{"id":"d1660cd9-ac00-4d97-add5-b900f062e889","arxiv_id":"2502.00280","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper argues that tuning the mother wavelet frequency in Wav-KANs controls NTK eigenvalue decay and mitigates spectral bias, but the supporting eigenvalue derivation is flawed.","lead":"This paper claims that the frequency of the mother wavelet in Wavelet Kolmogorov-Arnold Networks controls the decay of neural tangent kernel eigenvalues, letting the networks learn high-frequency components faster. It applies this idea to physics-informed variants on Poisson, Heat, Helmholtz, and Wave equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rank-one NTK analysis cannot support the claimed frequency-control mechanism: a rank-one kernel has only one nonzero eigenvalue, and Proposition 3's x_r-dependent bound is not an eigenvalue statement.","rationale":"The reader's weakest-assumption identification is on target: the rank-one reduction and the NTK-limit transfer are the weak points. My stress-test sharpens this into a direct mathematical contradiction. Even before considering whether the rank-one model extends to the full architecture, the rank-one model itself does not contain the phenomenon it is invoked to explain. A rank-one kernel has a single nonzero eigenvalue, so there is no eigenvalue-decay spectrum whose rate can be modulated. Moreover, Proposition 2's claimed eigenfunction form is false for general scale parameters, and Proposition 3's x_r-dependent expression for λ is not an eigenvalue. The empirical plots, which show many nonzero NTK eigenvalues, therefore come from the finite-width network's actual NTK, not from the kernel analyzed in the theory. This does not prove the empirical trend is false, but it does mean the central theoretical claim—that wavelet frequency controls NTK eigenvalue decay and thereby mitigates spectral bias—is unsupported. I also note the absence of code, baselines, and error bars, but the decisive issue is the invalid theoretical derivation. Since the reader already recommended REJECT, my concern reinforces that verdict rather than moving it.","tokens_in":20507,"tokens_out":7704,"duration_ms":74452,"concrete_test":"Compute the exact eigen-decomposition of the integral operator with kernel K(x_r,x_s)=e^{-x_r^2/2}cos(b x_r) e^{-x_s^2/2}cos(b x_s) on [0,1] for b=1,5,10,15,25 with S=1. The operator has rank one, so report the single nonzero eigenvalue λ(b)=∫_0^1 e^{-x^2}cos^2(bx)dx and compare it with Proposition 3's bound and with Figs. 3(b)/4. If (as expected) all other eigenvalues are exactly zero and λ(b) is roughly constant in b, the analyzed kernel cannot produce the plotted decaying spectrum, and the claimed frequency control of eigenvalue decay is not established. Independently, plug g=ψ_1^{1/S} into the eigenvalue equation for S=2; the equality fails unless ψ_1 is constant.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the NTK eigenvalue spectrum of a Wav-KAN be reshaped by the mother-wavelet frequency. The proof vehicle is the rank-one kernel K(x_r,x_s)=ψ_1(x_r)ψ_1(x_s) from Eq. (10) and Proposition 2. A rank-one integral operator has exactly one nonzero eigenvalue, λ=∫_C ψ_1^2(x) dx; there is no eigenvalue-decay spectrum whose rate could be controlled. The paper's Figs. 3(b), 4, and 11(b) plot a full spectrum of many eigenvalues, which cannot come from this kernel. Proposition 2 is also algebraically wrong for S≠1: substituting g=C_1 ψ_1^{1/S} into ∫ψ_1(x_r)ψ_1(x_s)g(x_s)dx_s=λg(x_r) gives a left side proportional to ψ_1(x_r), not to g(x_r); the correct eigenfunction is ψ_1 itself. Consequently Proposition 3's formula (39), in which λ is expressed as an x_r-dependent quantity, is not a valid eigenvalue equation, and the bound (14), 1/4 e^{-4(b x_r - T/S)^2} ≤ λ, cannot describe a single eigenvalue. The empirical trend in Sec. 3.3 may be genuine, but it is not explained by this analysis; the theoretical mechanism asserted in the abstract is unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies spectral bias in Wavelet-based Kolmogorov-Arnold networks (Wav-KANs) through the lens of the neural tangent kernel (NTK). It claims that the decay rate of the NTK eigenvalues of a Wav-KAN can be controlled by the frequency of the mother wavelet, thereby mitigating spectral bias and accelerating learning of high-frequency components. The theoretical analysis is carried out in Section 3.2 for a simplified rank-one kernel and is then used to motivate experiments in which the mother-wavelet frequency b is varied while approximating u(x)=sin(2πx)+0.1 sin(50πx). The paper also replaces the neural network inside PINNs by Wav-KANs, forming 'Wav-KINNs', and reports experiments on Poisson, heat, Helmholtz, and wave equations with high-frequency components.","tokens_in":20853,"tokens_out":7067,"duration_ms":68034,"significance":"If the central claim were established, the paper would provide a useful design principle for Wav-KANs: choose the mother-wavelet frequency to reshape the NTK spectrum and thereby control the convergence of high-frequency components in both function approximation and physics-informed settings. The empirical trend reported in Section 3.3—larger b slows eigenvalue decay and improves fitting of the 50π component—is plausible and visible in the figures, and the paper is commendable for testing the approach on several PDEs and for explicitly acknowledging limitations such as loss-term imbalance and the computational difficulty of Wav-KINN NTKs. However, the theoretical propositions that are supposed to explain this trend are invalid, and the experiments are not designed as an independent test of the theory. As it stands, the manuscript does not provide a sound theoretical foundation for the claimed frequency-control mechanism, so its significance is currently limited to an empirical observation.","major_comments":[{"comment":"The rank-one kernel K(x_r,x_s)=ψ1(x_r)ψ1(x_s) has exactly one nonzero eigenvalue, λ=∫_C ψ1^2(x_s) dx_s, with eigenfunction ψ1; every function orthogonal to ψ1 has eigenvalue zero. Consequently there is no eigenvalue-decay spectrum whose rate could be controlled by b. Proposition 2's assertion that g=C1 ψ1^{1/S} is an eigenfunction is also algebraically wrong for S≠1: substituting it into the integral equation gives ψ1(x_r) times a constant on the left-hand side, not λ ψ1^{1/S}(x_r). The many-eigenvalue spectra plotted in Figures 3(b), 4, 11(b), 12(b), and 13(b) therefore cannot be produced by the kernel analyzed in Proposition 2, and the theoretical mechanism asserted in the abstract is not supported.","section":"Sec. 3.2, Eq. (10), Proposition 2"},{"comment":"Equation (39) is not an eigenvalue equation: it expresses λ as a function of x_r, whereas an eigenvalue of the rank-one kernel is a constant independent of x_r. For S=1 the actual nonzero eigenvalue is ∫_0^1 e^{-(x_s-T)^2} cos^2(b(x_s-T)) dx_s, which does not contain the factor e^{-2(x_r-T)^2} cos^2(b(x_r-T)) appearing in Eq. (39). The lower bounds (14) and (15) therefore do not bound any eigenvalue, and the conclusion that 'the decay rate of the eigenvalues can be modulated by adjusting the frequency of the selected mother wave function' is not established by this proposition.","section":"Sec. 3.2, Proposition 3, Eqs. (36)-(39)"},{"comment":"The empirical validation is partially circular. The text states that 'the frequency parameter b can be empirically selected based on learning curves to optimize performance,' and the experiments then show that b=15 works well for target frequency 50π. The theoretical bound in Proposition 3 does not predict which b should be used, does not involve the target frequency, and does not predict the observed transition near b≈15. Thus the experiments demonstrate a fitted observation rather than a successful prediction of the proposed spectral-bias mechanism.","section":"Sec. 3.3"},{"comment":"The NTK dynamics in Eqs. (4)-(5) require the infinite-width gradient-flow limit in which the kernel remains constant during training, but the experiments use finite-width networks (e.g., [1,35,1], [2,15,15,1], [2,20,20,20,1]) trained with Adam and LBFGS. The paper explicitly assumes that 'similar behavior will extend to the general architecture of Wav-KANs,' but this is the load-bearing transfer step from the rank-one toy model to the actual networks whose spectra are plotted. No argument, convergence check, or width-scaling study is provided to justify this transfer, so the connection between the theory and the measured finite-width eigenvalue spectra is not established.","section":"Sec. 3.2, Eqs. (4)-(5); Sec. 4"}],"minor_comments":[{"comment":"The Morlet wavelet is written as ψ(x)=e^{-x^2} cos(bx) in the Figure 3 caption but as e^{-x^2/2} cos(bx) earlier in the text and as e^{-(x-T)^2/(2S^2)} cos(b(x-T)/S) in Eq. (38); the normalization should be made consistent throughout.","section":"Fig. 3 caption vs. Eq. (38)"},{"comment":"Equation (14) writes the exponent as -4(bx_r - T/S)^2, while the proof's Eq. (40) derives -4b^2(x_r-T)^2; these are not equivalent. Eq. (15) is also not implied for arbitrary T∈[0,1], since for T<1/2 the minimum over x_r∈[0,1] of the exponent occurs at x_r=0, not at x_r=1.","section":"Proposition 3, Eqs. (14) and (15)"},{"comment":"The Helmholtz equation and the forcing q(x,y) contain the parameter k, but k is never specified. Since the solution and the forcing depend on k through the k^2 u term, the problem is incompletely defined as stated.","section":"Sec. 4, Eq. (21)"},{"comment":"The claim that increasing the number of hidden units avoids overfitting while still learning high-frequency components is supported only by training curves and eigenvalue plots; no test error or other generalization metric is reported for the hidden-unit experiment.","section":"Sec. 3.4"},{"comment":"There are numerous typos and minor wording issues, including 'scalars' for 'scales' in Section 3.3, 'hiden units' in Figure 6(b), 'freq' in Figure 3(c), and '100 data points x ∈ [0,1] that are spaced' in Section 3.3, which should be 'evenly spaced'.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The paper appears to be an early-stage manuscript with a promising empirical observation but with a theoretical core that is mathematically unsound. The rank-one kernel analysis cannot support the claimed spectrum control, and the experiments are not presented as a predictive test. In addition, the empirical comparisons with Fourier-feature methods and MFF rely on previously reported results rather than direct baselines, and the Helmholtz setup is underspecified. These issues go beyond local presentation and would require a substantially revised theoretical framework and experimental design to be resolved within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short take: the paper is an empirical demonstration of a possibly real effect, but the theoretical wrapper is mathematically broken. The claim that mother-wavelet frequency controls NTK eigenvalue decay in Wav-KANs is not established by Proposition 3.\n\nWhat's new: as far as I know, this is the first NTK analysis of Wav-KANs, and the question is worth asking. The figures do show a clear trend: higher b (Morlet frequency) slows the eigenvalue decay and helps fit a 50π term, and the hidden-unit experiments suggest a similar effect. The authors are also honest that Helmholtz and Wave equations needed manual loss balancing, and they flag the limitation themselves.\n\nThe soft spot is fatal for the theory. For the rank-one kernel K(x_r,x_s)=ψ(x_r)ψ(x_s), there is exactly one nonzero eigenvalue, λ=∫ψ^2, which does not depend on x_r. Proposition 3 instead derives an x_r-dependent expression and calls it a lower bound on λ. That is not an eigenvalue statement. The bound 1/4 e^{-4(b x_r - T)^2} ≤ λ cannot hold uniformly in x_r unless λ is something else. Proposition 2 is also algebraically off for S≠1: the proposed eigenfunction doesn't satisfy the integral equation. And the paper then plots dozens of eigenvalues in Figures 3 and 11, which cannot come from the rank-one kernel. So the central mechanism, that you can shape the NTK spectrum by tuning b, is unsupported.\n\nEmpirically, the paper would need baselines, error bars, and code for the trend to be convincing; it has none of those. The b values are selected after looking at learning curves, so the 'prediction' is partly a fitted choice. The PDE results are encouraging but unquantified against standard PINNs or MFF.\n\nThat said, the core empirical observation may survive a rewrite: something about higher wavelet frequency helps with high-frequency content. But the theoretical analysis, as written, is not just incomplete; it's wrong in a load-bearing way. A referee could help the authors either fix the math or reframe the paper as purely empirical, but the current version doesn't support the abstract's claim.\n\nWho is this for: people studying KAN spectral bias might want to know about the empirical trend, but I wouldn't cite it as theory. If I were the editor, I'd send it to a mathematically careful referee, because the empirical observation is interesting enough to be worth the effort; a desk reject is also defensible, but I'd want a second pair of eyes on the NTK derivation first.","headline":"A plausible empirical trend wrapped in an invalid NTK argument; the frequency-control claim is not supported by the math.","tokens_in":21373,"tokens_out":3476,"would_cite":false,"duration_ms":34685,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","42C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that the frequency of the mother wavelet in a Wav-KAN controls the decay rate of the network's neural tangent kernel eigenvalues, giving a tunable mechanism to counteract spectral bias in function approximation and in…","keywords":["spectral bias","neural tangent kernel","Wavelet Kolmogorov-Arnold networks","physics-informed neural networks","frequency control","Morlet wavelet","Fourier features","multi-scale PDEs"],"falsifier":"Compute the neural tangent kernel eigenvalues of the full two-layer $[1, 35, 1]$ Wav-KAN at initialization over the 100 training points for $b = 1, 5, 10, 15, 25$ on repeated random initializations, and check whether the spectral decay rate is monotonically decreasing in $b$. Because Proposition 3 is proven only for the rank-one kernel with fixed $S$ and $T$, any failure of that monotonicity in the full kernel — or evidence that hidden-unit count, not $b$, dominates the decay — would falsify the claimed frequency-control mechanism.","tokens_in":20285,"feed_emoji":"🌊","tokens_out":17898,"duration_ms":138823,"temperature":0.7,"pith_summary":"This paper claims that spectral bias in Wavelet Kolmogorov-Arnold Networks (Wav-KANs) — the tendency to learn low-frequency function components before high-frequency ones — can be controlled by tuning the frequency of the mother wavelet. Analyzing the network's neural tangent kernel (the kernel that governs training dynamics under gradient flow), the authors derive an eigenvalue lower bound for a Morlet-wavelet network showing that eigenvalue decay slows as the wavelet frequency parameter $b$ grows. They argue this gives a direct mechanism to accelerate learning of high-frequency components, and they verify it numerically on a multiscale target function and on Poisson, heat, wave, and Helmholtz equations solved by physics-informed versions of the network (Wav-KINNs). The payoff, if the analysis transfers to full networks, is a parameter-efficient way to mitigate spectral bias without the domain partitioning or extra hyperparameters that Fourier-feature methods require.","feed_headline":"Wavelet frequency dial controls how fast nets learn high frequencies","feed_subtitle":"Theory and experiments show wavelet frequency controls how fast high-frequency details are learned in PINNs.","key_machinery":"The load-bearing object is the neural tangent kernel of a single-layer Wav-KAN, $K(x_r, x_s) = \\sum_i \\psi_i(x_r) \\psi_i(x_s)$, obtained by fixing the translation and scale parameters $T$ and $S$ so that only the weights $W$ train (equation 10). In one dimension this becomes the rank-one kernel $\\psi_1(x_r)\\psi_1(x_s)$; Proposition 2 shows the eigenfunctions take the form $g(x_r) = C \\psi_1^{1/S}(x_r)$, and Proposition 3 derives the eigenvalue lower bound $\\lambda \\geq \\tfrac{1}{4} e^{-4b^2(x_r - T)^2}$ for the Morlet wavelet with $S = 1$. This bound is the whole mechanism: it expresses how the wavelet frequency $b$ controls the kernel's spectral decay, which through the gradient-flow solution $f(X; t) = (I - e^{-tK_{\\mathrm{ntk}}})Y$ determines how fast each frequency component of the target is learned. The same NTK equations (4)-(7) connect larger eigenvalues to faster convergence, so slowing the decay transfers directly into faster high-frequency learning.","core_discovery":"The paper's central claim is that the decay rate of a Wav-KAN's neural tangent kernel eigenvalues is controlled by the frequency of the mother wavelet, giving a handle on spectral bias. Working in the single-layer case with translation and scale parameters fixed so that only the weights train, the kernel becomes rank-one, $K(x_r, x_s) = \\psi_1(x_r)\\psi_1(x_s)$ (equation 10), and for the Morlet wavelet $\\psi(x) = e^{-x^2/2}\\cos(bx)$ with $S = 1$ and $T \\in [0,1]$ the authors prove the eigenvalue satisfies $\\lambda \\geq \\tfrac{1}{4} e^{-4b^2(x_r - T)^2}$ (Proposition 3). Larger $b$ keeps the bound from dropping as fast, which they interpret as slower eigenvalue decay and therefore faster convergence on high-frequency components. Numerically, on the target $u(x) = \\sin(2\\pi x) + 0.1\\sin(50\\pi x)$, raising $b$ from 1 to 15 lets a two-layer $[1, 35, 1]$ Wav-KAN capture the high-frequency detail, while $b = 25$ overfits; increasing the hidden-unit count achieves a similar effect at fixed $b$. The paper then embeds Wav-KANs in physics-informed networks (Wav-KINNs) and reports that they approximate solutions to Poisson, heat, and wave equations with high-frequency content without domain separation, and that the Helmholtz and wave equations require balancing the domain and boundary terms in the loss.","pith_inferences":["The rank-one argument proves the eigenvalue bound for a single wavelet neuron, so the transfer to deep, wide networks with trainable $T$ and $S$ is an empirical question that the paper's own experiments only partially cover; a direct test would be measuring the full network's NTK spectrum at initialization for several $b$ values.","If the mechanism holds generally, the wavelet frequency behaves like a bandwidth knob, which suggests an automatic tuning scheme: choose $b$ so that the NTK eigenvalue decay rate matches the target function's known or estimated frequency content, something the paper does not develop.","The overfitting observed at $b = 25$ implies a bias-variance trade-off, so for targets with unknown frequency ranges, cross-validating $b$ against the NTK spectrum or the validation loss may be a cheaper alternative to grid-searching Fourier-feature scales.","The authors note that an NTK analysis of Wav-KINNs (where the loss includes PDE residuals) is computationally prohibitive; an approximate or empirical kernel for the residual loss could turn the observed loss-imbalance problem into a principled weighting rule instead of manual tuning."],"forward_implications":["Practitioners can treat the mother wavelet's frequency $b$ as a tunable control over learning speed across frequency bands: raising $b$ accelerates convergence on high-frequency components, at the cost of overfitting when $b$ is too large (the paper observes this at $b = 25$).","Wav-KINNs approximate solutions of elliptic, parabolic, and hyperbolic PDEs with high-frequency content without partitioning the domain or adding Fourier-feature hyperparameters, and the paper reports they do so with fewer parameters than multiple-Fourier-feature methods.","Increasing the number of hidden units at fixed wavelet frequency flattens the NTK eigenvalue spectrum in the same direction as raising $b$, giving an architecture-based alternative to frequency tuning that the paper finds does not overfit.","Loss terms for Wav-KINNs must be balanced (the paper uses weights $\\lambda_D$ and $\\lambda_{\\partial D}$) to reach accurate solutions on Helmholtz and wave equations; without balancing, training stalls on one term.","Fixing the translation and scale parameters ($T \\in [0,1]$, $S = 1$) leaves only the weights trainable, which the paper shows reproduces the accuracy of the full model and reduces the parameter count."],"supporting_citations":[{"why":"Supplies the NTK-eigenvalue methodology and the Fourier-features baseline whose hyperparameter burden this paper aims to avoid.","marker":"[19]"},{"why":"Provides the neural tangent kernel and the gradient-flow dynamics that tie eigenvalue size to per-frequency convergence speed.","marker":"[25]"},{"why":"Defines the Wav-KAN architecture whose spectral behavior is the object of study.","marker":"[24]"},{"why":"Introduces Fourier features as the standard remedy for high-frequency learning that the wavelet-frequency control is compared against.","marker":"[10]"},{"why":"Establishes KANs inside physics-informed networks (KINNs), the framework that Wav-KINNs extend and compare with on spectral bias.","marker":"[27]"},{"why":"Provides prior evidence that Wav-KANs perform well inside PINNs, motivating the Wav-KINN experiments.","marker":"[28]"},{"why":"Supplies the spectral-bias theory used to justify the eigenvalue-to-convergence link and to discuss loss-term imbalance.","marker":"[11]"}],"fun_headline_variants":["Wavelet frequency dial tunes how fast KANs pick up high-frequency details","Want high-frequency learning in KANs? Crank the wavelet frequency","The wavelet's frequency is the key to beating spectral bias in KANs","New theory: wavelet frequency controls NTK decay, easing spectral bias"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of frequency control covers only a single wavelet neuron with fixed translation and scale parameters, and the paper assumes the same behavior carries over to full multi-layer Wav-KANs with all parameters trainable.","fun_headline_variants_meta":{"raw":{"variants":["Wavelet frequency dial tunes how fast KANs pick up high-frequency details","Want high-frequency learning in KANs? Crank the wavelet frequency","The wavelet's frequency is the key to beating spectral bias in KANs","New theory: wavelet frequency controls NTK decay, easing spectral bias"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00085,"raw_usage":{"total_tokens":3783,"prompt_tokens":1116,"completion_tokens":2667,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":732,"completion_tokens_details":{"reasoning_tokens":2596}},"tokens_in":732,"tokens_out":2667,"duration_ms":19739,"temperature":1.0,"reasoning_tokens":2596,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T19:34:25.881909+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the neural tangent kernel eigenvalues of the full two-layer $[1, 35, 1]$ Wav-KAN at initialization over the 100 training points for $b = 1, 5, 10, 15, 25$ on repeated random initializations, and check whether the spectral decay rate is monotonically decreasing in $b$. Because Proposition 3 is proven only for the rank-one kernel with fixed $S$ and $T$, any failure of that monotonicity in the full kernel — or evidence that hidden-unit count, not $b$, dominates the decay — would falsify the claimed frequency-control mechanism.","supporting_citations":[{"cited_title":"Neural tangent kernel: convergence and generalization in neural networks","cited_arxiv_id":null,"evidence_quote":"Provides the neural tangent kernel and the gradient-flow dynamics that tie eigenvalue size to per-frequency convergence speed."},{"cited_title":"Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T","cited_arxiv_id":null,"evidence_quote":"Introduces Fourier features as the standard remedy for high-frequency learning that the wavelet-frequency control is compared against."},{"cited_title":"Kolmogorov arnold informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on kolmogorov arnold networks, 2024","cited_arxiv_id":null,"evidence_quote":"Establishes KANs inside physics-informed networks (KINNs), the framework that Wav-KINNs extend and compare with on spectral bias."},{"cited_title":"Bondar, and Abhijit Sen","cited_arxiv_id":null,"evidence_quote":"Provides prior evidence that Wav-KANs perform well inside PINNs, motivating the Wav-KINN experiments."},{"cited_title":"Towards understand- ing the spectral bias of deep learning, 2020","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral-bias theory used to justify the eigenvalue-to-convergence link and to discuss loss-term imbalance."}],"review_version":1}