Sampling Theory for Super-Resolution with Implicit Neural Representations

Gregory Ongie; Mahrokh Najaf

REVIEW 2 major objections 1 minor 57 references

Sampling Theory for Super-Resolution with Implicit Neural Representations

T0 review · 2 major / 1 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that shallow ReLU Fourier-feature INRs are exactly recoverable from low-pass Fourier samples once the sampling cutoff scales with the network width or feature frequency, and it derives those scaling laws.

desk verdict Theorem 2's proof has a genuine gap — Lemma 10 is false as stated for s=1 and the theorem omits a distinctness hypothesis the proof assumes — but the core sample-complexity results are new, largely sound, and worth refereeing. read the letter →

arxiv 2506.09949 v1 pith:4Y2KW6JC submitted 2025-06-11 eess.IV cs.CV

Mahrokh Najaf , Gregory Ongie This is my paper

classification eess.IVcs.CV MSC 94A2068T0742A10

keywords implicitneuralrepresentationssuper-resolutionFouriersamplingReLUnetworksweightdecayregularizationconvexoptimizationexactrecoverytrigonometricpolynomials

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 establishes sampling conditions under which a continuous image realized by a single-hidden-layer ReLU network with a Fourier-features layer is exactly recoverable from its low-pass Fourier coefficients, when training uses a generalized weight-decay penalty. For a width-1 image built from a single rectified trigonometric polynomial, roughly $O(K_0^d)$ samples---the low-pass set containing all frequencies with $\|k\|_\infty \le 3K_0$---suffice. For a 2D image that is a positive sum of $s$ rectified trigonometric polynomials, $O(s^2)$ samples, namely all frequencies with $\|k\|_1 \le 2s$, suffice for every student width $W \ge s$. The proof equates the nonconvex parameter-space problem with a convex problem over measures and then constructs an explicit dual certificate. These appear to be the first exact-recovery sampling bounds for Fourier-feature INRs, and they give a template for other linear inverse problems.

What carries the argument

The load-bearing object is the convex measure reformulation of the training problem: after rescaling, a shallow network $f_\theta$ corresponds to a signed measure $\mu=\sum_i a_i\delta_{w_i}$ on the unit sphere, and the generalized weight-decay objective becomes the weighted total-variation norm $\|\mu\|_{TV,\eta}$. The training problem with the Fourier data constraint $F_\Omega f=y$ is then equivalent to the convex program $\min \|\mu\|_{TV,\eta}$ subject to $K_\Omega\mu=y$, and optimality is certified by a dual function $\Phi(w)=\langle F_\Omega[w^\top\gamma(\cdot)]_+,z\rangle$ that is bounded by $\eta(w)$ everywhere and interpolates it on the support. For the width-$s$ theorem the certificate is built from the second-order operator $(\Delta+4\pi^2)[\tau]_+$, whose distributional support is the zero curve $Z(\tau)$ with density $\|\nabla\tau\|$; the proof then uses algebraic facts about zero sets of two-variable trigonometric polynomials—minimal polynomials, finitely many critical points, and divisibility.

What would settle it

Take $s=2$, distinct weight vectors $w_1,w_2\in\mathbb{R}^4\setminus V$ with $w_i\ne\pm w_j$, positive coefficients, and $\Omega=\{k\in\mathbb{Z}^2:\|k\|_1\le 4\}$. A numerical or algebraic search that finds two different four-point measures on the unit sphere producing identical Fourier samples over $\Omega$ would falsify the injectivity lemma behind the theorem; conversely, verifying on a fine spherical grid that the constructed dual function satisfies $|\Phi(w)|\le\eta(w)$ everywhere would directly confirm the certificate for that instance.

Watch

Extended reading notes

Core claim

The paper's central claim is that exact recovery of a shallow Fourier-feature INR from low-pass Fourier samples is governed by a sampling cutoff, not by parameter counting. For the width-$s$ theorem, if $f=\sum_{i=1}^s a_i[w_i^\top\gamma(\cdot)]_+$ with $a_i>0$ and $w_i\in\mathbb{R}^4\setminus V$, and if $\Omega$ contains every $k\in\mathbb{Z}^2$ with $\|k\|_1\le 2s$, then $f$ is the unique function-space minimizer of the INR training problem with $\eta(w)=\int_{\mathbb{T}^2}[w^\top\gamma(x)]_+\,dx$, for every width $W\ge s$. The width-1 theorem gives an analogous guarantee with a general integer Fourier-feature embedding and the weighting $\eta(w)=\|F_\Omega[w^\top\gamma(\cdot)]_+\|_2$ once $\Omega$ contains all frequencies with $\|k\|_\infty\le 3K_0$. The positive-coefficient assumption and the exclusion of the measure-zero set $V$ are exactly the hypotheses that make the dual certificate and the injectivity argument work.

Load-bearing premise

The width-$s$ guarantee rests on the assumption that the zero sets of the two-variable trigonometric polynomials involved behave like algebraic curves: each must be irreducible or a minimal polynomial, each must have only finitely many critical points, and each must divide any trigonometric polynomial that vanishes on that zero set; these facts are imported from the paper's reference [35] rather than proved here.

Editorial extensions

If this is right

With $\|k\|_1\le 2s$ low-pass samples, a 2D positive width-$s$ image is uniquely recovered as the function-space minimizer for every fitting width $W\ge s$.
With $\|k\|_\infty\le 3K_0$ samples, a width-1 image with general integer Fourier features is uniquely recovered for every fitting width $W\ge 1$.
The nonconvex INR training problem has exactly the global optima of a convex measure program, so optimality can be certified independently of the optimization trajectory.
The two specialized weight-decay penalties are sufficient for the proofs, and the paper's experiments indicate standard weight decay behaves at least as well in practice.

Reading between the lines

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

Extension: the same dual-certificate construction should transfer to any linear forward operator whose adjoint maps into trigonometric polynomials, so non-uniform Fourier samples and tomographic projections are natural next cases.
Extension: the positivity and $d=2$ restrictions look like artifacts of the certificate proof; a signed-coefficient or higher-dimensional version of the certificate would be a direct test of whether the $O(s^2)$ law is the real boundary.
Extension: the depth experiments hint that deep ReLU INRs with weight decay carry a piecewise-constant bias; if so, a deep sampling theory might stack the shallow result layer by layer, which this paper does not attempt.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read this one for the theorems, but check the proofs carefully. The paper gives the first sample-complexity bounds for exact recovery of shallow Fourier-feature ReLU INRs from low-pass Fourier data: O(K_0^d) for width-1 with general features, O(s^2) for width-s in 2D with restricted features. That is a real step beyond [46], which only characterized function-space regularity, and beyond off-the-grid super-resolution results for sparse measures or piecewise-constant images. The convexification via measures is not new, but the dual-certificate construction and the algebraic zero-set arguments are applied to a genuinely new model class. The paper is also honest about the main caveat: the guarantees hold for a generalized weight decay, not standard weight decay, and the experiments show standard WD works as well or better in practice.

Now the soft spots. The stress-test is right: Lemma 10 is false as stated. Take s=1 and Ω={0}. Any irreducible τ_1 outside V has zero mean because γ omits the constant feature, so the measure ν=δ_{w_1}−δ_{−w_1} satisfies K_Ων=F_Ω(τ_1)=0, yet f_ν=τ_1≠0. The lemma's own hypothesis is satisfied. The proof's final step invokes Γ_1⊂Ω, which fails for Ω={0}. With the stronger Ω in Theorem 2 (containing {k: ||k||_1≤2}), the conclusion is recoverable, so the main theorem is patchable. The second issue: Lemma 10 and Lemma 11 assume w_i≠±w_j for i≠j, but Theorem 2's statement only says w_i∉V. If two weight vectors are opposites, the zero-set argument collapses because ψ_j contains a squared copy of the same τ and the integral in (27) vanishes for the wrong reason. Either add the distinctness hypothesis to the theorem or prove the opposite-pair case separately. This is a gap in the written proof, not obviously a counterexample to the theorem. A minor issue: in the supplementary proof of Lemma 8, the coefficient-matching equations for c_1 and c_2 appear to have a typo; the stated factorization still follows after correcting the comparison.

Net: the central claims are new and probably true, but the paper should not go out in this form. The authors need to fix Lemma 10, patch the proof of Theorem 2, and reconcile the statement with the hypotheses the proof actually uses. The experiments and code are useful. I'd send it to referees — the issues are serious but localized and fixable. The right call is revise, not reject.

Referee Report

2 major / 1 minor

Summary. The paper studies exact recovery of continuous-domain images from low-pass Fourier samples when the image is represented by a shallow ReLU INR with an initial Fourier-features layer and the training objective is regularized by a generalized weight-decay penalty. The authors show that function-space minimizers of the nonconvex INR problem coincide with minimizers of a convex measure-space program, and then prove two sampling theorems. Theorem 1 treats width-1 images with a general integer-frequency Fourier-features embedding and shows uniqueness when the sampling set contains the cube of radius 3K0, using the regularizer η(w)=||FΩ[w^Tγ]_+||2. Theorem 2 treats width-s images in d=2 with the restricted four-feature embedding, positive outer coefficients, and generic weight vectors outside the measure-zero set V, showing uniqueness when the sampling set contains the ℓ1 ball of radius 2s, using η(w)=∫[w^Tγ]_+ dx. The proofs construct explicit dual certificates and rely on algebraic facts about zero sets of bivariate trigonometric polynomials imported from Ongie-Jacob. The experiments map empirical exact-recovery probabilities and compare standard and modified weight decay on phantom super-resolution.

Significance. If the theorems hold as stated after the fixes described below, this appears to be the first exact-recovery sampling bound for shallow Fourier-feature INRs, and it gives a clean connection between INR training and off-the-grid compressed sensing. The main theorems are genuinely parameter-free: the constants 3, 2, and 2s are fixed by the proofs, and the dual certificates are explicit rather than fitted to the data. The paper also contains complete proofs of the supporting lemmas, releases code for the experiments, and is careful to distinguish global-minimizer guarantees from the nonconvex optimization actually performed. The use of modified, rather than standard, weight decay is clearly stated and the conjecture that standard weight decay behaves similarly is supported only empirically, which is an honest limitation.

major comments (2)

[B.2, Lemma 10] Lemma 10 is false as stated for s=1. With Ω={k∈Z²: ||k||₁≤0}={(0,0)}, choose any w₁∈R⁴\V such that τ₁=w₁^Tγ is irreducible and set ν=δ_{w₁}-δ_{-w₁}. Because the restricted embedding γ in (12) has no constant feature, ∫τ₁ dx=0, and since [t]_+-[-t]_+=t, one has K_Ων=F_Ω([τ₁]_+-[-τ₁]_+)=F_Ω(τ₁)=0. Yet f_ν=[τ₁]_+-[-τ₁]_+=τ₁≠0, contradicting the lemma's implication K_Ων=0 ⇒ f_ν=0. The proof's final step "since Γ₁⊂Ω" is false under the lemma's own hypothesis when s=1. The conclusion is recoverable under Theorem 2's actual sampling assumption Ω⊇{k: ||k||₁≤2s}, so this is a localized but load-bearing gap: Lemma 10 supplies the injectivity condition used in Lemma 4 for Theorem 2.
[B.2, Theorem 2 and Lemmas 10–11] Theorem 2 states only w_i∈R⁴\V, but the proofs of Lemma 10 and Lemma 11 require the additional hypothesis w_i≠±w_j for i≠j. Lemma 7(a) gives finiteness of Z(τ_i)∩Z(τ_j) only when τ_i and τ_j are not scalar multiples, and such pairs are not excluded by V (for example, w=(1,1,2,1) and -w are both outside V). Without distinctness, ψ_j=∏_{i≠j}τ_i² vanishes identically on Z(τ_j) when τ_i=±τ_j, so the integral argument forcing a_j+b_j=0 in Lemma 10 collapses, and the divisibility step in Lemma 11 also breaks. The theorem's conclusion remains plausible for generic weights after adding w_i≠±w_j, which is still a measure-zero exceptional set, but the statement and proof must be aligned.

minor comments (1)

[Supplementary Material F, Lemma 8 proof] In the coefficient-matching display, the line defining c₁ contains a typo: the coefficient of z₁²z₂ is a₁b₃+a₂b₄, and the z₂-coefficient is a₀b₂+a₂b₀; the displayed expression a₁b₂+a₂b₄ does not match the expansion of the product. The subsequent argument does not appear to depend on this erroneous equality, but the display should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the INR sampling theorems are proven via explicit convex dual certificates, and the cited algebraic facts from [35] are independent, parameter-free support rather than a self-referential input.

full rationale

The paper's central claims are exact-recovery sampling theorems for shallow Fourier-feature ReLU INRs. The proof reduces the non-convex INR training problem to a convex measure problem (P_mu), proves equivalence via Lemma 1, and then constructs explicit dual certificates. No parameter in the theorems is fitted to the target data: the regularizers eta(w)=||F_Omega[w^T gamma(.)]_+||_2 (Theorem 1) and eta(w)=int [w^T gamma(.)]_+ dx (Theorem 2) are fixed functions of the known sampling operator Omega and the known Fourier-feature map, not of y. The dual certificates are existential objects built from the assumed ground-truth weights, following the standard off-the-grid compressed-sensing argument; this is not an input-output identity. The algebraic facts imported from Ongie-Jacob [35] in Lemma 7 are published, parameter-free statements about zero sets of two-variable trigonometric polynomials, and although one current author co-authored [35], these facts do not themselves contain the INR recovery result, so they constitute independent mathematical support rather than circularity. The paper also explicitly acknowledges that the generalized weight-decay forms arise from proof techniques and conjectures extension to standard weight decay, further indicating that no fitted quantity is being relabeled as a prediction. The proof of Theorem 2 has a statement-to-proof gap in that Lemma 10 assumes w_i != +/- w_j while Theorem 2 does not state that hypothesis; however, this is a soundness concern rather than a self-referential reduction, so it does not affect the circularity score. The experimental component is validated against independently defined phantoms (Shepp-Logan, MRI brain phantoms) and random teacher networks, so it is not a re-labeling of the derived quantities.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The theoretical claims are derived with no fitted constants: the constants 3, 2, and 2s are fixed by the proofs, and no parameters are tuned to the data in the theorems. The main axioms are standard convex-analysis and algebraic-geometry results plus the modeling restrictions of the INR architecture. The most paper-specific inputs are the two generalized weight decay penalties, which are chosen to make dual-certificate proofs work; the paper acknowledges standard weight decay is only conjectural.

assumptions (7)

standard math Representer theorem of Bredies and Carioni [6] guarantees sparse minimizers for (P_mu) when W >= |Omega|.
Invoked in Lemma 2, Section A.2, to bound support size by |Omega|.
standard math Fenchel-Rockafellar strong duality for the measure-space primal and its predual (Lemma 3, Section A.3).
The proof's continuity claim at 0 may be questionable because C has empty interior, but the main theorems can bypass it via direct dual-certificate comparison.
standard math Zero-set algebra of 2-variable trigonometric polynomials from [35] (Lemma 7) and the classification in Lemma 8.
Used in Section B.2 to prove injectivity and to construct the dual certificate; Lemma 8 is proven in the supplement.
domain assumption Weighting function eta is admissible: positive, 1-homogeneous, and zero only for zero ReLU units.
Definition in Section 2.2; needed for Lemma 1 and for S_eta to be well behaved.
domain assumption Fourier features use integer frequencies and the image is periodic on the torus; architecture is single-hidden-layer ReLU.
Section 2.1; this restricts the scope to periodic images, not general random-feature INRs.
ad hoc to paper For Theorem 2, outer weights a_i > 0 and w_i not in V (measure-zero exceptional set).
Stated in Theorem 2; the dual certificate construction needs positivity, and V is excluded for algebraic reasons.
ad hoc to paper The generalized weight decay regularizers eta(w) = ||F_Omega [w^T gamma]_+||_2 and eta(w) = integral [w^T gamma]_+ dx are the regularizers used in the guarantee.
Section 3.3; these are nonstandard and depend on the forward operator in Theorem 1. The paper explicitly conjectures standard weight decay.

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Cite this review

Pith. "Pith review of Sampling Theory for Super-Resolution with Implicit Neural Representations." pith.science (2026). https://pith.science/paper/4Y2KW6JC

@misc{pith2026250609949,
  author       = {Pith},
  title        = {Pith review of: Sampling Theory for Super-Resolution with Implicit Neural Representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4Y2KW6JC}},
  note         = {Machine review of arXiv:2506.09949}
}

read the original abstract

Implicit neural representations (INRs) have emerged as a powerful tool for solving inverse problems in computer vision and computational imaging. INRs represent images as continuous domain functions realized by a neural network taking spatial coordinates as inputs. However, unlike traditional pixel representations, little is known about the sample complexity of estimating images using INRs in the context of linear inverse problems. Towards this end, we study the sampling requirements for recovery of a continuous domain image from its low-pass Fourier samples by fitting a single hidden-layer INR with ReLU activation and a Fourier features layer using a generalized form of weight decay regularization. Our key insight is to relate minimizers of this non-convex parameter space optimization problem to minimizers of a convex penalty defined over an infinite-dimensional space of measures. We identify a sufficient number of Fourier samples for which an image realized by an INR is exactly recoverable by solving the INR training problem. To validate our theory, we empirically assess the probability of achieving exact recovery of images realized by low-width single hidden-layer INRs, and illustrate the performance of INRs on super-resolution recovery of continuous domain phantom images.

Figures

Figures reproduced from arXiv: 2506.09949 by the authors.

**Figure 1.** The shallow INR architecture considered in this work represents an image as a linear combination of rectified trigonometric polynomials. We study the sample complexity of estimating an image of this type from its low-pass Fourier samples. In a noise-free scenario, the regularization parameter λ in (2) needs to be very small to ensure data consistency is tightly enforced. As a model for this situation, we will focus … view at source ↗

**Figure 2.** Example of a random width 3 “teacher” INR used in our exact recovery experiments to validate Theorem 1 (top panel) and Theorem 2 (bottom panel). The image f is a weighted sum of the three randomly generated rectified trigonometric polynomials [τi ]+ = [w⊤ i γ(·)]+ shown on the right. Note that the two settings consider INRs defined with Fourier features embeddings γ(·) having different frequencies. exact recovery wi… view at source ↗

**Figure 3.** Empirical probability of exactly recovering an random image representable by a single hiddenlayer INR of width-W from low-pass Fourier samples with maximum sampling frequency K by solving (Pθ,W ). The left panel corresponds to the setting of Theorem 1, comparing standard weight decay (standard WD) with modified weight decay regularization proposed for Theorem 1 (modified WD-I) and the right panel corresponds to the… view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: Recovery of continuous domain DOT and SL phantoms from low-pass Fourier samples. We compare the zero-filled IFFT reconstruction with a shallow INR trained without regularization (without WD), with standard weight decay regularization (shallow INR, std WD), the modifie…

**Figure 5.** Figure 5: Recovery of continuous domain PWC-BRAIN and PWS-BRAIN phantoms from lowpass Fourier samples. We compare the zero-filled IFFT reconstruction with a shallow INR trained without regularization (without WD), with standard weight decay regularization (shallow INR, std WD),…

**Figure 6.** Figure 6: Effect of regularization on the PWC BRAIN phantom reconstruction and unit sizes of a trained depth 2 INR. In the first row of each panel, we compare the zero-filled IFFT reconstruction with the reconstructions obtained with a shallow INR trained with standard/modified …

**Figure 7.** Figure 7: Visualization of the four largest active units for the PWC-BRAIN phantom, obtained with a shallow INR. The INR is trained with standard weight decay regularization (std WD), The modified weight decay regularization proposed in Theorem 1 (mod WD-I), and the modified wei…

**Figure 8.** Figure 8: Effect of INR depth on SL and PWC-BRAIN phantom recovery. The top row shows comparisons of reconstructions of an INR of width 100 across depths 2, 5, 10, 15, and 20 trained with no regularization (i.e., with the weight decay hyperparameter λ = 0). The bottom row shows …

**Figure 9.** Figure 9: SL recovery using different INR architectures. Top row: we compare the zero-filled IFFT reconstruction with the phantom recovery from a Hash Encoding INR, depth 5 SIREN (SIREN), depth 14 SIREN (deep SIREN), and depth 20 ReLU INR (deep ReLU) with random Fourier features…

**Figure 10.** Figure 10: PWC-BRAIN recovery using different INR architectures. Top row: we compare the zerofilled IFFT reconstruction with the phantom recovery from a Hash Encoding INR, depth 5 SIREN (SIREN), depth 14 SIREN (deep SIREN), and depth 20 ReLU INR (deep ReLU) with random Fourier …

**Figure 11.** Figure 11: PWS-BRAIN recovery using different INR architectures. Top row: we compare the zero-filled IFFT reconstruction with the phantom recovery from Hash Encoding, depth 5 SIREN (SIREN), depth 14 SIREN (deep SIREN), and depth 15 INR (deep INR) with random Fourier features and…

**Figure 12.** Figure 12: Impact of regularization on the DOT phantom reconstruction and unit sizes of a trained shallow INR. In the first row of each panel, we compare the zero-filled IFFT reconstruction with the reconstructions obtained with a shallow INR trained with standard/modified weigh…

**Figure 13.** Figure 13: Visualization of the 4 largest active units for the DOT phantom, obtained with a shallow INR. The INR is trained with standard weight decay regularization (std WD), modified weight decay regularization proposed in Theorem 1 (mod WD-I), and the modified weight decay re…

**Figure 14.** Figure 14: The effect of varying the maximum frequency for Fourier features K0, on the SL phantom recovery trained using a shallow INR with standard WD, modified WD-I, and modified WD-II regularization. Observe that the lowest MSE is achieved for K0 = 10 for modified WD-I and WD…

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write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.check...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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