparse convex-combination argument in §3 to derive and display the closed-form bound δ_{2s} < f(a,p). We will then compute lim_{a→∞} f(a,1) coefficient by coefficient, confirming that every weighting factor converges exactly to the corresponding term in the Zhang-Li argument and that no a-dependent slack remains, thereby recovering δ_{2s} < √2/2 precisely when p=1. revision: yes

  2. Referee: [§3.1] §3.1 (Null-space property or convex-combination step): The translation from the TLp penalty to the RIP condition appears to introduce an auxiliary factor that depends on a; an explicit computation showing this factor tends to 1 (or to the precise Zhang-Li multiplier) as a→∞ is required to substantiate the sharp-bound recovery.

    Authors: We appreciate this observation. The auxiliary factor arising from the TLp-to-RIP translation is indeed a-dependent. In the revision we will insert an explicit limit computation in §3.1 that tracks this factor through the convex-combination weights and shows that it converges to 1 (equivalently, to the exact Zhang-Li multiplier) as a→∞. The calculation will be carried out by taking the pointwise limit inside the relevant inequalities and verifying the resulting constants match those of the sharp bound. revision: yes

Circularity Check

0 steps flagged

No circularity: RIP bounds derived independently and shown to recover external sharp constants

full rationale

The derivation chain begins with the definition of the TLp penalty (two explicit parameters a and p), applies the sparse convex-combination technique to obtain RIP-based recovery conditions whose thresholds depend on a and p, and then proves a mathematical limit statement showing that as a→∞ the thresholds approach the independently established sharp bounds of Zhang and Li (and recover δ_{2s}<√2/2 for p=1). No equation equates a derived quantity to a fitted parameter inside the paper, no self-citation supplies the load-bearing uniqueness or ansatz, and the limit is presented as an explicit reduction rather than an assumption. The central claims therefore remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claim rests on the new TLp penalty definition, the RIP assumption on the sensing matrix, and the sparse convex-combination technique; a and p are design parameters chosen by the user rather than fitted constants.

free parameters (2)
  • a
    Transformation parameter in the TLp penalty that controls how closely it approximates ℓ0 behavior.
  • p
    Exponent parameter in (0,1] that interpolates between ℓ1 and ℓ0-like penalties.
axioms (1)
  • domain assumption The sensing matrix satisfies the restricted isometry property of order 2s with constant δ_{2s} below a threshold depending on a and p.
    Invoked to obtain exact and stable recovery guarantees via the sparse convex-combination technique.
invented entities (1)
  • Transformed ℓ_p (TLp) penalty function no independent evidence
    purpose: Non-convex separable penalty that promotes sparsity with two tunable parameters.
    Newly defined in the paper; no independent evidence outside the model is supplied.

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