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Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI

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Pith's one-line read Three ill-posed inverse problems—deconvolution, elliptic source recovery, and MRI-style Fourier sampling—are shown to admit stable sparse recovery with explicit sample complexity.

desk verdict A serious applications paper with a real MRI theorem, but the central abstract result is quoted from an unavailable companion paper, so the conditional verdict is right. read the letter →

arxiv 2501.01929 v1 pith:A7U2CW67 submitted 2025-01-03 math.FA cs.ITmath.ITmath.OC

Giovanni S. Alberti , Alessandro Felisi , Matteo Santacesaria , S. Ivan Trapasso This is my paper · ORCID
classification math.FAcs.ITmath.ITmath.OC MSC 42C4094A2035R30
keywords compressedsensinginverseproblemssparserecoverywaveletsquasi-diagonalizationcoherencesamplecomplexitymagneticresonanceimaging
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

This paper claims that one abstract compressed-sensing framework, originally built for the sparse Radon transform, governs three distinct ill-posed inverse problems: deconvolving a wavelet-sparse signal from pointwise samples of a blurred version, recovering a sparse source term of an elliptic partial differential equation from pointwise samples of its solution, and reconstructing a wavelet-sparse signal from ill-posed Fourier samples of the kind motivated by magnetic resonance imaging. For each problem the authors verify two structural conditions—quasi-diagonalization of the forward map and a coherence bound tied to the sampling density—and then read off recovery guarantees from a single theorem. They also introduce a balancing property that controls the error of truncating the measurement range to a finite band, and they show that choosing the sampling density to match the coherence profile improves sample complexity. A sympathetic reader should take away a unified explanation of when sparse recovery survives ill-posedness, with the ill-posedness parameter entering explicitly in the bounds.

What carries the argument

The load-bearing mechanism is the pair of structural estimates known here as quasi-diagonalization and coherence. Quasi-diagonalization (Assumption 2.1) says the forward map $F$ behaves, on the wavelet dictionary $(\phi_{j,n})$, like a diagonal operator that attenuates scale $j$ by $2^{-bj}$, so the parameter $b$ quantifies ill-posedness. The coherence bound (Assumption 2.2) ties each measurement operator $F_t$ to the sampling density $f_\nu$ via $\|F_t\phi_{j,n}\|\le B\sqrt{f_\nu(t)}2^{dj}$; when the density is chosen proportional to the minimal function $g_\nu=\sup_{j,n}2^{2dj}\|F_t\phi_{j,n}\|^2$, the constant $B$ is controlled and sample complexity improves. The balancing property (Definition 2.10), $\|P^\perp U\Phi^*\iota_{\le j_0}\|\le \theta 2^{-2bj_0}$, controls the error of projecting the natural forward map $U$ onto the measurable band, and Proposition 2.11 shows that quasi-diagonalization of $U$ plus balancing of $P$ yields weak quasi-diagonalization of the truncated map $F=P\circ U$. These three conditions feed directly into Theorem 2.4.

What would settle it

Compute the actual $\|F_t\phi_{j,n}\|$ for a concrete compactly supported wavelet basis (for example, Daubechies-8) on the lattice $Z$ and compare with the bound $B_0 2^{-bj}(1+|t|^2)^{1/2}$; a counterexample violating this inequality at some scale would invalidate Lemma 5.4 and hence Theorem 5.6. Alternatively, simulate the weighted $\ell^1$ recovery for a signal whose tail is sparse but not $H^{1-b}$-regular and check whether the error exceeds the right-hand side of Theorem 5.6.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 2.4: if the forward map $F$ satisfies the quasi-diagonalization bound $c\sum_{(j,n)} 2^{-2bj}|x_{j,n}|^2 \le \|F\Phi^*x\|_{L^2_\mu}^2 \le C\sum 2^{-2bj}|x_{j,n}|^2$ and the coherence bound $\|F_t\phi_{j,n}\| \le B\sqrt{f_\nu(t)}2^{dj}$ with $0\le d\le b$, then the minimizer $\hat x$ of the constrained weighted $\ell^1$ program with weight $W=\operatorname{diag}(2^{bj})$ reconstructs the wavelet coefficients of the signal with error $\|W^{-\zeta}(x^\dagger-\hat x)\|_2 \le C_1\sigma_s(W^{-\zeta}P_{\le j_0}x^\dagger)_1/\sqrt{s}$ plus explicit noise and truncation terms, provided the number of random samples $m$ satisfies $m\ge C_0\tau\max\{\log^3\tau\log M_{\le j_0},\log(1/\gamma)\}$ with $\tau = B^2 2^{2(b-d)j_0} 2^{2(1-\zeta)bj_0}s$. The paper then shows that the three applications satisfy these hypotheses, with the ill-posedness parameter $b$ equal to the Bessel-potential order for deconvolution, $b=2$ for the elliptic source problem, and the modulation decay $b$ for MRI-style Fourier sampling.

Load-bearing premise

The recovery guarantees hold only when the noise and the truncated high-frequency tail of the signal satisfy the weighted bounds (1)-(4); in the MRI application this reduces to a regularity assumption on the tail in $H^{1-b}$, which sparsity alone does not imply.

Editorial extensions

If this is right

  • Sparse deconvolution with a Bessel kernel admits stable random-sampling recovery with $m\gtrsim j_0^2 2^{2j_0}s$ samples, up to logarithmic factors; the paper notes this is not a subsampling result but a stable sampling strategy with explicit noise robustness.
  • For cartoon-like images the deconvolution bound becomes $\|u^\dagger-\hat u\|_{L^2}\le C\beta^{1/(2b+1)}\log^2(1/\beta)$ up to logarithmic factors when $m$ is chosen of order $\beta^{-2(2b+3)/(2b+1)}$, connecting the abstract theorem to a concrete image class.
  • For the elliptic inverse source problem, wavelet-sparse sources are recoverable from pointwise samples of the solution with $b=2$ and $m\gtrsim \alpha^{-2}2^{2(1+\alpha)j_0}s$, showing the framework works without translation invariance.
  • For MRI-style Fourier sampling, variable density $f_\nu(t)\propto(1+|t|^2)^{-1}$ yields recovery with $m\gtrsim j_0 s$ up to logarithmic factors, requiring the balancing property and a regularity assumption on the high-frequency tail.

Reading between the lines

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

  • The same template should apply to any forward map whose wavelet-domain symbol decays like $2^{-bj}$: one can predict sample complexity $\tau = B^2 2^{2(b-d)j_0}2^{2(1-\zeta)bj_0}s$ without redoing the argument, so the paper effectively provides a recipe for future inverse problems.
  • The optimized sampling densities described in Section 2.5 are a candidate explanation for why variable-density MRI works in practice; a numerical study comparing the predicted density $(1+|t|^2)^{-1}$ with learned sampling patterns would test this directly.
  • The bounds do not specify how to choose $j_0$ given a noise level; optimizing the error bounds over $j_0$ could give practical guidance on the resolution level and sample count, a step the paper leaves implicit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops an abstract compressed sensing recovery framework for ill-posed inverse problems and applies it to three concrete settings: deconvolution with a Bessel kernel, a sparse inverse source problem for an elliptic PDE, and a moderately ill-posed Fourier sampling problem motivated by MRI. The central abstract result, Theorem 2.4, gives weighted l1 recovery bounds under quasi-diagonalization, coherence, balancing, and noise assumptions, in both uniform and nonuniform noise models. Sections 3, 4, and 5 verify the required assumptions for the three applications and state explicit sample complexity and error bounds, with Section 3 additionally deriving rates for cartoon-like images. Proofs of the application-level verifications are collected in Section 6, and wavelet background is provided in Appendix A.

Significance. If the abstract theorem is fully established, this paper offers a useful unified treatment of sparse recovery for several ill-posed inverse problems, with carefully verified assumptions and explicit sample complexities. The deconvolution result with an optimized nonuniform sampling density, the elliptic source recovery without translation equivariance, and the MRI-motivated Fourier sampling analysis are nontrivial and go beyond the companion paper's Radon transform focus. The paper is also explicit about constants and dependencies, and Section 3.6 gives a concrete rate for cartoon-like images. The main caveat is that the abstract uniform-noise case of Theorem 2.4 is not proved in the manuscript but delegated to the authors' companion paper; because all three applications inherit this result, the significance is contingent on that external proof.

major comments (2)
  1. [Section 6.1, proof of Theorem 2.4] The uniform-noise case of Theorem 2.4 is stated to follow from [9, Theorem 3.11] after dropping the assumption sup_t ||F_t|| <= C_F and replacing it with sup_t ||F_t Phi* P^perp_{<=j0} x^dagger|| <= r. The justification is a reference to [9, Proposition 5.8] and [9, Remark 5.9], but neither the statement nor the proof of that proposition is included in this manuscript. Since the nonuniform-noise case is then derived by reducing it to the uniform case via the reweighted operators F'_t = f_nu(t)^{-1/2} F_t, every recovery guarantee in Theorems 3.2, 3.5, 4.3, and 5.6 depends on this unproved extension. This is a load-bearing point: the modification concerns exactly the truncation-error hypothesis, and it is not self-evident from the finite-net arguments in [9] that the bound on Phi^*P^perp_{<=j0}x^dagger alone suffices. I ask the authors to include a complete proof of the weakened uniform-noise theorem, or at least to reproduce the relevant statement and proof of [9, Proposition 5.8] in an appendix.
  2. [Section 5.4 / Lemma 5.3, Theorem 5.6] Lemma 5.3 requires N >= theta^{-1/b} C^{1/(2b)} 2^{j0}, but Section 5.3 writes N = C_0 2^{j0} with C_0 = theta^{-1/b} C^{1/b}, and Theorem 5.6 states N = C^{1/b}2^{j0} without the theta factor. The exponent on C and the theta dependence are inconsistent between the lemma and its application. This does not affect the recovery theorem if the constants are allowed to absorb the discrepancy, but the statements should be made uniform.
minor comments (4)
  1. [Definition 2.10] In equation (11), the displayed operator norm should have domain l2(Lambda_{<=j0}), not H1, since the operator being bounded is P^perp U Phi^* iota_{<=j0} from l2(Lambda_{<=j0}) to H.
  2. [Lemma 5.4] The proof states that F_0 phi_{j,n} = 0 for all dictionary elements, but this is not true for the low-frequency scaling functions in the dictionary; the low-frequency part is said to be handled analogously, so the statement should be restricted to the wavelet components or justified for the scaling functions as well.
  3. [Theorem 5.6] The measurement model writes y_k := (1+|t_k|^2)^{-b/2} \hat u^dagger(t_k) + epsilon_k, but the forward operator F_t is defined via the normalized frame elements \tilde\psi_t = exp(2pi i t \cdot x)/|B_R|^{1/2}; the normalization factor |B_R|^{-1/2} appears to be missing from the displayed formula for y_k.
  4. [Section 3.6] The value j0 := floor(2/(1+2b) log(1/beta)) is used as a scale index, but j0 should be a nonnegative integer; the floor notation is present, yet the subsequent sample complexity statement could state more explicitly that all estimates hold up to rounding.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the application theorems independently verify concrete estimates, and the cited abstract result is a genuine prior theorem.

full rationale

The load-bearing Theorem 2.4 is taken from the authors' earlier work [9], but the present paper's contributions are the concrete verifications in Sections 3–5. The uniform-noise case of Theorem 2.4 is explicitly delegated to [9, Theorem 3.11] with a weakened truncation hypothesis; this is a citation to a prior theorem with stated assumptions, not a reduction of the conclusion to the hypotheses of the present paper. The nonuniform-noise case is obtained by the legitimate change of measurement operators F'_t = f_nu(t)^{-1/2}F_t and a change of measure dµ' = f_nu dµ; the paper verifies that quasi-diagonalization and coherence are preserved under this transformation. The application sections do not fit parameters and then relabel them as predictions: each theorem (3.2, 3.5, 4.3, 5.6) verifies the specific quasi-diagonalization, balancing, and coherence estimates through independently established lemmas involving Bessel-kernel decay, elliptic regularity, and wavelet/frame Littlewood-Paley bounds. The sampling densities are chosen from the derived coherence bounds by construction, and the recovery estimates follow from the abstract theorem rather than from the definitions. The skeptical concern about whether [9, Proposition 5.8] really supports the weakened truncation assumption is a correctness risk about the prior theorem, not a circularity in this paper; per the review rules it does not raise the circularity score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The main theoretical load is carried by standard wavelet, frame and PDE tools, plus the authors' own companion theorem [9]. The free parameters α and η are tuning parameters in the coherence interpolations; they are not fitted to data.

free parameters (2)
  • α
    Interpolation parameter in the coherence bound for the elliptic problem (Lemma 4.2). It appears in the sample complexity of Theorem 4.3 as α^{-2}2^{2(1+α)j0}s and is a free choice in (0,1]; the paper does not optimize it.
  • η
    Parameter in Theorem 3.5 controlling the exponential sampling density and the distortion of the coherence bound; η ∈ (0,b-1] is a free choice affecting τ = η^{-2}2^{2(1+η)j0}s.
assumptions (6)
  • standard math Existence of compactly supported r-regular orthonormal wavelets (Meyer's construction, Proposition A.5/A.6).
    Used to define the wavelet dictionary and Littlewood-Paley estimates throughout; the regularity r is assumed in each section (r>2 for deconvolution/PDE, r>1 for MRI).
  • domain assumption Beurling frame theorem: the exponential system (tilde ψ_t)_{t∈Z} is a frame of L2(B_R) for a δ-dense, η-separated set Z (Definition 5.1, cited to [13]).
    Needed in Section 5 to make U a bounded map and to establish quasi-diagonalization (Proposition 5.2 and Lemmas 6.1-6.4).
  • domain assumption Elliptic regularity: for C^2 domain Ω and σ∈C^1 with σ≥λ, solutions to (22) lie in H^2(Ω) and satisfy ||Fu||_{H^2}≤C||u||_{L2}.
    Used in Section 4 to justify pointwise measurements (Sobolev embedding H^2⊂C) and the two-sided norm equivalence of Lemma 4.1.
  • ad hoc to paper [9, Theorem 3.11] and [9, Propositions 5.8, 5.12] from the authors' companion paper (to appear).
    The uniform-noise case of Theorem 2.4 is stated as 'up to minor modifications' of this theorem, and the proof in Section 6.1 refers to it rather than reproducing it.
  • standard math Littlewood-Paley characterization of Sobolev norms for r-regular wavelets (Proposition A.7).
    Bridges the quasi-diagonalization constants to Sobolev norms; used in Sections 3, 4, 5.
  • standard math Complex interpolation of Sobolev and weighted-sequence spaces (Bergh-Löfström).
    Used in Lemma 4.2 and Lemma 6.4 to interpolate between integer-order estimates.

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Pith. "Pith review of Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI." pith.science (2026). https://pith.science/paper/A7U2CW67

@misc{pith2026250101929,
  author       = {Pith},
  title        = {Pith review of: Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7U2CW67}},
  note         = {Machine review of arXiv:2501.01929}
}
read the original abstract

This paper extends the sample complexity theory for ill-posed inverse problems developed in a recent work by the authors [`Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform', J. Eur. Math. Soc., to appear], which was originally focused on the sparse Radon transform. We demonstrate that the underlying abstract framework, based on infinite-dimensional compressed sensing and generalized sampling techniques, can effectively handle a variety of practical applications. Specifically, we analyze three case studies: (1) The reconstruction of a sparse signal from a finite number of pointwise blurred samples; (2) The recovery of the (sparse) source term of an elliptic partial differential equation from finite samples of the solution; and (3) A moderately ill-posed variation of the classical sensing problem of recovering a wavelet-sparse signal from finite Fourier samples, motivated by magnetic resonance imaging. For each application, we establish rigorous recovery guarantees by verifying the key theoretical requirements, including quasi-diagonalization and coherence bounds. Our analysis reveals that careful consideration of balancing properties and optimized sampling strategies can lead to improved reconstruction performance. The results provide a unified theoretical foundation for compressed sensing approaches to inverse problems while yielding practical insights for specific applications.

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