{"id":"19796c45-b816-4ed4-a3f0-5c0807a6b062","arxiv_id":"2505.19618","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Rotation-equivariant convolutions improve self-supervised image denoising, and an adaptive fusion of equivariant and standard networks achieves the best reported PSNR.","lead":"Researchers show that replacing the convolution layers in self-supervised denoising networks with rotation-equivariant versions improves denoising quality on standard benchmarks, and an adaptive fusion of the equivariant and standard networks gives the best results. This is worth reading because it tests whether baking rotational symmetry into the architecture helps when no clean training data is available, extending the known benefit of translation symmetry in CNNs.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's smoothness assumption on latent feature maps is incompatible with the ReLU activation it explicitly invokes; the O(R1h+R2h^2) equivariance bound may be vacuous for the actual network, so the 'reliable rotation equivariant' guarantee is not established.","rationale":"The reader's weakest assumption is the smoothness of feature maps and filters, and I agree that this is the most load-bearing concern. My reading sharpens it: the issue is not just that R1, R2 might be large or hard to estimate, but that condition (15) is structurally incompatible with the ReLU activation named in Theorem 3. A ReLU feature map is not C^2, so the Hessian bound used to define R1 and R2 does not exist as a finite quantity; hence the theorem's guarantee is vacuous for the actual network. This directly undermines the paper's strongest claim that 'we can indeed achieve a reliable rotation equivariant network' through the proved bound. The empirical results, however, are substantial and internally consistent across N2N, N2V, and R2R, and the reported equivariance errors are finite and small, so the correct response is not rejection but a conditional acceptance requiring the authors to either repair the theorem (e.g., with a Lipschitz-gradient or BV-based argument), remove the ReLU/smoothness inconsistency, or explicitly downgrade the theoretical claim. The parameter-count mismatch and missing proofs are secondary: the proofs are deferred, but the primary logical flaw is the assumption-activation conflict, not the deferral itself. Since the reader already issued a conditional verdict based on essentially the same weakest point, my read does not move the verdict.","tokens_in":14849,"tokens_out":4793,"duration_ms":57234,"concrete_test":"Examine the supplementary proof of Theorem 3 and locate every use of the Hessian bounds in (15), especially any Taylor-expansion or interpolation lemma applied to feature maps. Then check whether the proof can accommodate rc(x) = ReLU(u^T x), whose Hessian is the distribution u u^T delta(u^T x). If any C^2 Taylor remainder or pointwise Hessian argument is invoked, Theorem 3 fails for the stated ReLU U-Net. A complementary numerical check: train the N2N-EQ model and measure the equivariance error in (16) at input resolutions 128, 256, and 512. If the error does not approximately halve as the resolution doubles, the predicted O(h) decay is not observed even where the bound is finite.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim is Theorem 3: replacing all convolutions with Fconv layers makes a ReLU U-Net approximately rotation equivariant, with error bounded by R1h + R2h^2. The proof conditions in (15) require the latent channel functions rc and filters phi_l to be C^2 with uniformly bounded Hessians: |nabla^2 rc(x)| <= H0 and |nabla^2 phi_l(x)| <= Hl. But a U-Net with ReLU activations produces, after the first nonlinearity, continuous piecewise-linear feature maps. For such a map, the Hessian is zero almost everywhere but has Dirac-delta singularities along the kink hyperplanes, so no finite pointwise or sup-norm Hessian bound exists. Consequently, the constants R1 and R2, which are explicitly allowed to depend on H0 and Hl, are either infinite or ill-defined, and inequality (16) does not yield a finite equivariance-error guarantee for the trained network. This is not merely a missing proof: Theorem 3 states ReLU as the activation and simultaneously assumes smoothness that ReLU networks do not possess. Theorems 1-2 only need gradient bounds and are plausible for ReLU features, but Theorem 3 propagates through repeated nonlinearities. The empirically measured equivariance errors are small, so the experimental claim may survive, but the paper's headline contribution of a 'rigorous theoretical analysis' establishing a reliable rotation-equivariant U-Net is unsupported as stated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript proposes to incorporate rotation-equivariant convolutions (Fconv) into self-supervised image denoising networks. It states theorems bounding the equivariance error introduced by downsampling and upsampling operations (Theorems 1 and 2) and by a full U-Net with Fconv layers and ReLU activations (Theorem 3 and Corollary 1). It then introduces AdaReNet, which fuses a vanilla CNN branch and an equivariant branch through a learned mask and a self-correcting module, with a loss combining the main denoising objective and per-branch regularizers. Experiments on Noise2Noise, Noise2Void, and R2R report PSNR/SSIM improvements over the vanilla baselines on several Gaussian-noise benchmarks, along with ablations on the choice of equivariant convolution and on rotation augmentation. The code is publicly released.","tokens_in":15196,"tokens_out":9070,"duration_ms":85496,"significance":"If the theoretical claims can be repaired, the paper makes a useful contribution by introducing rotation-equivariant priors into self-supervised denoising at the network architecture level. The empirical results are broadly positive, the ablations across equivariant convolution families are informative, and the public code is a practical asset. The adaptive fusion idea is a reasonable response to the fact that natural images are only locally rotation-equivariant. However, the central theoretical guarantee, which is presented as the paper's headline contribution, is not established as stated because the smoothness assumptions of Theorem 3 are incompatible with the ReLU activation used in the experiments; this issue must be resolved before the 'rigorous theoretical analysis' claim can be accepted.","major_comments":[{"comment":"The notation in Eq. (4) treats the feature map as a function on E(2) = R2 ⋊ O(2), while Eq. (3) defines it on R2 × S. The relation between these two domains, and the precise definition of the action πE_\\tilde{A} on e(x,A), should be stated more carefully, because the subsequent error bounds depend on which function space and transformation action is used. In particular, after discretization, the sampled feature map is only defined on a finite set of rotations S, and the continuous rotation of such a sample is not unique; the paper should specify how \tilde{π}^E_\\tilde{A}(F) is computed in the analysis.","section":"Section 3.1.1, Eq. (4) and (5)"}],"minor_comments":[{"comment":"The claim of being the 'first time' with 'comprehensive theoretical analysis' should be tempered, since the theoretical analysis as stated is not valid for ReLU networks; the novelty claim can stand, but it should not be tied to an unproven guarantee.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuinely promising empirical component: the PSNR improvements across N2N, N2V, and R2R, the ablation in Table 4, and the released code all suggest that rotation-equivariant convolutions can help self-supervised denoising. However, the theoretical centerpiece, Theorem 3, is not valid as stated because its C^2 smoothness assumptions on latent feature maps cannot hold for a ReLU network. This is the main reason for a major revision rather than acceptance. The authors should be asked to either adjust the theory to a smooth activation, relax the assumptions to a regularity class that ReLU networks satisfy, or substantially weaken the theoretical claims. They should also correct the overclaims in Sections 4.1 and the conclusion regarding consistent superiority, given the counterexamples in Tables 3 and 5. I would be willing to review a revised version focusing on these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Take this as a conditional accept with real reservations. The empirical core is probably sound: swapping vanilla convolutions for Fconv layers in N2N, N2V, and R2R consistently improves PSNR by roughly 0.1-1 dB, and the adaptive fusion mask adds a bit more. That is a legitimate, modest contribution, and the code being public helps.\n\nWhat's genuinely new: the first use of rotation-equivariant convolutions in self-supervised denoising, and the first attempt to quantify equivariance error from up/downsampling in U-Nets. The mask-based fusion of an equivariant and a vanilla branch is a sensible design, and Fig. 4 is a nice sanity check that the network learns to trust the vanilla branch near high-frequency edges.\n\nThe soft spots are in the theory, and they're not minor. Theorem 3 assumes the latent feature maps r_c and filters φ_l have uniformly bounded Hessians (conditions 15), then concludes the U-Net's equivariance error is O(h) + O(h^2). But the paper explicitly sets the activation to ReLU. A ReLU network produces piecewise-linear features whose second derivative is a sum of Dirac deltas on the kink hyperplanes; there is no finite sup-norm Hessian bound. So the constants R1, R2 in Theorem 3 are either infinite or ill-defined, and the 'reliable rotation equivariant network' guarantee is unsupported. The proof is deferred to supplementary, but the statement in the main text already shows the incompatibility. The same issue affects the claimed O(h) rates for maxpooling and stride downsampling, whose continuous definitions (6)-(7) also look ad hoc; they don't match how these operators are actually implemented.\n\nThat said, the empirically measured equivariance errors are small (e.g., 0.068 vs 0.233 for N2V), so the experimental conclusion may survive even if the theorem doesn't. The paper would be much stronger if the authors either proved the bound under weaker, ReLU-compatible assumptions (e.g., Lipschitz or BV norms) or dropped the 'rigorous guarantee' framing. I'd also like error bars, and an ablation that isolates the self-correcting ResNet module from the mask fusion; the current ablations mainly compare equivariant backbones. The citation pattern is fine; building on the same group's Fconv is legitimate.\n\nBottom line: a reasonably well-executed empirical paper with an overreaching theory. It deserves peer review, but a serious referee should demand a corrected Theorem 3 or a toned-down claim, plus a few more ablations. If the authors deliver that, it becomes a solid conference contribution.","headline":"A useful empirical finding about rotation-equivariant self-supervised denoising, but the stated Theorem 3 guarantee is vacuous for ReLU networks.","tokens_in":15715,"tokens_out":4386,"would_cite":true,"duration_ms":42677,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","68U10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that replacing every convolution in a U-Net with rotation-equivariant Fconv layers yields an approximately rotation-equivariant denoiser whose error vanishes as resolution grows, and that this built-in rotation prior…","keywords":["rotation equivariance","self-supervised image denoising","U-Net","Fconv","equivariance error bounds","adaptive fusion","Noise2Noise","Noise2Void"],"falsifier":"Train the N2N-EQ network and measure its relative rotation equivariance error $\\|UNet(\\tilde{\\pi}_\\theta(X))-\\tilde{\\pi}_\\theta(UNet(X))\\|/\\|UNet(\\tilde{\\pi}_\\theta(X))\\|$ at increasing input resolutions. The bound in Theorem 3 predicts the error should decline at the rate $O(h)$ plus $O(h^2)$; if the measured error does not shrink as the resolution increases, the theoretical guarantee does not describe the trained network.","tokens_in":14669,"feed_emoji":"🔄","tokens_out":10428,"duration_ms":84017,"temperature":0.7,"pith_summary":"This paper sets out to add a rotation-equivariance prior to self-supervised image denoising by changing the network architecture itself. It claims that replacing every convolution in the U-Net with a rotation-equivariant Fconv layer turns the whole U-Net into an approximately rotation-equivariant denoiser, with rigorous bounds on the equivariance error coming from the downsampling, upsampling, and convolutional layers. If true, self-supervised methods like Noise2Noise, Noise2Void, and Recorrupted-to-Recorrupted inherit the same kind of built-in symmetry that translation-equivariant convolutions already provide for translations, without requiring clean data or rotation augmentation. The paper reports consistent PSNR improvements from the equivariant networks and further gains from an adaptive fusion of equivariant and vanilla branches, selected per image region.","feed_headline":"Rotation-equivariant U-Nets sharpen self-supervised denoising","feed_subtitle":"The paper proves the equivariance error shrinks with resolution, then fuses equivariant and vanilla branches for extra PSNR.","key_machinery":"The load-bearing object is the Fconv rotation-equivariant convolution, a Fourier series expansion based filter parametrization that makes convolutions equivariant to rotations in the continuous domain and only approximately equivariant after discretization. The proof machinery decomposes the U-Net into blocks, one equivariant convolution plus a downsampling operator per downsampling block, and one upsampling operator plus two equivariant convolutions per upsampling block, and bounds each block's equivariance error using first-order smoothness bounds on feature maps. These block errors are then accumulated into the whole-network bound. The adaptive part is a MaskNetwork that produces a per-pixel mask fusing the vanilla branch output and the equivariant branch output element-wise, followed by a residual self-correcting block.","core_discovery":"The core claim is that the U-Net, despite its upsampling and downsampling layers, can be made a reliable rotation-equivariant network simply by replacing all convolutions with Fconv layers. Theorem 1 bounds the equivariance error of max-pooling and stride downsampling by $2\\sqrt{2}Gh$, Theorem 2 bounds nearest-neighbor and bilinear upsampling by $2(\\sqrt{2}+1)Gh$, and Theorem 3 accumulates blockwise errors into a whole-network bound $R_1h + R_2h^2$ under smoothness conditions on feature maps and filters; Corollary 1 extends the bound to arbitrary rotation angles. The empirical discovery is that the resulting N2N-EQ, N2V-EQ, and R2R-EQ networks consistently improve PSNR over the vanilla baselines, and that the AdaReNet mask, which mixes the equivariant and vanilla outputs, improves them further. The paper also claims that among tested equivariant convolution designs, Fconv's Fourier-series parametrization is the one that preserves enough representation accuracy for low-level vision.","pith_inferences":["Beyond the tested methods, the same convolution-replacement recipe should transfer to other U-Net-based self-supervised restoration tasks, such as inpainting or medical-image reconstruction, because the theoretical argument concerns the U-Net skeleton rather than the denoising loss.","The mask's observed preference for vanilla output near high-frequency edges implies a testable correlation: mask values should track local gradient magnitude, and forcing the mask to be uniform should reproduce the behavior of the equivariant branch alone.","Because the proof assumes smooth activations while ReLU is piecewise linear, one could test whether smoother activations such as SiLU give smaller measured equivariance errors, which would indicate whether the bound governs real trained networks.","The specific constants $R_1$, $R_2$, and $R_3$ are stated in the supplementary material, so the size of the bound, not just its $O(h)$ scaling, can be checked there."],"forward_implications":["A U-Net with all convolutions replaced by Fconv layers is approximately rotation equivariant, with whole-network error bounded by $R_1h + R_2h^2$, which vanishes as the input resolution grows.","Downsampling and upsampling layers contribute equivariance error only at order $O(h)$; because the equivariant convolutions contribute $O(h^2)$, the sampling operators become the dominant error source at finite resolution.","The equivariant versions of Noise2Noise, Noise2Void, and Recorrupted-to-Recorrupted beat their vanilla baselines on Gaussian denoising (for instance, N2V-EQ raises BSD500 PSNR from 28.17 to 29.05 dB at $\\sigma=25$).","A learned mask that fuses the equivariant and vanilla branch outputs yields further consistent gains, using the vanilla branch more near high-frequency edges and the equivariant branch in smooth regions.","Without rotation augmentation, the adaptive equivariant network produces the largest improvements, indicating that the architecture-level prior partially substitutes for data augmentation."],"supporting_citations":[{"why":"Supplies the Fconv rotation-equivariant convolution, with Fourier-series filter parametrization, that the paper substitutes into every convolution layer.","marker":"[49]"},{"why":"Supplies the prior analysis of global rotation equivariance error for image-processing networks that this paper extends to U-Net up/downsampling blocks.","marker":"[11]"},{"why":"Defines Noise2Noise, the self-supervised framework whose U-Net architecture is used for the main equivariance analysis.","marker":"[26]"},{"why":"Defines Noise2Void, the blind-spot self-supervised method whose U-Net version is made equivariant in N2V-EQ.","marker":"[21]"},{"why":"Defines Recorrupted-to-Recorrupted, the self-supervised method whose DnCNN baseline is made equivariant in R2R-EQ.","marker":"[33]"},{"why":"Baseline group-equivariant network in the ablation; the comparison supports choosing Fconv for low-level vision.","marker":"[8]"},{"why":"Baseline steerable-filter equivariant network in the ablation; the comparison supports the claim that representation accuracy matters more than exact equivariance.","marker":"[45]"},{"why":"Baseline PDO-eConv equivariant network in the ablation; comparison supports Fconv as the most effective equivariant convolution in the denoising setting.","marker":"[39]"}],"fun_headline_variants":["Rotation-equivariant nets sharpen self-supervised denoising","Self-supervised denoising gains rotation-equivariant edge","Adaptive rotation fusion lifts self-supervised denoising","First rotation-equivariant self-supervised denoiser"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The error bounds assume the network's internal feature maps and filters vary smoothly, with bounded slopes and curvatures; trained ReLU networks are piecewise linear, so the constants in the bound may not be small or even finite for the network actually trained.","fun_headline_variants_meta":{"raw":{"variants":["Rotation-equivariant nets sharpen self-supervised denoising","Self-supervised denoising gains rotation-equivariant edge","Adaptive rotation fusion lifts self-supervised denoising","First rotation-equivariant self-supervised denoiser"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000529,"raw_usage":{"total_tokens":2596,"prompt_tokens":1038,"completion_tokens":1558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":654,"completion_tokens_details":{"reasoning_tokens":1492}},"tokens_in":654,"tokens_out":1558,"duration_ms":12838,"temperature":1.0,"reasoning_tokens":1492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:10:26.334463+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the N2N-EQ network and measure its relative rotation equivariance error $\\|UNet(\\tilde{\\pi}_\\theta(X))-\\tilde{\\pi}_\\theta(UNet(X))\\|/\\|UNet(\\tilde{\\pi}_\\theta(X))\\|$ at increasing input resolutions. The bound in Theorem 3 predicts the error should decline at the rate $O(h)$ plus $O(h^2)$; if the measured error does not shrink as the resolution increases, the theoretical guarantee does not describe the trained network.","supporting_citations":[{"cited_title":"Fourier series expansion based filter parametrization for equivariant convolutions","cited_arxiv_id":null,"evidence_quote":"Supplies the Fconv rotation-equivariant convolution, with Fourier-series filter parametrization, that the paper substitutes into every convolution layer."},{"cited_title":"Rotation equivariant proximal operator for deep unfolding methods in image restoration","cited_arxiv_id":null,"evidence_quote":"Supplies the prior analysis of global rotation equivariance error for image-processing networks that this paper extends to U-Net up/downsampling blocks."},{"cited_title":"Noise2void-learning denoising from single noisy images","cited_arxiv_id":null,"evidence_quote":"Defines Noise2Void, the blind-spot self-supervised method whose U-Net version is made equivariant in N2V-EQ."},{"cited_title":"Recorrupted-to-recorrupted: unsupervised deep learning for image denoising","cited_arxiv_id":null,"evidence_quote":"Defines Recorrupted-to-Recorrupted, the self-supervised method whose DnCNN baseline is made equivariant in R2R-EQ."},{"cited_title":"Group equivariant convo- lutional networks","cited_arxiv_id":null,"evidence_quote":"Baseline group-equivariant network in the ablation; the comparison supports choosing Fconv for low-level vision."},{"cited_title":"General e (2)-equivariant steerable cnns","cited_arxiv_id":null,"evidence_quote":"Baseline steerable-filter equivariant network in the ablation; the comparison supports the claim that representation accuracy matters more than exact equivariance."},{"cited_title":"Pdo-econvs: Partial differential operator based equivari- ant convolutions","cited_arxiv_id":null,"evidence_quote":"Baseline PDO-eConv equivariant network in the ablation; comparison supports Fconv as the most effective equivariant convolution in the denoising setting."}],"review_version":1}