Structural Kolmogorov-Arnold Convolutions: Learnable Function on the Values or the Filter Shape as Parameter-Efficient Alternative to Per-Edge Convolutional KANs

Abstract: Convolutional Kolmogorov--Arnold Networks (KANs) replace the fixed weights of a convolutional kernel with learnable univariate functions. The dominant formulation attaches one such function to every kernel entry and lets it act on pixel values, expressive but parameter-heavy and prone to overfitting. We argue that the learnable functions are better placed in the \emph{structure} of the convolution than on each edge, and we organise the design space along a single axis: whether the function acts on the pixel \emph{values} or on the filter \emph{shape}. We study three realisations. SV-KAN applies one shared univariate function to the values and leaves the spatial filter free and static, aa classical convolution with a single learnable shared activation. AG-KAN keeps the shared value function but supplies the spatial structure through a content-adaptive Gaussian gate. RF-KAN instead moves the learnable functions onto the filter shape, building each filter from oriented ridge profiles expanded in a localised oscillatory (Morlet) wavelet basis with content-adaptive amplitudes. Under a matched four-layer protocol with in-run references and three seeds, RF-KAN and SV-KAN reach $88.47\pm0.10\%$ and $88.20\pm0.31\%$ on CIFAR-10 and $64.40\pm0.19\%$ and $64.57\pm0.30\%$ on CIFAR-100, at about $0.4$M parameters. At this matched scale the shape model and the simplest value model meet at the top, both above a plain convolution and every per-edge KAN we tested, including the official Gram variant, at roughly a fifth of the parameters. A controlled study attributes the RF-KAN gain to an intrinsically localised oscillatory basis and to content adaptivity, and an ablation that removes the learned shape entirely, leaving only the shared value function, collapses accuracy by over forty points, identifying the learned shape as the load-bearing ingredient at this scale.

• The paper introduces SV-KAN, AG-KAN, and RF-KAN, which place learnable functions on shared pixel values or structured filter shapes instead of every convolutional edge.
• RF-KAN reaches 88.47% on CIFAR-10 with 0.40 million parameters, outperforming a matched plain convolution by 1.87 points and per-edge KAN baselines by up to 4.77 points.
• The results show that learned filter shape is essential and function multiplicity adds little at compact scale, while open questions include performance on deeper backbones, larger datasets, and latency-sensitive deployments.

Motivation and central claim

Convolutional Kolmogorov–Arnold Networks (KANs) replace the fixed scalar weights of a convolutional kernel with learnable univariate functions, typically attaching one independent function to every kernel entry. This per-edge formulation, introduced by Bodner et al. and extended to multiple bases by Drokin, is expressive but expensive: the parameter count scales as $C \cdot C_{\text{out}} \cdot P^2 \cdot (D{+}1)$ per layer, inflating both parameters and inference time, and at matched budgets on natural images convolutional KANs generally match or trail plain convolutions. The paper's contribution is a structural reframing: a convolutional patch carries both pixel values and their spatial positions, and a learnable univariate function can act on either. The authors organise the design space along this single value–shape axis and argue that placing the learnable structure in the geometry of the convolution — not on every edge — is the parameter-efficient alternative.

Three architectures instantiate the axis. SV-KAN (Shared-Value KAN) is the minimal value model: a classical convolution whose fixed activation is replaced by one shared learnable function $\psi(t)$ applied to pixel values, with a free, static, learnable filter bank. AG-KAN (Adaptive-Gate KAN) keeps the shared value function but supplies spatial structure through a content-adaptive anisotropic Gaussian gate whose orientation and two scales are predicted per patch by a small routing network. RF-KAN (Ridge-Function KAN) moves the learnable functions onto the filter shape itself: each filter is a sum of $R$ oriented ridge profiles, where each profile is expanded in a Morlet wavelet basis with learnable amplitudes and content-adaptive corrections, and the pixel values enter the aggregation linearly.

Method

All operators are cast in a common "spatial KAN convolution" form, $z_{s,c} = \sum_u w_s(u)\,\rho(x_c(u))$, followed by a linear projection, batch normalisation, ReLU and pooling. The shared univariate function is written in residual form, $$\psi(t) = t + \alpha \sum_m a_m \kappa((t-\mu_m)/\sigma_m)$$, with a scalar gate $\alpha$ initialised at zero so the layer starts at the identity; the kernel $\kappa$ is either a Gaussian (RBF variant) or a real Morlet wavelet $\cos(\omega_0 z)e^{-z^2/2}$.

RF-KAN's construction is the paper's technical core. Each filter is $$w_s(u) = \sum_r \psi_{s,r}(\langle u, n_{s,r} \rangle)$$ with learnable orientations $n_{s,r}$; the inner product collapses the two-dimensional coordinate to a scalar, so each profile remains univariate while producing a ridge constant orthogonal to its orientation. Because values enter linearly, RF-KAN is linear in the patch — doubling the input doubles the output — a formal property distinguishing it from all value-side formulations. The authors are careful to disclaim any claim of realising the Kolmogorov–Arnold representation theorem: the Morlet expansion is smooth and finite, whereas the theorem's inner functions need not be; what RF-KAN provides is a controlled regularity bias, since kernel entries are tied to samples of a few smooth continuous profiles rather than free parameters. Filters are rendered on a finer grid and folded into the $\psi(t)$0 kernel through a fixed bilinear matrix, which is exact and removes the dominant computational cost. Content adaptivity enters through a routing network that predicts additive corrections to the base wavelet amplitudes; the routing output is initialised near, but not exactly at, zero to avoid a gradient dead-lock in which the adaptive branch never activates — a practical necessity the authors report.

Experimental results

All comparisons use a matched four-layer backbone (channels $\psi(t)$1) on CIFAR-10 and CIFAR-100, three seeds, and in-run references, since run-to-run GPU non-determinism contributes 0.1–0.5 points at this scale.

Model	CIFAR-10 acc. (%)	Params (M)	Δ vs. CNN
RF-KAN (wavelet, adaptive)	88.47 ± 0.10	0.40	+1.87
SV-KAN (shared value, free shape)	88.20 ± 0.31	0.39	+1.60
AG-KAN (shared value + gate)	86.87 ± 0.17	0.43	+0.27
Plain convolution	86.60 ± 0.17	0.39	0.00
Per-edge KAN, Gram (official)	85.95 ± 0.05	1.94	−0.65
Adaptive Gabor (non-KAN)	85.77 ± 0.39	0.40	−0.83
Per-edge KAN, Legendre	85.39 ± 0.41	1.94	−1.21
Per-edge KAN, Chebyshev	83.70 ± 0.34	1.94	−2.90
SV-KAN, shape removed (uniform sum)	46.14 ± 0.12	0.39	−40.46

On CIFAR-100 the ordering holds with wider margins: SV-KAN reaches $\psi(t)$2 and RF-KAN $\psi(t)$3, roughly 2.8 points above the plain convolution, while the strongest per-edge baseline (Gram, $\psi(t)$4) trails by about four points at five times the parameters. The headline empirical claim is that the shape model (RF-KAN) and the simplest value model (SV-KAN) meet at the top from opposite ends of the value–shape axis: at this scale, the placement of the learnable function matters more than its multiplicity. Every structural variant also exceeds all per-edge baselines, including the official Gram implementation, at roughly a fifth of the parameters — a statement about accuracy per parameter in precisely the compact natural-image setting where convolutional KANs have been least convincing.

Ablations and design boundaries

The RF-KAN gain decomposes into two largely independent contributions. Replacing the Morlet basis with a radial-basis ridge costs about 1.3 points ($\psi(t)$5); removing content adaptivity costs a further 0.8 points ($\psi(t)$6). A routing gain of comparable size appears independently for the RBF ridge and for a static Gabor filter, indicating adaptivity is a general lever. The Morlet central frequency is the most influential hyperparameter (static wavelet ridge: $\psi(t)$7, $\psi(t)$8, $\psi(t)$9 at $R$0), while the atom count saturates early, so $R$1 is fixed.

Two further controls sharpen the interpretation. Against a canonical Gabor filter granted identical content-adaptive routing, the free Morlet ridge wins by about 2.7 points on CIFAR-10 and 3.5 on CIFAR-100, showing the advantage comes from the free basis rather than adaptive geometry alone. Within SV-KAN, sharing one value function, one per filter, and one per channel are statistically equivalent ($R$2, $R$3, $R$4), so diversifying the function adds parameters without benefit. Most strikingly, the ablation removing the learned shape entirely — replacing the filter bank with a fixed uniform sum — collapses CIFAR-10 accuracy to about 46%, a drop of more than forty points, identifying the learned shape as the load-bearing ingredient at this scale.

Limitations and open questions

The evidence is confined to a compact four-layer backbone on CIFAR-scale data; the authors state plainly that evaluation on deeper residual backbones, Tiny ImageNet and ImageNet scale is underway rather than established. The interpretability benefit of the structural placement — one shared function plus a readable gate versus on the order of $R$5 per-edge functions — is offered explicitly as a hypothesis, not a demonstrated result. The effect of the ridge count $R$6, a rank-controlled measurement of the expressivity ceiling imposed by the smooth-profile constraint, and a full FLOPs and latency accounting are left open. The authors also expect RF-KAN's additional latent capacity to separate it from SV-KAN only at larger scale, which remains untested.

Conclusion

The paper reorganises convolutional KAN design around a single question — whether the learnable univariate function acts on pixel values or on the filter shape — and shows that both placements, realised as RF-KAN and SV-KAN, dominate per-edge convolutional KANs at a matched four-layer budget while beating a plain convolution of equal size, at roughly a fifth of the per-edge parameter count. The ablation identifying the learned shape as indispensable, and the finding that function multiplicity adds nothing at this scale, together support the central thesis that structural placement, not per-edge expressivity, is what buys performance in compact convolutional KANs.