EfficientKAN: Optimized Spline-Based KAN

• EfficientKAN is an optimized spline-based Kolmogorov-Arnold Network that replaces scalar edge weights with learnable functions for improved efficiency and scalability.
• Its innovations include a reordered spline computation, fused linear evaluations, and adaptive regularization techniques that significantly reduce memory usage and computational overhead.
• Empirical benchmarks reveal that EfficientKAN frequently outperforms conventional KAN variants under resource-constrained conditions while serving as a robust baseline for further efficiency enhancements.

Searching arXiv for papers on EfficientKAN and related KAN efficiency variants. EfficientKAN is a computationally optimized, spline-based Kolmogorov-Arnold Network (KAN) variant and implementation lineage that preserves the defining KAN idea of replacing scalar edge weights with learnable univariate functions, while reorganizing how those functions are evaluated so that memory usage and computation are reduced relative to the original formulation (Liu et al., 2024, Hoever et al., 16 Jun 2026). In later literature, it functions both as a concrete spline-based baseline—often associated with the open-source Efficient-KAN repository—and as a reference point for broader “efficient KAN” redesigns in which basis functions, parameter sharing, grouping, or hardware mappings are modified to make KANs more practical on real workloads (Thu et al., 19 Mar 2026, Aung et al., 26 Nov 2025, Shi et al., 18 Apr 2025).

1. KAN background and the problem EfficientKAN addresses

KANs were introduced as alternatives to multilayer perceptrons in which fixed node activations are replaced by learnable univariate functions on edges. In the standard layered form, a KAN layer computes

$$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$

with each $\phi_{l,j,i}$ typically parameterized as a spline (Liu et al., 2024). This construction is motivated by the Kolmogorov-Arnold representation theorem and was presented as offering strong accuracy and interpretability, especially on function fitting and scientific tasks (Liu et al., 2024).

The difficulty is that the original spline-centric formulation is expensive. Later work summarizes the bottleneck plainly: KANs are usually $10\times$ slower than MLPs, given the same number of parameters, because B-spline computation is recursive and must be carried out for many edge-wise functions (Qiu et al., 2024). EfficientKAN emerges precisely at this engineering bottleneck. In comparative studies it is described as a version of KAN that reduces memory usage and computational overhead through a reordered processing sequence, more efficient B-spline handling, and revised regularization and initialization choices (Ta et al., 2024).

A common misconception is that EfficientKAN is simply “KAN, but smaller.” The literature instead presents it as a reformulation of how spline-based KANs are computed. The functional form remains spline-based, but the implementation avoids some of the tensor expansion and per-activation overhead that made early KAN implementations difficult to scale (Hoever et al., 16 Jun 2026).

2. Core formulation and implementation mechanics

The spline-based EfficientKAN discussed across several papers retains the standard residual KAN edge form

$$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$

where $b(x)$ is a base function such as SiLU, $B_i(x)$ are B-spline basis functions, and the coefficients are learned (Ta et al., 2024, Hoever et al., 16 Jun 2026). In classification-head and benchmarking studies, EfficientKAN is also written in the broader KAN form

$$f(\mathbf{x}) = \sum_{q=1}^{2n+1}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right),$$

with $\phi_{q,p}(x)=\mathrm{spline}(x;\mathbf{w}_{q,p})$ (Thu et al., 19 Mar 2026, Aung et al., 26 Nov 2025).

Its practical distinctiveness lies in the evaluation order. EfficientKAN applies activation functions on the input first and then linearly combines them, rather than expanding the input tensor to $(n_{batch}, n_{outputs}, n_{inputs})$ as in less efficient realizations. KANLib’s analysis of EfficientKAN identifies several signature techniques: direct computation of B-spline basis values from the inputs without input expansion; fused spline and residual computation via coarse-grained torch.nn.functional.linear calls; $L_1$ regularization over coefficients or weights instead of the original per-activation regularization; adaptive grid rescaling; and reduced intermediate activation storage (Hoever et al., 16 Jun 2026). A Burmese news classification study adds two further details: EfficientKAN applies $\phi_{l,j,i}$0 regularization directly to the weights that combine spline activations, and it may include optional learnable scaling for a speed–expressiveness trade-off (Aung et al., 26 Nov 2025).

Other benchmark-oriented summaries attribute additional implementation changes to EfficientKAN, including learnable scaling parameters for activation functions and a change in initialization from Xavier to Kaiming Uniform (Ta et al., 2024). Taken together, these changes do not alter the basic spline basis family, but they substantially change the runtime profile. This suggests that EfficientKAN is best understood as a systems-level refactoring of spline KANs rather than a new approximation theorem or a departure from the edge-function paradigm.

3. Reported empirical behavior across benchmark settings

The empirical record for EfficientKAN is broad but heterogeneous. It is repeatedly competitive, often parameter-efficient, and frequently easier to train than naive spline KANs, but it is not uniformly the best-performing KAN or the best model family on every task.

Setting	Reported EfficientKAN result	Source
myMNIST digit recognition	Precision 0.9841, Recall 0.9898, F1-Score 0.9869, Accuracy 0.9918 with $\phi_{l,j,i}$1	(Thu et al., 19 Mar 2026)
MNIST benchmark	$\phi_{l,j,i}$2 average validation accuracy, $\phi_{l,j,i}$3 F1, 120 s average time	(Ta, 2024)
Fashion-MNIST benchmark	88.98 validation accuracy, 88.91 F1, 247.85 s	(Ta et al., 2024)
California Housing	RMSE $\phi_{l,j,i}$4, $\phi_{l,j,i}$5, 4050 params, 143.60 $\phi_{l,j,i}$6 1.90 ms inference	(Hoever et al., 16 Jun 2026)
Burmese news classification	fastText + EfficientKAN reached F1-score 0.928; mBERT + EfficientKAN reached 0.917 F1	(Aung et al., 26 Nov 2025)
Tennessee Eastman fault detection	EfficientKAN-AE reached $\phi_{l,j,i}$7 FDR with only 500 samples	(Villagómez et al., 4 Aug 2025)

On image-style flattened-vector benchmarks, EfficientKAN is usually a meaningful baseline rather than the dominant model. In the myMNIST benchmark it outperformed the raw MLP, but CNN remained best overall with F1 $\phi_{l,j,i}$8 and Accuracy $\phi_{l,j,i}$9, while PETNN (GELU) reached F1 $10\times$0 and Accuracy $10\times$1 (Thu et al., 19 Mar 2026). On MNIST and Fashion-MNIST comparisons, EfficientKAN was faster than BSRBF-KAN but had slightly lower validation accuracy and F1, and FC-KAN variants also outperformed it on both datasets (Ta, 2024, Ta et al., 2024).

On software-framework benchmarks, the picture is different. KANLib reports that EfficientKAN and KANLib’s B-spline implementation have similar inference time and predictive accuracy, and both are about $10\times$2 faster than PyKAN on California Housing (Hoever et al., 16 Jun 2026). This result is important because it shows EfficientKAN’s contributions persisting when transplanted into a modular framework rather than remaining isolated in a bespoke codebase.

4. Applications in language and industrial monitoring

EfficientKAN has been used as a lightweight but expressive head in low-resource language classification. In Burmese news classification, only the classification head is trained while the encoder is frozen, and EfficientKAN is compared with MLP, FourierKAN, and FasterKAN. Under fastText embeddings, EfficientKAN achieved the highest F1-score, 0.928; with mBERT embeddings, it reached 0.917 and matched or slightly outperformed MLPs (Aung et al., 26 Nov 2025). The same study reports that EfficientKAN is slower than MLP in head-only training—24.6 s versus 18.1 s in the fastText setting—but attributes to it higher expressiveness for localized nonlinear feature relationships and improved sparsity via $10\times$3 regularization (Aung et al., 26 Nov 2025).

A separate industrial-monitoring line uses EfficientKAN inside autoencoders. EfficientKAN-AE on the Tennessee Eastman Process employs a $10\times$4 architecture, SiLU + B-spline expansion, 6 basis functions per edge, and $10\times$5, entropy-based, and weight regularization, with 11,550 parameters (Villagómez et al., 4 Aug 2025). In this setting EfficientKAN-AE reached $10\times$6 fault detection rate with only 500 samples and was described as decisively superior to the Orthogonal Autoencoder and other KAN-AE variants when data is scarce (Villagómez et al., 4 Aug 2025). The same paper also reports that WavKAN-AE eventually becomes the top performer as training size grows, while OAE surpasses EfficientKAN-AE for very large datasets above 50k samples (Villagómez et al., 4 Aug 2025).

These applications clarify where EfficientKAN has been especially useful: frozen-encoder classification heads, compact tabular or vector models, and low-data industrial detection. This suggests that the architecture’s principal advantage is not universal accuracy dominance, but a favorable balance of adaptive nonlinearity, moderate parameter count, and practical trainability in settings where full-scale spatial or sequence inductive biases are unavailable or intentionally frozen.

5. Extensions, reinterpretations, and naming ambiguity

The term “EfficientKAN” is not used uniformly across the literature. In the spline-based line, it refers to the optimized B-spline implementation just described. In point-cloud analysis, however, “Efficient-KANs” in PointKAN-elite denote a different construction: B-splines are replaced by rational functions,

$10\times$7

and parameter growth is controlled by grouped activation sharing rather than by EfficientKAN’s fused B-spline evaluation strategy (Shi et al., 18 Apr 2025). PointKAN-elite reports 3.1M parameters, 2.3G FLOPs, 93.3 ModelNet40 OA, and 84.1 ScanObjectNN PB_T50_RS OA, while being about $10\times$8 smaller and about $10\times$9 less FLOPs than PointMLP (Shi et al., 18 Apr 2025). That architecture is therefore an efficient KAN in a generic sense, but not the same object as the Blealtan-style spline EfficientKAN.

KANLib makes this distinction explicit by unifying PyKAN, EfficientKAN, and FastKAN within a single framework. It adopts EfficientKAN’s optimized B-spline basis evaluation, fused linear computation, and coefficient-based $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$0 regularization, but extends them with grid extension, optional LayerNorm, and support for both B-spline and GRBF layers (Hoever et al., 16 Jun 2026). KANLib’s B-spline configuration achieved RMSE $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$1, $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$2, 4050 parameters, and 142.03 $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$3 1.65 ms inference, very close to the EfficientKAN reference numbers (Hoever et al., 16 Jun 2026).

EfficientKAN also sits within a rapidly expanding family of efficiency-oriented KAN modifications. LSS-SKAN argues for an Efficient KAN Expansion Principle in which parameters should expand network scale rather than basis complexity, and reports that LSS-SKAN outperformed all compared popular KAN variants in execution speed on MNIST (Chen et al., 2024). LTBs-KAN introduces linear-time B-spline computation and product-of-sums factorization, reaching 203,378 parameters, 0.9632 accuracy, and 244 s on MNIST, compared with EfficientKAN’s 254,080 parameters, 0.9438 accuracy, and 367 s in the same comparison (Merin-Martinez et al., 23 Apr 2026). PowerMLP goes further by replacing recursive B-spline evaluation with non-iterative spline representations and reports training speed about 40 times faster than KAN in various tasks (Qiu et al., 2024).

6. Limitations, contested claims, and ongoing research directions

EfficientKAN should not be treated as a synonym for state of the art. On myMNIST it trailed CNN, PETNN, and JEM (Thu et al., 19 Mar 2026). On MNIST, BSRBF-KAN achieved $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$4 average validation accuracy versus EfficientKAN’s $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$5, and FC-KAN variants reached 97.91 on MNIST and 89.99 on Fashion-MNIST versus EfficientKAN’s 97.34 and 88.98 in that benchmark (Ta, 2024, Ta et al., 2024). Even in the Burmese news setting where EfficientKAN gave the highest fastText result, FasterKAN was presented as offering the best trade-off between speed and accuracy (Aung et al., 26 Nov 2025).

A second misconception is that EfficientKAN solves the computational problem of spline KANs once and for all. Later work shows that substantial headroom remains. Quantization studies report that B-splines can be quantized to 2–3 bits with negligible loss in accuracy, that precomputed 8-bit lookup tables improve GPU inference speedup by up to $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$6, and that ResKAN18 achieves a $$\phi(x) = w_b\, b(x) + w_s\, \mathrm{spline}(x), \qquad \mathrm{spline}(x) = \sum_i c_i B_i(x),$$7 reduction in BitOps without loss of accuracy using low-bit-quantized B-spline tables (Errabii et al., 18 Mar 2026). Linear-time spline algorithms, grouped transforms, single-parameter basis functions, and MLP-hybrid approximations all continue to compete with or supersede the particular efficiency gains delivered by EfficientKAN (Merin-Martinez et al., 23 Apr 2026, Chen et al., 2024, Qiu et al., 2024).

This suggests that EfficientKAN is historically important not because it closed the subject, but because it converted spline KANs from a largely proof-of-concept architecture into a reusable experimental baseline. Its enduring role in the literature is therefore twofold: as a specific optimized B-spline KAN implementation, and as the reference point against which later KAN efficiency proposals—grouped, rational, factorized, quantized, or hybrid—are measured (Hoever et al., 16 Jun 2026, Shi et al., 18 Apr 2025).