{"id":"aa088573-e045-43ee-aaba-46e09fabe70e","arxiv_id":"2508.05160","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"This paper constructs rotation-equivariant encoder and implicit neural representation modules for arbitrary-scale super-resolution, achieving exact equivariance for 90-degree rotations and bounded error otherwise.","lead":"An image super-resolution method is redesigned so that rotating the input rotates the output the same way, end to end, rather than distorting repeated textures and edges. The rotation-aware modules can be dropped into existing arbitrary-scale super-resolution models, giving modest quality gains with fewer parameters.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"B-Conv bicubic basis is not closed under 45°/22.5° rotations, so the p8/p16 equivariance bounds of Theorem 3/Corollary 4 do not apply to the proposed encoder.","rationale":"The reader's weakest assumption correctly identifies that the encoder's exact equivariance is the load-bearing premise for the later theorems, and that the grid-invariance issue makes p8/p16 only approximate. My concern sharpens one specific, checkable mechanism: even in the continuous-domain idealization, the proposed B-Conv filter parameterization is not closed under 45°/22.5° rotations, so the encoder is not exactly equivariant for t=8/16 and the paper's theoretical bound for approximate equivariance is missing a term. This is a concrete technical gap rather than a disagreement with consensus: the p4 exactness result appears sound, but the advertised p8/p16 approximate guarantees are not supported by the presented mathematics. The reader also flagged the LTE formula inconsistency in Eq (33); that is real and important for reproducibility, but it is likely a typo and does not threaten the core p4 framework as directly as the encoder closure issue. I therefore recommend keeping the verdict CONDITIONAL: the paper should be accepted only after the authors either restrict their approximate-equivariance claims to p4, or replace B-Conv with a steerable basis for t>4 and provide the missing representation-error analysis.","tokens_in":29479,"tokens_out":16737,"duration_ms":196076,"concrete_test":"Numerically compute the L2 error of the best approximation of a 45°-rotated tensor-product cubic B-spline by the B-Conv basis span of Eq (35) over the filter support. If the residual is nonzero, the filter family is not closed under p8 rotations. Then measure the equivariance error of a single randomly initialized B-Conv layer for a 45° rotation on a smooth test image at decreasing mesh sizes δ; if the error plateaus to a positive value as δ→0, the missing basis-closure error is confirmed and Corollary 4 cannot cover the proposed encoder.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theoretical guarantee assumes the encoder Φ^EQ_CNN is exactly rotation equivariant with respect to the discrete group S of size t (Theorem 3). The proposed B-Conv encoder (Sec 3.3.1) parameterizes every filter as a linear combination of tensor-product cubic B-splines on a fixed square grid, Eq (35): φ(x) = Σ w_ab φ_Bic(x1/δ - a) φ_Bic(x2/δ - b). This function space is rotation-invariant only for rotations by multiples of 90°. Rotating a basis element by 45° gives φ_Bic((x1+x2)/(√2δ)-a)·φ_Bic((-x1+x2)/(√2δ)-b), which is not a separable spline on the original grid and is not in the span of Eq (35). Consequently, for t=8 or t=16 the B-Conv layer cannot represent the exactly rotated filters required for group equivariance, so the premise of Theorem 3 fails for the actual architecture. Corollary 4's bound Cδ + Ĉ/t tracks only the error from approximating an arbitrary continuous rotation by a neighboring group element in an exactly equivariant network; it does not include the B-Conv filter-representation error, which is independent of δ and does not vanish as the mesh size goes to zero. Table 2 therefore reports an equivariance error that conflates discretization/group-approximation effects with an unquantified basis-closure error. The p4 exactness claim (Corollary 3) is not affected, because the bicubic basis is closed under 90° rotations, but the paper's approximate p8/p16 guarantee is not established for the proposed encoder.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a rotation-equivariant arbitrary-scale image super-resolution (ASISR) framework. The INR module is redesigned into group-convolution-style input, intermediate, and output layers with parameter sharing over a discrete rotation group, and the encoder is constructed with a bicubic-spline-parameterized rotation-equivariant convolution (B-Conv), plus a simplified Rot-E transformer. The authors prove (or state proofs to be in a supplement) that the local INR is exactly rotation equivariant on its group, that the global INR has equivariance error bounded by Cδ, and that the full network has error bounded by Cδ for discrete rotations and Cδ + Ĉ/t for arbitrary angles, with exact equality for the p4 (90-degree) group. Experiments on DIV2K and standard benchmarks report large reductions in equivariance error, consistent PSNR improvements across LIIF, OPE, and LTE backbones, parameter reductions, and generalization to hyperspectral and thermal images. Code is provided.","tokens_in":29870,"tokens_out":9432,"duration_ms":103649,"significance":"If the claims hold, this is a useful advance: it appears to be the first ASISR method with end-to-end rotation equivariance, and the modular design allows existing INRs to be converted to Rot-E versions in a plug-and-play manner. The p4 exactness result is clean and the empirical equivariance-error tables are striking (e.g., Table 1: NMSE drops from ~1.0 to ~2e-4). The release of code and the parameter-sharing structure are also strengths. However, the p8/p16 approximate equivariance claim is not established for the proposed B-Conv encoder, and one of the three concrete INR instantiations (Rot-E LTE) is written incorrectly in Eq. (33). Because the central theoretical contribution is the equivariance analysis and its architectural realization, these gaps need to be repaired before the paper can be accepted.","major_comments":[{"comment":"The B-Conv basis is closed under p4 rotations but not under p8/p16 rotations. The span of tensor-product cubic B-splines on a square lattice is invariant under 90-degree rotations only; rotating a basis element by 45 degrees gives a spline with diagonal break lines that is not in the span of Eq. (35). Therefore a B-Conv layer cannot represent the exactly rotated filters required for t=8 or t=16, and the premise of Theorem 3 (an exactly equivariant encoder) fails for the proposed encoder at those groups. Corollary 4's O(Ĉ/t) term accounts only for approximating an arbitrary continuous rotation by a nearby group element in an exactly equivariant network; it does not include this basis-closure error, which is independent of δ and does not vanish as the mesh size goes to zero. Table 2's nonzero p8/p16 errors are consistent with this gap, but the theory does not quantify it. Please add the m","section":"Sec. 3.3.1, Eq. (35); Theorem 3 / Corollary 4"},{"comment":"The Rot-E LTE formula does not follow from the input-layer definition in Eq. (7). In Eq. (7), the coordinate argument is A^{-1}x and the feature index is the group element A (up to reindexing). Eq. (33), as printed, uses x instead of A^{-1}x and uses a double sum over A,B with F^{B^{-1}A}; this is not the composition of the proposed equivariant modules. Consequently, Theorem 1 and Corollary 1 do not apply to the Rot-E LTE instantiation as written, and since LTE-EQ is one of the three architectures evaluated experimentally, the claim that the tested method satisfies the equivariance theorem is not supported. Please correct Eq. (33) to match Eq. (7), or explain the intended index convention.","section":"Sec. 3.2.4, Eq. (33)"},{"comment":"The main proofs are deferred: the text repeatedly states \"please see the supplementary material,\" but the arXiv version contains no supplementary file. Theorems 1-3 and Corollaries 2-4 are load-bearing for the paper's central claims, and the current manuscript does not allow the reader to verify the equivariance-error bounds or the p4 exactness. Please include the full proofs in the submission, either in the main text or in a complete supplement.","section":"Sec. 3.2.3, Theorems 1-3"}],"minor_comments":[{"comment":"The index set for k is written as \"k=0,2,...,t-1\"; it should be \"k=0,1,...,t-1\".","section":"Eq. (1), Sec. 3.1"},{"comment":"The label \"f^{E-OPE}\" for the Rot-E LIIF example appears to be a typo; it should be \"f^{E-LIIF}\".","section":"Eq. (27), Sec. 3.2.4"},{"comment":"Some table entries appear corrupted or duplicated, e.g., the EDSR/LIIF x2 PSNR is listed as 32.18 for both Urban100 and BSD100, and several columns in Tables 3 and 4 are misaligned. Please regenerate the tables.","section":"Sec. 4.1, Table 3/4"},{"comment":"Typos: \"AISSR\" should be \"ASISR\" (Sec. 3.2), \"ERSD\" should be \"EDSR\" (Sec. 4.1), \"texting set\" should be \"testing set\" (Sec. 4.2), and \"represention\" appears in a figure caption.","section":"Sec. 3.2 and Sec. 4.1"},{"comment":"References [18] and [69] appear to describe the same work (\"Rotation equivariant proximal operator for deep unfolding methods in image restoration\") in different venues; please distinguish or de-duplicate them.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main correctness risk is the p8/p16 gap: the proposed B-Conv encoder is not exactly equivariant under those groups, so the approximate guarantee is currently unsubstantiated for the implemented system. This is fixable by quantifying the basis-closure error or by limiting the claim, but it must be addressed. Also ensure the missing supplement is included. The p4 contribution itself appears sound and the empirical gains are credible."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the p4 result is real; the p8/p16 theory is not. This paper deserves referee time, but the authors need to fix the theoretical overreach before it is publishable.\n\nThe genuinely new piece is the Rot-E INR layer stack—coordinate lifting with group-indexed weights (Eqs. 7–9) applied to the implicit function part of ASISR. That is a clean composition of standard group-equivariant layers and gives the first plug-in rotation equivariance for arbitrary-scale SR with exact 90° behavior. The B-Conv bicubic filter parametrization is a reasonable alternative to Fourier/harmonic bases, and the experiments show that replacing only the encoder or only the INR is not enough—you need both. The parameter savings are real and the empirical equivariance errors on p4 are close to zero, matching Corollary 3.\n\nThe soft spots are significant. The stress-test note is correct: the B-Conv basis is the tensor-product cubic B-spline on a fixed square grid, which is closed under 90° rotations but not under 45° or 22.5°. So for t=8 or t=16, the actual encoder is not exactly equivariant, and the premise of Theorem 3 fails. Corollary 4's Cδ + Ĉ/t bound tracks only the group-approximation error in an exactly equivariant network; it does not include the B-Conv representation error, which does not vanish with δ. Table 2's numbers conflate these two sources. The p4 exactness (Cor. 3) is unaffected. That is a load-bearing flaw in the stated guarantee, though not in the main p4 contribution.\n\nThere is also a concrete bug: Eq. (33), the Rot-E LTE instantiation, uses x where Eq. (7) requires A^{-1}x, and it has an extra sum over B. As printed, that formula does not realize the equivariance. The proofs are in a supplement that is not present in the arXiv version. The claim that the method 'consistently improves' performance does not survive Table 4—several ×12 entries go slightly backwards, which the text does acknowledge. No rotation-augmentation baseline is included, so part of the gain could come from regularization, though that is a minor point given the architectural argument.\n\nThe paper is worth engaging with: the p4 architecture is a solid composition, the plug-and-play results are mostly positive, and the flaws are addressable in revision. I would send it to peer review, but the authors must fix the theory to match the architecture or restrict the guarantee to p4, correct Eq. (33), include the supplement, and soften the overclaims.","headline":"Useful p4 plug-in for rotation equivariant ASISR, but the p8/p16 theory overreaches because the B-Conv basis isn't closed under 45° rotations.","tokens_in":30364,"tokens_out":4913,"would_cite":true,"duration_ms":56083,"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 builds the first arbitrary-scale super-resolution network that is rotation equivariant end to end: for 90-degree rotations the equivariance error is exactly zero, for other angles it is bounded, and the reworked INRs gain PSNR wi","keywords":["rotation equivariance","arbitrary-scale super-resolution","implicit neural representation","equivariance error analysis","filter parameterization","bicubic interpolation","plug-and-play INR","group-equivariant networks"],"falsifier":"Recompute the p4 equivariance error on 100 DIV2K test images with a trained Rot-E LTE model: if the NMSE between $\\Phi(\\pi_A(I))$ and $\\pi_A(\\Phi(I))$ is not near the reported $10^{-4}$–$10^{-3}$ (while the original LTE sits near 0.3), the central claim fails. Independently, implement Eq. (33) literally with $x$ rather than $A^{-1}x$ and test on p8: if the error does not follow the predicted $C\\delta+\\hat{C}/t$ bound, the printed formula does not realize the claimed equivariance.","tokens_in":29375,"feed_emoji":"🔁","tokens_out":10518,"duration_ms":98106,"temperature":0.7,"pith_summary":"Arbitrary-scale super-resolution (ASISR) builds a continuous image function from a low-resolution input so that any magnification can be sampled from it. This paper's claim is that the entire ASISR pipeline—encoder plus the implicit neural representation (INR)—can be made rotation equivariant for the first time: if the input image is rotated by 90 degrees, running the network gives the same result as running the network and rotating the output, with a guaranteed error bound for other angles. To get there, the paper redesigns the INR into three rotation-equivariant layer types whose group-indexed feature shifts mirror coordinate rotations, and it replaces the harmonic or Fourier filter bases of rotation-equivariant convolutions with bicubic-interpolated kernels. Applied to the standard INRs LIIF, OPE, and LTE, the method lowers equivariance error by three orders of magnitude, raises PSNR on in-scale and out-scale benchmarks with fewer parameters, and transfers to hyperspectral and thermal imagery.","feed_headline":"First rotation-equivariant arbitrary-scale super-resolution network","feed_subtitle":"Rotating the input image and upscaling now equals upscaling first and rotating the output, with proven error bounds.","key_machinery":"The load-bearing object is the group-indexed feature tensor $F$ of size $h\\times w\\times n\\times t$, whose fourth index labels the rotation group, and the local INR operator built from three equivariant layers: the input layer $H_{ij}(x,B)=\\sum_A \\varphi(W_{in}^{B^{-1}A}, F_{ij}^{A}, A^{-1}x)$, the intermediate layer $\\hat{H}_{ij}(x,A)=\\sum_B W^{A^{-1}B}\\cdot H_{ij}(x,B)$, and the output layer $f_{ij}(x)=\\psi\\bigl(W_{out}^{2}(\\sum_A W_{out}^{1}\\cdot \\hat{H}_{ij}(x,A))\\bigr)$. Each layer shifts the group index by the inverse rotation precisely when the coordinate is rotated by that rotation, which is the property that makes composition over $L$ layers equivariant (Theorem 1). The encoder side","core_discovery":"The paper's central claim is that the INR—the module inside an arbitrary-scale super-resolution network that maps local latent codes and continuous coordinates to pixel values—can be rebuilt so that the whole network is rotation equivariant end to end. The construction matches a rotation of the input coordinate with a cyclic shift of the feature tensor along the rotation-group dimension, so the input, intermediate, and output layers of the INR are equivariant by construction (Theorem 1). The paper proves that with a 2- or 4-element rotation group, the global SR map commutes exactly with the rotation (Corollary 3), and that for arbitrary rotation angles the equivariance error is at most $C\\de","pith_inferences":["One consequence the authors do not quantify is that end-to-end equivariance acts as a continuous data augmentation: the network's outputs are exactly consistent under the training group, so the effective sample count for rotationally symmetric patterns should be larger than for a comparable non-equivariant network; measuring training-set-size curves would test this.","The construction is modular in the choice of $\\varphi$ and $\\psi$, so any current or future INR whose core operation is a function of concatenated features and coordinates can likely be wrapped in the same input/intermediate/output layers; the paper only demonstrates LIIF, OPE, and LTE.","Self-flagged scope limit: the paper's conclusion concedes that for images whose local features are not rotationally symmetric (e.g., rainy images), strict fixed-angle equivariance may not improve and could hurt, so real deployments should assess the rotational symmetry of the target domain before adopting the method.","The general-group claim for the LTE instantiation rests on Eq. (33), which as printed uses $x$ rather than the $A^{-1}x$ required by Eq. (7); a literal implementation with $x$ would not be equivariant for $t>4$, so that formula is worth checking against the intended derivation."],"forward_implications":["With a p4 rotation group, the method guarantees that rotating the LR input by 90 degrees and upsampling equals upsampling and then rotating the HR output exactly; measured equivariance error drops from roughly 1.0 NMSE for standard LIIF/OPE/LTE to $10^{-3}$–$10^{-4}$.","Existing ASISR methods can be converted to rotation-equivariant versions by swapping in the three-layer Rot-E INR scheme; the paper validates this on LIIF, OPE, and LTE, with the best SwinIR-LTE gaining 0.2 dB on Urban100 $\\times2$ and trimming parameters from 13.7M to 7.7M.","For arbitrary rotation angles, the equivariance error is bounded by $C\\delta+\\hat{C}/t$, so finer low-resolution grids and larger rotation groups push the error toward zero.","The bicubic-basis filter parameterization (B-Conv) represents filter kernels more accurately than harmonic- or Fourier-basis approaches, producing the strongest SR results among the compared parameterizations, and can be swapped into existing rotation-equivariant convolution encoders.","Trained on natural images, the Rot-E versions improve SR reconstruction when transferred to hyperspectral (CAVE) and thermal datasets without retraining."],"supporting_citations":[{"why":"Defines the LIIF local-implicit-image-function baseline whose latent-code-plus-coordinate input the Rot-E input layer generalizes.","marker":"[6]"},{"why":"Provides the LTE baseline with learned amplitude/frequency codes; the paper's Rot-E conversion of it (Eq. 33) is the strongest demonstrated INR.","marker":"[7]"},{"why":"Introduces the parameter-free orthogonal-position-encoding INR, showing the Rot-E scheme also works with no learnable INR weights.","marker":"[8]"},{"why":"Supplies the Fourier-basis equivariant convolution (F-Conv) and the Rot-E encoder backbone that B-Conv modifies and outperforms in Table 9.","marker":"[25]"},{"why":"Establishes the group-indexed cyclic-shifting feature structure and steerable E(2)-equivariant layers that the Rot-E INR reuses.","marker":"[16]"},{"why":"Gives the PDO-eConv approximate-equivariance error analysis that the paper's error-bound theory parallels.","marker":"[17]"},{"why":"Provides the harmonic-basis steerable-filter parameterization, the second baseline for B-Conv in the encoder ablation.","marker":"[58]"},{"why":"Supplies the universal equivariant linear-layer construction that underlies the intermediate layer of the Rot-E INR.","marker":"[68]"},{"why":"Defines the cubic (bicubic) interpolation function used as the basis for B-Conv's kernel parameterization.","marker":"[70]"}],"fun_headline_variants":["First end-to-end rotation-equivariant super-resolution","Rotating then upscaling equals upscaling then rotating","Plug-and-play rotation equivariance for super-resolution","Bounded error rotation-equivariant super-resolution"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"The end-to-end guarantee assumes that rotating the local image patch acts on the feature tensor $F_{ij}$ exactly as a cyclic shift along the rotation-group dimension together with spatial rotation; this is exact only when the sampling grid is invariant under the rotation (the p4 group with $t=4$), so for p8 and p16 grids the approximation error is not tightly quantified.","fun_headline_variants_meta":{"raw":{"variants":["First end-to-end rotation-equivariant super-resolution","Rotating then upscaling equals upscaling then rotating","Plug-and-play rotation equivariance for super-resolution","Bounded error rotation-equivariant super-resolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000664,"raw_usage":{"total_tokens":2911,"prompt_tokens":828,"completion_tokens":2083,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":2022}},"tokens_in":572,"tokens_out":2083,"duration_ms":16519,"temperature":1.0,"reasoning_tokens":2022,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T23:31:12.437736+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the p4 equivariance error on 100 DIV2K test images with a trained Rot-E LTE model: if the NMSE between $\\Phi(\\pi_A(I))$ and $\\pi_A(\\Phi(I))$ is not near the reported $10^{-4}$–$10^{-3}$ (while the original LTE sits near 0.3), the central claim fails. Independently, implement Eq. (33) literally with $x$ rather than $A^{-1}x$ and test on p8: if the error does not follow the predicted $C\\delta+\\hat{C}/t$ bound, the printed formula does not realize the claimed equivariance.","supporting_citations":[{"cited_title":"Learning continuous image representation with local implicit image function","cited_arxiv_id":null,"evidence_quote":"Defines the LIIF local-implicit-image-function baseline whose latent-code-plus-coordinate input the Rot-E input layer generalizes."},{"cited_title":"Local texture estimator for implicit representation function","cited_arxiv_id":null,"evidence_quote":"Provides the LTE baseline with learned amplitude/frequency codes; the paper's Rot-E conversion of it (Eq. 33) is the strongest demonstrated INR."},{"cited_title":"Ope-sr: Orthogonal position encoding for designing a parameter-free upsampling module in arbitrary-scale image super-resolution","cited_arxiv_id":null,"evidence_quote":"Introduces the parameter-free orthogonal-position-encoding INR, showing the Rot-E scheme also works with no learnable INR weights."},{"cited_title":"Fourier series expansion based filter parametrization for equivariant convolu- tions.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the Fourier-basis equivariant convolution (F-Conv) and the Rot-E encoder backbone that B-Conv modifies and outperforms in Table 9."},{"cited_title":"General e (2)-equivariant steerable cnns.Advances in Neural Information Processing Systems, 32, 2019","cited_arxiv_id":null,"evidence_quote":"Establishes the group-indexed cyclic-shifting feature structure and steerable E(2)-equivariant layers that the Rot-E INR reuses."},{"cited_title":"Pdo-econvs: Partial differential operator based equivariant con- volutions","cited_arxiv_id":null,"evidence_quote":"Gives the PDO-eConv approximate-equivariance error analysis that the paper's error-bound theory parallels."},{"cited_title":"Universal equivariant multilayer percep- trons","cited_arxiv_id":null,"evidence_quote":"Supplies the universal equivariant linear-layer construction that underlies the intermediate layer of the Rot-E INR."},{"cited_title":"Image interpolation techniques in digital im- age processing: an overview.International Journal of Engineering Research and Applications, 4(10):70–73, 2014","cited_arxiv_id":null,"evidence_quote":"Defines the cubic (bicubic) interpolation function used as the basis for B-Conv's kernel parameterization."}],"review_version":1}