Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction

Mintu Pathak; Meshv Patel; Mohendra Roy; Jwngsar Brahma

cs.LGOct 13, 2026

Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction

Authors: Mintu Pathak, Meshv Patel, Mohendra Roy, Jwngsar Brahma

Organizations: Department of Applied Sciences, Tezpur University, Napaam, Tezpur, 784028, Assam, India. · Department of ICT, Pandit Deendayal Energy University, Raysan, Gandhinagar, 382007, Gujarat, India.

Abstract

The Kolmogorov-Arnold Network (KAN) challenges traditional MLPs by placing learnable activation functions on network edges. However, scaling KANs to spatial data exposes limitations in default B-Splines. While hybrid architectures mix multiple basis families on a single edge to resolve this, monolithic superposition causes geometric overlap, parameter explosion, and severe numerical ill-conditioning ( $κ(G) \approx 10^{16}$ ).To overcome this, we propose Parallel BesselKAN, which isolates B-Spline and low-order Bessel branches into parallel processing pipelines integrated via a parameter-free quadratic interaction. This decoupled architecture acts as an efficient internal ensemble, bypassing monolithic stability bottlenecks. It compresses the parameter footprint by $35\%$ while achieving $97.99\%$ accuracy on MNIST. Although standard MLPs retain a hardware speed advantage for pure vision tasks, Parallel BesselKAN provides a robust, parameter-efficient framework stabilizing mixed-basis networks and paving the way for scalable spatial adaptations like Convolutional KANs.

Figures & tables

Figure 1: Difference between MLP and KAN. Figure adapted from Liu et al. ( Liu et al., 2025 ) under a Creative Commons Attribution 4.0 International License.

Config	$\kappa(\Theta)$	$\kappa(G)$	Rank( $\Theta$ )	Rank( $G$ )
Spline only (n=5)	$5.50\times 10^{0}$	$3.02\times 10^{1}$	5/5	5/5
Bessel only ( $\nu=0$ – $3$ )	$2.89\times 10^{2}$	$8.39\times 10^{4}$	4/4	4/4
Hybrid ( $s=5,b=4$ )	$1.46\times 10^{7}$	$2.13\times 10^{14}$	9/9	9/9
Hybrid ( $s=5,b=6$ )	$1.73\times 10^{8}$	$3.30\times 10^{16}$	11/11	10/11
Bessel horror ( $\nu=0$ – $7$ )	$6.09\times 10^{7}$	$3.67\times 10^{15}$	8/8	7/8

Table 1: Empirical Conditioning and Effective Rank Across Basis Configurations

Figure 2: Spectral characterization and approximation bounds of the hybrid architecture 1: ( a ) Gram matrix condition numbers scale exponentially in hybrid configurations. ( b ) The hybrid model collapses reconstruction RMSE on complex oscillatory targets like $\sin(5x)$ .

Figure 3: Spectral characterization and approximation bounds of the hybrid architecture 2: ( c ) The symmetric geometric overlap between the B-spline partition and low-order Bessel functions over $[-1,1]$ drives the primary near-null space.

Figure 4: Expressive capacity versus numerical stability trade-off. Expanding the Bessel truncation order $q$ beyond $q=2$ yields negligible improvement in mean reconstruction error (RMSE) while causing the design matrix condition number $\kappa(\Theta)$ to grow exponentially, eventually exceeding double-precision limits.

Dataset	Samples (train / test)	Input	Dim.	Classes
Iris	120 / 30	4 tabular features	4	3
MNIST	60,000 / 10,000	$28\times 28$ grayscale, flattened	784	10
Fashion-MNIST	60,000 / 10,000	$28\times 28$ grayscale, flattened	784	10
CIFAR-10	50,000 / 10,000	$32\times 32\times 3$ RGB, flattened (C,H,W)	3072	10

Table 2: Classification benchmarks. Iris uses a random train/test split of its 150 samples.

Figure 5: Basis functions of the two parametric paths. (a) The five cubic B-spline basis functions on $[-1,1]$ with knot vector $(-1,-1,-1,-1,0,1,1,1,1)$ , evaluated for $u\in[-1,1)$ (at exactly $u=1$ the implemented basis is zero; Section 6.2 ). (b) The Bessel functions $J_{0}$ – $J_{3}$ used in the Bessel path, shown for $z\in[-10,10]$ .

Dataset	Model	Epochs	LR	Batch	Mixing	Seeds
Iris	BesselKAN	60	$3\times 10^{-3}$	32	no	5
MNIST	BesselKAN	25	$3\times 10^{-3}$	64	no	5
Fashion-MNIST	BesselKAN	35	$3\times 10^{-3}$	64	yes	5
CIFAR-10	BesselKAN	25	$3\times 10^{-3}$	64	no	5
MNIST / F-MNIST	MLP	25 / 35	$10^{-3}$	64	–	5
MNIST	Parallel fusion	25	$10^{-3}$	64	no	1

Table 3: Training configuration. All runs use AdamW (weight decay

10^{-4}

), gradient-norm clipping at 1.0 and a cosine schedule.

Dataset	Model	Params	Accuracy	Epochs	Time (s)
MNIST	BesselKAN (ours)	516,640	$97.53\pm 0.07$	25	40.3
MNIST	MLP (param-matched)	516,760	$98.55\pm 0.03$	25	14.8
MNIST	EfficientKAN Ta (2025)	–	$\approx 97.0$	25	–
MNIST	BSRBF-KAN Ta (2025)	–	97.59	25	–
MNIST	FC-KAN (DoG+BS) Ta et al. (2026)	–	97.91	25	–
Fashion-MNIST	BesselKAN + mixing (ours)	516,714	$89.34\pm 0.30$	35	40.7

Table 4: Test accuracy (%). “Time” is seconds per epoch on the MPS GPU. Literature values were quoted, not re-run.

Figure 6: Test accuracy of the hybrid KAN and the parameter-matched MLP on MNIST and Fashion-MNIST (mean $\pm$ standard deviation over five seeds); dashed lines mark the accuracies reported for BSRBF-KAN and FC-KAN. Note the truncated vertical axes.

Configuration	Params	Accuracy (%)
Spline-only ( $P=0$ )	360	$86.0\pm 6.4$
Hybrid, $P=2$	520	$84.7\pm 6.5$
Hybrid, $P=4$	680	$80.7\pm 6.4$
Hybrid, $P=6$	840	$84.7\pm 3.8$
Hybrid, $P=4$ , mixing	691	$82.0\pm 6.1$
Single-spline request, $P=6$	784	$85.3\pm 6.5$

Table 5: Iris ablation (60 epochs, five seeds).

Target	Spline (5)	Bessel (4)	Hybrid (5+4)
$\sin(5x)$	0.3632	0.3535	0.0568
$J_{0}(3x)$	0.0124	0.0601	0.0006
$J_{1}(5x)+J_{2}(3x)$	0.1304	0.1322	0.0163
$x^{2}$	0.0447	0.0066	0.0000
$\|x\|$	0.0532	0.0669	0.0107
$\mathbb{1}[x>0]$	0.1926	0.1861	0.1504

Table 6: In-sample least-squares fit error (RMSE) on the 500 points.

Figure 7: Learned first-layer edge functions. Left: four shape templates ( $k$ -means, $k=4$ ) of the standardized edge functions $\hat{\phi}(u)$ over $u\in[-1.5,1.5]$ . Right: share of each template among active edge functions on MNIST and Fashion-MNIST.

Figure 8: Contribution of the spline and Bessel branches of the last layer on MNIST. Left: mean absolute (L1) magnitude per digit class. Right: spline-to-Bessel magnitude ratio per class; colours mark the digit groups (1, 4, 7) and (0, 6, 8).

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LTBs-KAN: Linear-Time B-splines Kolmogorov-Arnold Networks

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Cinvestav, Unidad Guadalajara · Av. del Bosque 1145, El Bajio · Zapopan, Jalisco, 45017, Mexico+3

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Kolmogorov-Arnold Networks (KANs) are a recent neural network architecture offering an alternative to Multilayer Perceptrons (MLPs) with improved explainability and expressibility. However, KANs are significantly slower than MLPs due to the recursive nature of B-spline function computations, limiting their application. This work addresses these issues by proposing a novel base-spline Linear-Time B-splines Kolmogorov-Arnold Network (LTBs-KAN) with linear complexity. Unlike previous methods that rely on the Boor-Mansfield-Cox spline algorithm or other computationally intensive mathematical functions, our approach significantly reduces the computational burden. Additionally, we further reduce model's parameter through product-of-sums matrix factorization in the forward pass without sacrificing performance. Experiments on MNIST, Fashion-MNIST and CIFAR-10 demonstrate that LTBs-KAN achieves good time complexity and parameter reduction, when used as building architectural blocks, compared to other KAN implementations.

Abstract

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