{"id":"5a63d0e2-be5c-4c83-b2d4-473e82b692ac","arxiv_id":"2602.05977","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"ClKAN extends complex-valued KANs to arbitrary Clifford algebras, and using scrambled Sobol-sequence grids cuts the parameter count in higher-dimensional spaces.","lead":"Kolmogorov-Arnold Networks are a neural-network design extended to Clifford algebras (quaternions, geometric algebra) using smart random grids to shrink the parameter count. The paper tests the architecture on synthetic and physics-inspired function-fitting tasks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma IV.2 conflates trained weights with point evaluations of a fixed g and omits the 1/n QMC normalization; the O(n^{-1}) expressivity claim is not established.","rationale":"The reader's weakest assumption is exactly the point that breaks Lemma IV.2. In QMC integration the weights are values of a fixed function g at sample points; in the trained network they are free parameters optimized on data and therefore depend on the grid and dataset. The missing 1/n factor is a further, independent algebraic slip: with w_g=g(g) the sum Φ is n times the QMC estimate, so the cited scrambled-net bound gives O(n) variance, not O(n^{-1}). Because the paper's 'solution to exponential scaling' is sold on this lemma, the theory must be repaired (e.g., by replacing the QMC claim with a standard RBF approximation error bound) or the claim should be downgraded to an empirical observation. The experiments still show interesting parameter reductions, so conditional acceptance remains appropriate.","tokens_in":12788,"tokens_out":6608,"duration_ms":68042,"concrete_test":"Take a smooth target h(x)=∫_{[0,1]^d} g(y)φ(x−y)dy with fixed g and φ (e.g., a Gaussian). For n=2^m, m=1..8, draw 200 scrambled Sobol nets. For each net, (A) set w_g=(1/n)g(g) and compute the network sum Σ w_g φ(x−g); (B) optimize w to fit h on a fixed set of x points, re-fitting for each net. Estimate over the 200 nets E∥Φ−h∥^2 and the bias ∥E[Φ]−h∥^2 as a function of n. If (A) shows O(n^{-1}) but (B) does not, or if the bias does not vanish, Lemma IV.2's application to the trained network is refuted. This isolates the fixed-g assumption from the normalization issue.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theoretical claim rests on Lemma IV.2. Its proof rewrites the network sum Φ(x)=Σ_{g∈G} w_g φ(x−g) as the scrambled-net estimator ĥ_n(x)=(1/n)Σ_i g(y_i)φ(x−y_i) by setting g(y_i)=w_g. This identification fails twice. First, the QMC estimator carries a 1/n factor; Φ does not, so under the cited bound Var(Φ) would be O(n), not O(n^{-1}). Second, Property IV.1 requires the integrand f_x(y)=g(y)φ(x−y) to be fixed before the sample is drawn. In the trained network, w_g are free parameters optimized on a dataset; they depend on the particular Sobol realization and on the target, so E[Φ(x)] over random scramblings need not equal ∫ g(y)φ(x−y)dy for any fixed g. Unbiasedness and the O(n^{-1}) variance therefore do not follow. The subsequent sentence 'training our KAN means training g(y) evaluated at the grid points' is a reinterpretation, not a proof. Consequently the paper's headline theoretical support for Sobol grids as a cure for exponential scaling is unsupported; only the empirical parameter-reduction observation remains.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Clifford Kolmogorov-Arnold Networks (ClKAN), an extension of complex-valued KANs to arbitrary Clifford algebras. The main architectural contributions are two types of RBFs (naive and 'Clifford' RBF), three batch-normalization strategies for Clifford-valued activations, and a randomized quasi-Monte Carlo (RQMC) Sobol grid for generating the RBF centers. The central theoretical claim is Lemma IV.2, which states that the Sobol-grid network sum is an unbiased estimator of a convolution integral with O(n^{-1}) variance, and the paper uses this to argue that Sobol grids mitigate the exponential parameter growth of full grids. The experiments compare ClKAN with CVKAN baselines on complex-valued synthetic tasks, a holography dataset, and higher-dimensional Clifford algebras (e.g., Cl(2), Cl(0,2), Cl(1,1), Cl(1,0,1)).","tokens_in":13086,"tokens_out":5363,"duration_ms":55760,"significance":"If Lemma IV.2 were correct, the paper would provide a principled, scalable way to avoid the exponential growth of grid points in hypercomplex KANs, which is a relevant practical problem for scientific and engineering applications. The architecture extension itself is natural, and the paper covers several Clifford algebras and datasets; the commitment to release code is a strength. However, the theoretical support is invalid as written, and the empirical evidence is weakened by comparison-protocol mixing, best-configuration selection without error bars, and high run-to-run variance in several reported configurations. With a corrected theoretical statement (or an explicit demotion of the Sobol-grid claim to a heuristic) and more rigorous experimental reporting, the paper could become a useful contribution.","major_comments":[{"comment":"The proof identifies the network sum Φ(x)=Σ_{g∈G} w_g φ(x−g) with the scrambled-net estimator ĥ_n(x)=(1/n)Σ_i g(y_i)φ(x−y_i) by setting g(y_i)=w_g. This identification fails twice. First, Φ contains no 1/n normalization; under the cited Property IV.1 the unscaled sum would have variance O(n), not O(n^{-1}). Second, Property IV.1 requires the integrand f_x(y)=g(y)φ(x−y) to be fixed before the sample is drawn; in a trained network the weights w_g are optimized on data and depend on the particular Sobol realization and the target function, so unbiasedness of Φ over random scramblings does not follow. The sentence 'training our KAN means training g(y) evaluated at grid points' is a reinterpretation, not a proof. The asserted O(n^{-1}) expressivity result is therefore unsupported.","section":"IV-D, Lemma IV.2"},{"comment":"The headline comparison mixes experimental protocols. The CVKAN baselines [7], [8] use learning rate 0.01 and report validation loss, while ClKAN uses learning rate 0.1, adds a separate test split, and reports test loss. Table I selects the best model across all RBF types, batch-normalization strategies, grid types, and architecture sizes, but gives no error bars, no number of configurations tried, and no significance testing. The claim that ClKAN 'achieves similar results' is therefore not established by this table; the improvement over [8] is confounded by the learning-rate increase.","section":"V-A, Table I"},{"comment":"The evidence for the higher-dimensional benefit of the Sobol grid is not robust. Several configurations in Table IV have standard deviations comparable to or exceeding the mean, e.g., square N_g=5: 0.102±0.199 and squaresquare N_g=8: 1.425±2.414. The conclusion that Sobol grids 'consistently' outperform the full grid is based on selected N_g values without multiple-comparison control. The reported instabilities undermine the parameter-reduction claim as a general statement; at minimum, per-fold results and a significance analysis are needed.","section":"VI-D, Table IV"}],"minor_comments":[{"comment":"The symbol n is used both for the vector-space dimension in Section III-A and for the grid size |G| in Section IV-D. The (t,m,d)-net definition notes this, but the ambiguity makes Lemma IV.2 and Eq. (5) hard to follow.","section":"III-A and IV-D"},{"comment":"Calling x φ(x) a 'radial basis function' is misleading, since the term is not radial (it depends on direction). Suggest a name such as 'Clifford-weighted RBF'.","section":"IV-A, Eq. (1b)"},{"comment":"The phrase 'grid points per dimension' for the Sobol grid is ambiguous. For the full grid, N_g^D grid points are used; it is unclear whether the same total is used for the Sobol grid or whether N_g denotes the number of Sobol samples. The parameter counts suggest the latter for some tables, but this should be stated explicitly.","section":"V-C and Tables II/IV"},{"comment":"The proof of Property IV.1 cites [35] and [37] without specifying the exact theorem or the precise regularity conditions on f_x beyond L^{1+ε}. A self-contained statement of the theorem would help.","section":"IV-D, Property IV.1"},{"comment":"The figures omit S-5 and S-7 'for better readability', but these are exactly the configurations that show high variance in Table IV. Please either include them or provide the omitted data in a supplementary table.","section":"VI-B and VI-D, Figs. 4-5"}],"recommendation":"major_revision","confidential_remarks":"The theoretical expressivity claim (Lemma IV.2) is presented as a key contribution, yet the proof is invalid as written: the missing 1/n normalization and the dependence of trained weights on the Sobol realization are not fixable by minor edits. The revision should either prove a correct statement (e.g., with appropriately scaled random weights and an analysis that accounts for optimization) or explicitly reframe the Sobol-grid contribution as an empirical parameter-reduction heuristic. The empirical section also needs more careful comparison reporting before the parameter-reduction claim can be considered supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: ClKAN is a reasonable, incremental extension of CVKAN to Clifford algebras, and the Sobol-grid parameter reduction appears to work in their experiments. But Lemma IV.2, which the paper leans on for its central expressivity claim, is not a valid proof as written. Treat this as an engineering contribution with an overstated theoretical justification.\n\nWhat's new and good: replacing the complex field with a general Clifford algebra in the edge functions is a natural move, and the scrambled-Sobol grid is a non-obvious practical tweak that lets the network use far fewer grid points than a uniform lattice. The three batch-normalization variants are straightforward but useful for practitioners. They also report a concrete finding: for some Clifford algebras (e.g., Cl(1,0,1)), a Sobol grid with a few hundred parameters can beat a full grid with hundreds of thousands on synthetic fitting tasks. That is worth knowing.\n\nSoft spots, in proportion: The main issue is Section IV-D. Lemma IV.2 claims Φ(x) is an unbiased QMC estimator of a convolution integral with variance O(n^{-1}). To get there, the proof sets g(y_i) = w_g, treating trained weights as point evaluations of a fixed function independent of the grid. In reality, the weights are optimized on a fixed scrambled net, so the randomization bound from Property IV.1 does not apply. The lemma also drops the 1/n factor that appears in the QMC estimator (though that could be absorbed into w_g). As written, the claim that Sobol-CliffordKAN 'can learn any function that can be written as a convolution with kernel φ with error O(n^{-1})' is not established. The paper also overstates the 'solution to exponential scaling': the Sobol grid does reduce parameters by making N_g the total point count, but it doesn't remove the dimension dependence of the underlying function approximation problem.\n\nEmpirically, the headline comparison in Table I picks the best configuration across many RBF, normalization, grid, and architecture choices, with no error bars, and uses a 10x higher learning rate than the original CVKAN baseline. That makes the numerical gains hard to interpret. Some Sobol-grid runs in Table IV have large variance (e.g., N_g = 8 for squaresquare). These are fixable by pre-registering configurations and reporting error bars, but they matter.\n\nNone of this makes the paper worthless. The parameter-reduction effect is visible in the higher-dimensional experiments, and the code release is a plus. I would send this to a serious referee, but with the expectation that Lemma IV.2 needs to be either corrected (more likely, replaced with a standard RBF-interpolation statement) or removed entirely. The paper is of most interest to people working on geometric/hypercomplex deep learning; a skeptical reader should focus on Section IV-D and Table I.","headline":"Useful empirical recipe with a flawed theoretical wrapper: the Sobol-grid expressivity lemma (IV.2) doesn't hold as stated, but the parameter-reduction observation is real enough to warrant serious review.","tokens_in":13574,"tokens_out":2621,"would_cite":false,"duration_ms":30387,"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":"By drawing its grid points from a scrambled Sobol sequence, a Clifford-valued Kolmogorov-Arnold Network becomes an unbiased estimator of a convolution with variance O(n^{-1}), learning any such function with error that shrinks as the grid g","keywords":["Kolmogorov-Arnold networks","Clifford algebra","randomized quasi-Monte Carlo","Sobol sequences","radial basis functions","hypercomplex neural networks","curse of dimensionality"],"falsifier":"Train a Sobol-CliffordKAN on a function of the form h(x)=∫ g(y)φ(x−y)dy with a known smooth g, and record the empirical squared error as the number of grid points n ranges from, say, 8 to 2000. If the error does not decay like O(n^{-1}) or the estimator shows systematic bias—or if the fitted weights w_g at the grid points differ substantially from g evaluated at those points—then the convolution-estimator identification in Lemma IV.2 does not transfer to the trained network.","tokens_in":12678,"feed_emoji":"📐","tokens_out":8447,"duration_ms":82130,"temperature":0.7,"pith_summary":"The paper introduces Clifford Kolmogorov-Arnold Networks (ClKAN), which move KAN-style learnable radial-basis activations from complex numbers into any Clifford algebra. To stop the number of grid points, and hence trainable weights, from growing exponentially with the algebra's dimension, the paper replaces the uniform lattice with a randomized quasi-Monte Carlo grid built from scrambled Sobol sequences. Its central claim is that on such a grid, the network's output Φ(x)=Σ w_g φ(x−g) is an unbiased estimator of the convolution h(x)=∫ g(y)φ(x−y)dy, with variance O(n^{-1}); therefore the model can learn any function that is a convolution of a smooth function with the radial kernel φ, using n grid points instead of 8^D. On complex-valued tasks ClKAN matches the CVKAN baseline, and on quaternion, Cl(2), Cl(1,1), and Cl(1,0,1) tasks the Sobol grid matches or outperforms the full grid while using a few percent of the parameters. The paper also proposes two RBF variants and three Clifford-aware batch-normalization schemes, with no single normalization strategy dominating in the experiments.","feed_headline":"Sobol grid shrinks Clifford KAN parameters from exponential to linear","feed_subtitle":"Its unbiased O(n^{-1}) convolution estimate lets sparse Sobol grids match the full grid.","key_machinery":"The Sobol grid—a scrambled (t,m,d)-net generated by a quasi-random Sobol sequence and scaled to a fixed range such as [−2,2]—is the object that carries the argument. It provides low-discrepancy grid points that cover the hypercube evenly in every realization, permitting a variance bound for scrambled nets. The central identity, Lemma IV.2, states that the RBF-sum Φ(x) over the Sobol grid forms an unbiased estimator of the convolution integral h(x), with variance O(n^{-1}); this lemma converts the network's expressivity into a numerical-integration statement. The two RBF variants (naive φ=exp(−∥x∥²) and Clifford φ=x exp(−∥x∥²)) determine whether the activation maps Cl→R or Cl→Cl, and the Clif","core_discovery":"The paper's key discovery is an identity between a KAN built on a scrambled Sobol grid and a randomized quasi-Monte Carlo integral estimate. Writing the network output as Φ(x)=Σ_{g∈G} w_g φ(x−g) and identifying the trained weights w_g with point evaluations of a function g(y) at the grid nodes, the authors prove that Φ(x) is an unbiased estimator of h(x)=∫_{[0,1]^d} g(y)φ(x−y)dy, with expected squared error bounded by Γσ²/n, i.e. O(n^{-1}). The immediate consequence is that the architecture can represent, with controlled error, every function that can be written as a convolution of a smooth g with the RBF kernel φ. Because the Sobol grid has a user-chosen number n of points, instead of the 8","pith_inferences":["A direct test of Lemma IV.2 would train Sobol-CliffordKAN on a known convolution h(x)=∫ g(y)φ(x−y)dy, measure the empirical squared error for increasing grid sizes n, and check whether the O(n^{-1}) rate and unbiasedness actually appear for the trained rather than the fixed function.","The transfer of the QMC variance bound to trained weights is the load-bearing assumption; a more defensible framing would treat the weights as a grid-dependent function g_n and analyze the realized estimator's bias-variance trade-off rather than fixed-function integration error.","Because the Clifford RBF φ(x)=x exp(−∥x∥²) is direction-preserving, the convolution interpretation suggests the architecture is specialized for signed, graded quantities; applying ClKAN to physics or robotics tasks with multivectors (for example Maxwell's equations or inverse kinematics) would test whether this geometric emphasis matters in practice.","The single-layer analysis could be extended compositionally: if each KAN layer is a convolution estimator, a deep stack composes or multiplies such smoothing operators; how the O(n^{-1}) error propagates through layers remains an open question worth testing."],"forward_implications":["If the convolution-estimator identity holds, a single ClKAN layer can approximate any convolution-kernel function in a Clifford algebra with error O(n^{-1}), roughly independent of the algebra dimension D.","On higher-dimensional algebras such as Cl(1,0,1), experiments show the Sobol grid reduces parameter count to about 2–6% of the full 8^D grid while matching or improving test error.","The kernel φ acts as a regularizing smoother: the network implicitly estimates the smoothed function g*φ, and in the β→∞, n→∞ limit the original g is recovered.","The three batch-normalization variants (node-wise, dimension-wise, component-wise) provide controls for keeping post-activation inputs inside the fixed grid range; without normalization training often collapses.","Re-sampling the Sobol grid for every cross-validation run yields an implicit ensemble effect, and an initial learning rate of 0.1 materially improves training stability in the reported experiments."],"fun_headline_variants":["Clifford KAN hits O(n^{-1}) error with Sobol grid","Sobol grid turns Clifford KAN convolution into unbiased estimate","Randomized QMC grid gives Clifford KAN unbiased convolution","Clifford KAN with Sobol grid: O(n^{-1}) error, linear parameters","Unbiased Clifford KAN: Sobol grid gives O(1/n) error"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The variance bound requires the trained weights w_g to be point evaluations of a fixed smooth function g(y) that is independent of the grid, but in the actual network the weights are free parameters optimized on the data, so the unbiasedness and O(n^{-1}) error need not hold for the trained model.","fun_headline_variants_meta":{"raw":{"variants":["Clifford KAN hits O(n^{-1}) error with Sobol grid","Sobol grid turns Clifford KAN convolution into unbiased estimate","Randomized QMC grid gives Clifford KAN unbiased convolution","Clifford KAN with Sobol grid: O(n^{-1}) error, linear parameters","Unbiased Clifford KAN: Sobol grid gives O(1/n) error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000676,"raw_usage":{"total_tokens":2853,"prompt_tokens":626,"completion_tokens":2227,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":370,"completion_tokens_details":{"reasoning_tokens":2137}},"tokens_in":370,"tokens_out":2227,"duration_ms":15384,"temperature":1.0,"reasoning_tokens":2137,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:02:46.739144+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a Sobol-CliffordKAN on a function of the form h(x)=∫ g(y)φ(x−y)dy with a known smooth g, and record the empirical squared error as the number of grid points n ranges from, say, 8 to 2000. If the error does not decay like O(n^{-1}) or the estimator shows systematic bias—or if the fitted weights w_g at the grid points differ substantially from g evaluated at those points—then the convolution-estimator identification in Lemma IV.2 does not transfer to the trained network.","supporting_citations":[],"review_version":1}