{"id":"410fbd85-1065-48ff-adde-493f04b27384","arxiv_id":"2507.13393","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"CDF normalization to quantiles improves MNIST test accuracy and training speed of Legendre-KAN compared with min-max scaling.","lead":"Replacing standard min-max scaling with Gaussian CDF normalization in Legendre-based KANs gives a 0.5-2 percentage point accuracy gain on MNIST and roughly 2x faster convergence. The paper also interprets KAN weights as moments of local joint distributions, moving toward explainable neural networks.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claim that CDF normalization alone improves Legendre-KAN is confounded: KAL_NET and CDFKAL_NET differ in SiLU count and LayerNorm placement, not only normalization.","rationale":"The reader's weakest assumption focused on the uniformity mechanism and its failure in Fig. 9. I agree that is a weakness, but I see a more load-bearing issue: the experimental design does not isolate normalization because the compared architectures differ in SiLU count and LayerNorm placement. This directly undermines the strongest claim, 'by just switching rescaling.' The reader's conditional verdict is still appropriate, but the condition should explicitly require a controlled one-variable comparison. Therefore the verdict remains CONDITIONAL, hence UNCHANGED. My alignment with the reader is partial because we identify different primary threats, though both point to insufficient evidence for the causal claim.","tokens_in":8082,"tokens_out":3948,"duration_ms":46385,"concrete_test":"Run an architecture-constant comparison: take the KAL_NET implementation and replace only the MinMax rescaling with per-feature Gaussian-CDF normalization (z-score followed by Phi), leaving double SiLU, LayerNorm placement, optimizer, learning rate, batch size, and data split exactly unchanged. Sweep Legendre degrees 3–7 on the same 20,000-image subset with at least 5 random seeds, and compare mean and standard deviation of test accuracy. If the CDF-only variant does not beat KAL_NET by roughly 0.5–2 pp on average, the paper's causal claim fails. A complementary check is to add the double SiLU to CDFKAL_NET and see whether the gap persists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest claim (Sec. VI.D) is that replacing MinMax scaling with Gaussian-CDF normalization in CDFKAL_NET yields a consistent 0.5–2 pp accuracy gain. However, the architecture comparison in Sec. VI.C changes multiple factors simultaneously. KAL_NET uses per-layer MinMax rescaling, double SiLU nonlinearities, and a final LayerNorm; CDFKAL_NET and CDFKAL_NET_FIXEDNORM use LayerNorm before the CDF transform and omit SiLU entirely; CDFKAL_SILU adds a single SiLU, not the double SiLU of KAL_NET. Thus the measured gain cannot be causally attributed to the normalization switch: it may come from removing one SiLU, from the changed LayerNorm placement, or from an interaction. The paper itself notes that the original LegendreKAN uses LayerNorm with trainable beta/gamma (Sec. V.C), and the CDF variants alter that placement and freeze or train these parameters differently. Without an architecture-constant comparison, the central causal claim is not established. The proposed uniformity mechanism is also only partially supported: Fig. 9 shows the first CDF-normalized layer remains far from uniform, so the stated reason for low-degree polynomial improvement is not directly observed on the data used. The lack of multiple seeds and error bars amplifies the uncertainty, but the confound is the primary correctness risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes replacing the MinMax rescaling used in Legendre-KAN with Gaussian-CDF normalization to quantiles, arguing that near-uniform inputs on [0,1] better match the orthonormal Legendre basis and thus improve generalization. Experiments on a 20,000-sample MNIST subset compare four variants (KAL_NET, CDFKAL_NET, CDFKAL_NET_FIXEDNORM, CDFKAL_SILU) across Legendre degrees 3–11, and the authors report a consistent 0.5–2 percentage-point accuracy gain and roughly 2× faster convergence for the CDF-based variants. The paper also presents an HCR (hierarchical correlation reconstruction) interpretation of polynomial KAN weights as moments and suggests extensions such as mutual information estimation and bidirectional propagation.","tokens_in":8358,"tokens_out":2953,"duration_ms":32373,"significance":"If the central causal claim were established, the result would be significant and practically useful: a simple, cheap preprocessing change that improves both accuracy and convergence speed for polynomial KANs, with a coherent interpretability framework. The manuscript has real strengths: the code is promised to be public, the comparison includes several closely related variants, and Figure 9 honestly shows that the first-layer features remain far from uniform, which is a useful diagnostic. However, the experimental design does not currently isolate the normalization choice from other architectural changes, so the headline claim is not yet supported. The HCR material is suggestive but secondary to the empirical claim.","major_comments":[{"comment":"The central comparison is confounded: KAL_NET and CDFKAL_NET differ in more than the normalization transform. According to the architecture list in Section VI.C(a), KAL_NET uses per-layer MinMax rescaling, double SiLU nonlinearities, and a final LayerNorm, while CDFKAL_NET uses LayerNorm before the CDF transform, omits SiLU entirely, and places LayerNorm before each layer (Section V.E). The 0.5–2 pp gain reported in Section VI.D can therefore be caused by removing one SiLU, by changing LayerNorm placement, or by an interaction, rather than by CDF normalization per se. The paper needs controlled ablations that vary only the normalization transform while holding activation counts and LayerNorm placement fixed; it should also include a plain z-score (without the CDF step) condition to show that the CDF step specifically is what helps.","section":"VI.C–VI.D"},{"comment":"All accuracy and convergence results appear to be based on a single run per configuration on a 20,000-image subset of MNIST. The claim that the gain is 'consistent' requires statistical support: report mean and standard deviation over multiple seeds (at least five) and, if appropriate, a paired significance test. Without this, the magnitude of the 0.5–2 pp effect cannot be distinguished from run-to-run variability.","section":"VI.B–VI.D"},{"comment":"The proposed mechanism—CDF normalization makes features nearly uniform, which is why low-degree Legendre polynomials generalize better—is only partially supported by the data. Figure 9 itself shows that the first-layer features after Gaussian-CDF normalization are still far from uniform on MNIST, and the text acknowledges this. The paper should provide quantitative uniformity diagnostics (e.g., histogram divergence from uniform, or a comparison with EDF normalization) and, ideally, show that uniformity correlates with the accuracy gain across degrees. Otherwise the mechanism remains a conjecture rather than an explanation of the observed improvement.","section":"V.C and Fig. 9"},{"comment":"The convergence claims are not quantitatively pinned down. The text states 'converges noticeably faster,' 'shaves roughly 5–10% off the wall-clock time,' and 'converge ≈ 2× faster,' but no convergence criterion is defined (e.g., epochs to reach a target test loss or test accuracy), and Figure 8 shows curves without error bars. Please define the criterion, report the relevant numbers with variance, and state whether the 2× figure refers to epochs to convergence or to wall-clock time; the two are not the same.","section":"VI.D(b)–VI.D(c)"}],"minor_comments":[{"comment":"The caption describes the experiments as 'polynomial regression tasks of degree 3 through 11,' but Section VI.B states that all models are trained on MNIST classification. Please correct the caption to match the actual task.","section":"Fig. 8 caption"},{"comment":"Reference [3] (Legendre-KAN) lacks venue, year, and arXiv identifier; this makes it difficult for readers to locate the baseline implementation. Please complete the citation.","section":"References"},{"comment":"The naming is inconsistent: 'CDFKAL_NET', 'CDFKAL NET', and 'CDFKAL' are all used for what appears to be the same model. Please standardize the notation.","section":"Throughout"},{"comment":"The entropy and mutual information approximations are introduced without stating the required normalization of the basis or the conditions under which the ln(1+a) ≈ a approximation is accurate. A brief derivation or a reference to where this is worked out would help the reader assess the range of validity.","section":"Section IV, Eq. (10)–(11)"},{"comment":"Figure 7 reports wall-clock time, but the caption says 'time required ... to reach convergence.' Since the models may reach different final accuracies at different times, please clarify whether the time is to a fixed epoch count or to a defined convergence point.","section":"Section VI.D(b)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central empirical claim is plausible and worth testing, but the current experiment is confounded. The authors should be able to fix this with additional controlled runs, so I see no reason to reject outright. The reliance on the authors' own prior work [2] for the HCR framing is disclosed and is not the main issue; the main risk is that the 'normalization matters' headline could evaporate once SiLU counts and LayerNorm placement are held fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The empirical headline is undersupported because the architecture comparison changes too many things at once, but the idea is sensible, the writing is clear, and the fixed-norm variant is a useful practical data point. Worth a serious referee if the authors tighten the comparison.\n\nWhat's actually new: applying Gaussian-CDF quantile normalization to Legendre-KAN and sweeping polynomial degree on MNIST, including a frozen-LayerNorm variant that shaves training time. The figures honestly show the accuracy gain shrinking at higher degrees, which is the right way to present the claim. The HCR interpretation is mostly restated from the authors' own prior work, but the argument that low-degree Legendre polynomials benefit from more uniform inputs is at least coherent.\n\nNow the soft spots. The stress-test concern lands. Section VI.D attributes the 0.5–2 pp gain to switching normalization, but KAL_NET and CDFKAL_NET differ in SiLU count, LayerNorm placement, and the presence of the final LayerNorm. That is a load-bearing confound. Without an ablation that holds the architecture constant (e.g., KAL_NET with CDF normalization, or CDFKAL_NET with double SiLU and final LayerNorm), the gain cannot be pinned on the normalization. Also missing: error bars, multiple seeds, a second dataset. The “≈ 2× faster” convergence claim is qualitative and not backed by a number with variance. The paper's own Fig. 9 shows the first CDF-normalized layer is still far from uniform, so the stated mechanism is only partially observed on the very data used. The self-citation pattern is heavy but not improper; the CDF method does come from [2], and the authors do compare against an external baseline.\n\nWho this is for: researchers working on KAN variants or on input normalization for polynomial activation networks. They will get a quick, readable case study and a reasonable hypothesis, but not a controlled test of it.\n\nRecommendation: send it to peer review, but with a referee brief asking for an architecture-constant ablation, error bars, and at least one additional dataset before the causal claim is accepted. The idea is plausible enough to merit that investment, and the current version is close enough that a revision could settle it.","headline":"The core claim about CDF normalization is plausible but the experiment changes too many architectural details to prove it; the idea is worth taking seriously and the paper deserves a careful referee.","tokens_in":8836,"tokens_out":1670,"would_cite":false,"duration_ms":19974,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"CDF normalization lifts Legendre-KAN accuracy by up to 2 points.","keywords":["Kolmogorov-Arnold networks","Legendre polynomials","CDF normalization","quantile normalization","data normalization","hierarchical correlation reconstruction","MNIST","copula"],"falsifier":"A concrete falsifier is to run the identical degree sweep on a dataset whose raw features are already independent and uniform on [0,1]: if CDF-normalized KAN does not beat MinMax-KAN there, the gain is not about matching the Legendre basis support but about correcting non-uniform inputs. A second check would replace the Gaussian CDF with an empirical CDF (rank transform) on MNIST; if accuracy fails to improve while the histograms become more uniform, the uniformity mechanism itself is in doubt.","tokens_in":1533,"feed_emoji":"📈","tokens_out":2290,"duration_ms":64496,"temperature":0.7,"pith_summary":"The paper claims that the choice of input normalization is a major driver of how well Legendre-polynomial KANs generalize. Replacing the MinMax rescaling used in the original Legendre-KAN with a two-step Gaussian-CDF transform, which maps each feature to an approximately uniform variable on [0,1], raises MNIST test accuracy by roughly 0.5 to 2 percentage points for all polynomial degrees below 8. The same swap also cuts the number of epochs needed to reach a given accuracy by about half. The authors argue that low-degree Legendre polynomials, which are orthonormal on [0,1], can represent near-uniform inputs with less overfitting, and they offer an interpretation of the resulting network as a hierarchical correlation reconstruction whose weights are mixed moments of a local joint distribution.","feed_headline":"CDF normalization lifts Legendre-KAN accuracy by up to 2 points","feed_subtitle":"Near-uniform inputs let low-degree polynomials generalize better and reach target accuracy in half the epochs.","key_machinery":"The central object is the Gaussian-CDF normalization layer: each feature is z-score standardized and then transformed by the standard normal cumulative distribution function, $u = \\frac{1}{2}(1+\\mathrm{erf}((x-\\mu)/(\\sigma\\sqrt{2})))$, mapping values into (0,1) with an approximately uniform distribution when the feature is roughly Gaussian. This output feeds rescaled Legendre polynomials, which are orthonormal on [0,1] with respect to the uniform weight, so the composition matches the basis's design assumption. The paper also develops an HCR (hierarchical correlation reconstruction) reading in which the trained coefficients are mixed moments of a local joint density model, which explains why uniform inputs make low-order polynomial terms sufficient and gives a route to interpretability and information-theoretic quantities.","core_discovery":"On the paper's own terms, the central discovery is that replacing MinMax scaling with Gaussian-CDF normalization, namely standardizing each feature by its batch mean and standard deviation and then passing the result through the standard normal CDF, $\\frac{1}{2}(1+\\mathrm{erf}(x/\\sqrt{2}))$, consistently improves Legendre-KAN test accuracy on MNIST by about 0.5 to 2 percentage points for all Legendre degrees lower than 8, while also roughly doubling the convergence speed in epochs. A variant with frozen affine parameters, CDFKAL NET FIXEDNORM, retains most of the accuracy gain and shaves about 5 to 10 percent off wall-clock training time relative to the baseline. The authors interpret the improvement as evidence that uniformizing inputs better matches the orthonormality assumption of the Legendre basis, so that low-degree polynomials, which generalize better, become sufficient.","pith_inferences":["A natural untested extension is to replace the Gaussian CDF with an empirical CDF (rank-based) normalization; the paper's own Figure 9 shows the first-layer features after the Gaussian transform are still far from uniform, so an EDF variant could plausibly push the accuracy gain further on non-Gaussian data.","The same uniformity argument should apply to other global orthonormal bases used in KANs, such as Chebyshev or DCT bases, since they also assume a fixed support; if confirmed, normalization choice rather than basis choice would be the dominant factor in low-degree generalization.","The mutual information approximation from the HCR reading could be turned into a training regularizer, such as an information-bottleneck loss, making the normalization insight a principled lever for controlling overfitting in deep polynomial networks."],"forward_implications":["If the claim holds, any Legendre-KAN implementation can improve both accuracy and convergence by swapping its input scaler to Gaussian-CDF normalization, with no other architectural change.","The fixed-norm variant shows that trainable affine parameters after the CDF are unnecessary, so the normalization can be made simpler and cheaper while keeping most of the benefit.","Low-degree polynomial KANs, which the paper shows generalize best, become more practical because CDF normalization removes the need to rely on high-degree terms that overfit.","The HCR interpretation suggests KAN coefficients can be read as moments of a local joint density, opening a path to explainability and to approximate entropy or mutual information from the network's weights."],"supporting_citations":[{"why":"The Legendre-KAN baseline that supplies the MinMax scaling and architecture which the paper replaces and compares against.","marker":"[3]"},{"why":"Batch normalization, which provides the z-score standardization step that the Gaussian-CDF transform builds on.","marker":"[4]"},{"why":"Earlier proposal of CDF normalization in the machine-learning context and the HCR neuron interpretation used throughout the paper.","marker":"[2]"},{"why":"Copula theory, which motivates normalization to quantiles and nearly uniform marginals, the conceptual source of the method.","marker":"[1]"},{"why":"The Kolmogorov-Arnold network framework that Legendre-KAN and the CDF variants instantiate.","marker":"[6]"},{"why":"HCR density parametrization, which provides the moment-based interpretation of the polynomial coefficients in Section IV.","marker":"[5]"}],"fun_headline_variants":["CDF normalization boosts Legendre-KAN by up to 2 points","Quantile-normalized inputs double Legendre-KAN speed and accuracy","Swap MinMax for CDF to gain 2 points on Legendre-KAN","CDF normalization: +2 pts accuracy, half the epochs for KAN","Near-uniform inputs make Legendre-KAN fit faster and better"],"cache_read_input_tokens":11008,"weakest_assumption_plain":"The claim rests on the assumption that Gaussian-CDF transformation makes each input feature nearly uniform on [0,1], which is the stated reason low-degree Legendre polynomials generalize better; the paper's own Figure 9 shows first-layer MNIST features after the transform are still far from uniform.","fun_headline_variants_meta":{"raw":{"variants":["CDF normalization boosts Legendre-KAN by up to 2 points","Quantile-normalized inputs double Legendre-KAN speed and accuracy","Swap MinMax for CDF to gain 2 points on Legendre-KAN","CDF normalization: +2 pts accuracy, half the epochs for KAN","Near-uniform inputs make Legendre-KAN fit faster and better"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000592,"raw_usage":{"total_tokens":2736,"prompt_tokens":869,"completion_tokens":1867,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":1771}},"tokens_in":485,"tokens_out":1867,"duration_ms":15329,"temperature":1.0,"reasoning_tokens":1771,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T16:54:10.391867+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete falsifier is to run the identical degree sweep on a dataset whose raw features are already independent and uniform on [0,1]: if CDF-normalized KAN does not beat MinMax-KAN there, the gain is not about matching the Legendre basis support but about correcting non-uniform inputs. A second check would replace the Gaussian CDF with an empirical CDF (rank transform) on MNIST; if accuracy fails to improve while the histograms become more uniform, the uniformity mechanism itself is in doubt.","supporting_citations":[{"cited_title":"Legendre-kan: High accuracy ka network based on legendre polynomials","cited_arxiv_id":null,"evidence_quote":"The Legendre-KAN baseline that supplies the MinMax scaling and architecture which the paper replaces and compares against."},{"cited_title":"Copula theory: an introduction,","cited_arxiv_id":null,"evidence_quote":"Copula theory, which motivates normalization to quantiles and nearly uniform marginals, the conceptual source of the method."}],"review_version":1}