KAN-based Autoencoders in Theory and Practice

• KAN-based autoencoders (KAN-AEs) are architectures that replace fixed node activations with learnable univariate edge functions inspired by the Kolmogorov–Arnold theorem.
• They are applied across domains such as image reconstruction, asset pricing, channel coding, fault detection, and audio, using basis families like polynomial, B-spline, and Fourier.
• Empirical studies reveal competitive gains in reconstruction, classification, and energy efficiency while illustrating trade-offs in scalability, interpretability, and hyperparameter tuning.

Searching arXiv for recent work on KAN-based autoencoders and closely related KAN representation papers. KAN-based autoencoders (KAN-AEs) are autoencoder architectures in which the encoder, the decoder, or both are built from Kolmogorov–Arnold Network (KAN) layers rather than conventional multilayer perceptron blocks with fixed node activations. Their defining structural move is to place learnable univariate functions on edges and to aggregate those edge-wise transforms additively at receiving nodes, an arrangement explicitly connected in the literature to the Kolmogorov–Arnold representation theorem. By 2024–2026, the term had come to cover several related families: deterministic representation-learning autoencoders for images and denoising, conditional latent-factor models in asset pricing, OFDM channel-coding autoencoders with symbolic regression, industrial fault-detection autoencoders using different basis families, and hypernetwork-driven audio implicit representations interpreted from an autoencoder perspective (Yu et al., 2024, Wang et al., 2024, Moradi et al., 2024, Perre et al., 4 Jan 2026, Villagómez et al., 4 Aug 2025, Marszałek et al., 4 Mar 2025).

1. Mathematical basis and defining characteristics

The common mathematical premise is that KANs operationalize the Kolmogorov–Arnold representation theorem through edge-wise univariate functions. One form reported in the literature is

$$f(x_1,\dots,x_n)=\sum_{q=0}^{2n}\phi_q\!\left(\sum_{p=1}^{n}\psi_{q,p}(x_p)\right),$$

while another widely used statement is

$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$

At layer level, the corresponding KAN update is written as an additive aggregation of learned univariate edge maps, for example

$y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$

or, in a more general multilayer notation,

$$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$

This differs from an MLP layer of the form $y=\sigma(Wx+b)$, where nonlinearity is node-local and fixed rather than edge-local and learned (Yu et al., 2024, Faroughi et al., 30 Jul 2025).

Within KAN-AEs, this structural change is used to replace the fixed-activation blocks of standard autoencoders. The autoencoder mapping remains conventional at the highest level—an encoder $E$ maps $x$ to a latent $z$, and a decoder $D$ reconstructs $\hat{x}$—but the internal parameterization becomes a composition of KAN layers. Reported KAN-AE variants instantiate the learnable univariate functions with polynomial, B-spline, Gaussian RBF, Fourier, wavelet, or mixed SiLU-plus-spline bases, depending on the application domain (Yu et al., 2024, Moradi et al., 2024, Villagómez et al., 4 Aug 2025, Perre et al., 4 Jan 2026).

The interpretability claim associated with KAN-AEs follows directly from this edge-function view. In principle, each learned $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$0 or $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$1 can be plotted and inspected independently, so the model exposes a per-edge transformation rather than only aggregate weight matrices. Several papers emphasize this property, although not all of them provide systematic interpretability analyses or formal interpretability metrics (Wang et al., 2024, Moradi et al., 2024).

2. Architectural forms

The KAN-AE label covers multiple architectures rather than a single canonical template.

Setting	Encoder/decoder structure	Basis or edge-function family
General representation learning	Symmetric KAN encoder and decoder, shallow three-layer topology	Polynomial edge functions with Sigmoid
Image reconstruction	KAN layer, ReLU, dense bottleneck; mirrored decoder	B-spline KAN via efficient implementation
Asset pricing	KAN beta network plus linear factor network	Learnable spline functions on edges
Channel coding	One KAN layer in encoder and one in decoder	B-spline plus SiLU, then SR and pruning
Fault detection	Symmetric $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$2 autoencoders	EfficientKAN, FastKAN, FourierKAN, WavKAN
Audio INR from an autoencoder perspective	Hypernetwork encoder predicts INR parameters; KAN INR decodes coordinates	SiLU plus B-spline mixture

The most explicit deterministic formulation is the Kolmogorov–Arnold Auto-Encoder (KAE), which replaces the MLP layers in both encoder and decoder with KAN layers whose edge functions are learnable polynomials. It uses a shallow topology $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$3 with $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$4. Its layer is

$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$5

with polynomial order $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$6, Sigmoid $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$7, and a learned constant matrix term $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$8. The same paper notes that, unlike a concurrent KAN-AE that interleaves KAN with dense/ReLU and uses B-splines, KAE eliminates dense layers and reduces hyperparameter overhead through the polynomial basis (Yu et al., 2024).

A different image-oriented design uses hybrid blocks rather than pure KAN stacks. In that formulation, the encoder is a KAN layer from input size to hidden size, followed by ReLU and a dense layer to the bottleneck; the decoder mirrors this structure in reverse. The paper gives an example encoder $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$9 via KAN, ReLU, then $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$0 via dense, and an example decoder $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$1 via dense, ReLU, then $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$2 via KAN. The KAN layers are implemented with B-spline basis functions through an efficient KAN library (Moradi et al., 2024).

In latent-factor asset pricing, the architecture departs from image reconstruction but still retains an encoder–decoder interpretation. The encoder, called the beta network, maps lagged firm characteristics $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$3 to factor exposures $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$4 using KAN layers built from learnable spline edge functions, wrapped by trainable linear embeddings $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$5 and $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$6. The decoder, called the factor network, is linear:

$y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$7

with $y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$8 obtained from a ridge regression compression of returns onto characteristics (Wang et al., 2024).

The channel-coding literature uses an OFDM autoencoder in which a single KAN layer is placed in the encoder and another in the decoder. There, edge activations are parameterized as

$y_k=\sum_{j=1}^{d_{\text{in}}}\phi_{k,j}(x_j)$9

with third-degree B-splines and five control points per activation. After training, these learned functions are converted to symbolic expressions by symbolic regression and then pruned, producing a symbolic-regression autoencoder closely matched to the trained KAN-AE (Perre et al., 4 Jan 2026).

Fault-detection studies compare several basis families under a fixed shallow autoencoder geometry $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$0: EfficientKAN-AE with B-splines and activity-based regularization, FastKAN-AE with Gaussian RBFs, FourierKAN-AE with truncated Fourier series, and WavKAN-AE with a single wavelet per edge (Villagómez et al., 4 Aug 2025). In audio, a broader “autoencoder perspective” is adopted: a hypernetwork $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$1 maps an audio segment to a task-specific parameter vector $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$2, and the decoder is an INR $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$3 evaluated on time coordinates. In that setting, the “latent” is the parameter vector itself (Marszałek et al., 4 Mar 2025).

3. Objectives, optimization, and evaluation protocols

The dominant objective in KAN-AEs is still reconstruction, but the exact loss is domain-specific. In KAE for representation learning, the model is trained as a deterministic autoencoder with mean squared reconstruction error only,

$$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$4

and the full objective is $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$5. The paper explicitly states that there are no KL divergences or variational terms, and no additional regularization such as smoothness or sparsity on edge functions. Denoising is evaluated by feeding noisy inputs to a model trained on clean data and measuring MSE under Gaussian noise $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$6 and salt-and-pepper noise with flip probability $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$7. The same work uses latent-space nearest-neighbor classification accuracy and Recall@10 for similarity search, with Euclidean distance in both input and latent spaces (Yu et al., 2024).

The asset-pricing formulation trains end-to-end on realized-return MSE,

$$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$8

without a separate pricing loss or factor-variance penalty. Its training protocol is rolling and temporally structured: train on 1957–1987, validate on 1987–1999, test on 2000–2016, with recursive yearly refits and validation-based tuning of the ridge parameter $$h_i^{(l)}=\sum_j f_{ij}^{(l)}(h_j^{(l-1)})+b_i^{(l)}.$$9 and other hyperparameters (Wang et al., 2024).

The channel-coding KAN-AE departs from reconstruction loss and instead minimizes a modified cross-entropy over decoder logits, with noise scheduling across $y=\sigma(Wx+b)$0 epochs. Training uses Adam with learning rate $y=\sigma(Wx+b)$1, batch size $y=\sigma(Wx+b)$2, and an AWGN variance schedule corresponding to $y=\sigma(Wx+b)$3 from $y=\sigma(Wx+b)$4 dB to $y=\sigma(Wx+b)$5 dB. Its post-training symbolic-regression stage fits candidate symbolic functions $y=\sigma(Wx+b)$6 to learned edge activations using a combined score

$y=\sigma(Wx+b)$7

where $y=\sigma(Wx+b)$8 is a non-linearity score based on the minimum piecewise-linear segmentation needed to satisfy a target approximation error (Perre et al., 4 Jan 2026).

Image-reconstruction KAN-AEs compared with CNN autoencoders use the standard MSE objective, AdamW with learning rate $y=\sigma(Wx+b)$9, weight decay $E$0, ten epochs, and additional L1 regularization on KAN weights in the efficient KAN implementation. Their downstream representation quality is assessed with a KNN classifier on the learned latent vectors, reporting accuracy and F1-score (Moradi et al., 2024).

In the audio hypernetwork setting, the objective is a two-term waveform and spectral loss,

$E$1

where $E$2 is temporal L1 and $E$3 is a multi-resolution mel-STFT loss. This is optimized as a meta-objective over tasks, with the hypernetwork producing $E$4 and thus $E$5 (Marszałek et al., 4 Mar 2025). A broader scientific-machine-learning synthesis further suggests KAN-AE extensions with variational objectives and physics-informed penalties such as $E$6; this should be read as adapted KAN-AE guidance rather than a single benchmarked autoencoder configuration (Faroughi et al., 30 Jul 2025).

4. Empirical performance across application domains

The strongest general-purpose representation-learning evidence comes from KAE. Across MNIST, FashionMNIST, CIFAR10, and CIFAR100, KAE improved reconstruction, retrieval, classification, and denoising over a Sigmoid AE and over B-spline, Fourier, and wavelet KAN baselines. For reconstruction, KAE with $E$7 achieved the best errors on all datasets and both latent sizes; examples include MNIST with $E$8, where AE yielded $E$9 and KAE $x$0, and CIFAR100 with $x$1, where AE yielded $x$2 and KAE $x$3. For similarity search, KAE with $x$4 was best overall; on MNIST with $x$5, Recall@10 rose from $x$6 to $x$7, and on CIFAR100 from $x$8 to $x$9. For nearest-neighbor classification in latent space, KAE with $z$0 gave the highest accuracy, including MNIST $z$1 versus $z$2, CIFAR10 $z$3 versus $z$4, and CIFAR100 $z$5 versus $z$6. Under Gaussian noise, MNIST denoising MSE at $z$7 improved from $z$8 to $z$9 (Yu et al., 2024).

The image-reconstruction comparison with CNN autoencoders is more mixed. On SVHN and CIFAR-10, AE-KAN consistently achieved lower MSE than AE-CNN across the tested bottlenecks; for example, on SVHN the paper reports $D$0 versus $D$1, $D$2 versus $D$3, and $D$4 versus $D$5, while on CIFAR-10 it reports $D$6 versus $D$7, $D$8 versus $D$9, and $\hat{x}$0 versus $\hat{x}$1. On MNIST, AE-KAN outperformed AE-CNN at small and medium bottlenecks but lost at the largest tested bottleneck, where AE-CNN with $\hat{x}$2 beat AE-KAN with bottleneck $\hat{x}$3 in both MSE and KNN metrics. This study therefore presents KAN-AEs as competitive rather than uniformly dominant (Moradi et al., 2024).

In asset pricing, the KAN-based conditional autoencoder achieved comparable total $\hat{x}$4 and better out-of-sample predictive $\hat{x}$5 than the MLP-based conditional autoencoder across all tested numbers of latent factors. Predictive $\hat{x}$6 values were $\hat{x}$7, $\hat{x}$8, and $\hat{x}$9 for KAN-CA at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$00, compared with $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$01, $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$02, and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$03 for the MLP baseline. Long–short Sharpe ratios were $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$04, $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$05, and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$06 for KAN-CA, versus $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$07, $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$08, and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$09 for the baseline, with the clearest gain at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$10 (Wang et al., 2024).

In communication systems, the KAN-AE paired with symbolic regression maintained BLER comparable to the MLP-AE under both AWGN and flat Rayleigh fading while reducing measured GPU energy. Reported hardware-based energy was $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$11 J for the MLP-AE and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$12 J for the KAN-based SR-AE, giving an energy ratio of approximately $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$13. The hardware-independent non-linearity score $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$14 was $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$15 for the MLP-AE and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$16 for the KAN-based SR-AE (Perre et al., 4 Jan 2026).

In fault detection on the Tennessee Eastman Process, WavKAN-AE achieved the highest overall FDR, reaching $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$17 using 4,000 training samples and remaining the top performer as other variants scaled. EfficientKAN-AE reached $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$18 FDR with only 500 samples and dominated the extreme low-data regime, while FastKAN-AE became competitive only at larger scales such as $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$19 samples. FourierKAN-AE consistently underperformed, and the Orthogonal Autoencoder baseline improved only gradually with more data (Villagómez et al., 4 Aug 2025).

For audio, the directly benchmarked KAN result is a KAN implicit representation rather than a standard bottleneck autoencoder, but the paper explicitly frames the hypernetwork-plus-INR pipeline as an autoencoder perspective. In standalone INR experiments, KAN achieved the best Log-Spectral Distance of $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$20 and the best PESQ of $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$21 for $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$22 s audio, and the best LSD of $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$23 for $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$24 s audio. In the hypernetwork setting, FewSound outperformed HyperSound, with a $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$25 reduction in MSE and a $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$26 increase in SI-SNR; among target networks, NeRF was best overall and KAN was a close second (Marszałek et al., 4 Mar 2025).

5. Interpretability, efficiency, and recurrent trade-offs

A recurring theme is that KAN-AEs trade fixed activations for inspectable edge functions. The clearest operational statement appears in asset pricing, where the learned spline functions $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$27 in the beta network can be visualized to understand how each characteristic contributes to each factor exposure, enabling partial-dependence visualization and sensitivity analysis at the granularity of characteristic-to-factor exposure channels (Wang et al., 2024). Related papers on image reconstruction and fault detection make similar interpretability claims, but they also note that explicit visualization or quantitative interpretability analysis is absent (Moradi et al., 2024, Villagómez et al., 4 Aug 2025).

Efficiency claims are highly application-dependent and should not be generalized indiscriminately. In KAE for representation learning, the polynomial design reduced hyperparameter overhead relative to spline-based KANs and achieved higher accuracy with fewer parameters than some competing KAN variants, with capacity reported at approximately $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$28–$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$29K parameters versus approximately $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$30K for KAN and $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$31K for FourierKAN, and training times of $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$32–$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$33 seconds (Yu et al., 2024). In channel coding, symbolic regression and pruning further reduced complexity and yielded lower measured energy than the MLP-AE baseline (Perre et al., 4 Jan 2026). In fault detection, WavKAN-AE used $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$34 parameters, fewer than OAE at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$35, EfficientKAN-AE at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$36, FastKAN-AE at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$37, and FourierKAN-AE at $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$38 (Villagómez et al., 4 Aug 2025). By contrast, the image-reconstruction study found AE-KAN to have dramatically more parameters than AE-CNN—on CIFAR-10, roughly $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$39–$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$40 million versus $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$41–$$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$42—even though reported runtimes were similar (Moradi et al., 2024). A common misconception is therefore that KAN-AEs are intrinsically parameter-efficient; the literature supports a narrower statement that efficiency depends on basis family, task geometry, and baseline choice.

Basis choice is another consistent trade-off. In KAE, polynomial order $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$43 tended to preserve local distances better and gave the best Recall@10, whereas $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$44 gave the best classification accuracy and slightly better reconstructions (Yu et al., 2024). In industrial fault detection, localized and multiresolution families performed best: EfficientKAN-AE was strongest in data-scarce regimes, WavKAN-AE was best overall, FastKAN-AE improved with scale, and FourierKAN-AE lagged, which the paper attributes to the mismatch between global periodic bases and localized nonstationary faults (Villagómez et al., 4 Aug 2025). In audio, spline-based KAN with a SiLU mixture improved perceptual fidelity relative to fixed-activation INR baselines, especially in LSD and PESQ (Marszałek et al., 4 Mar 2025).

This suggests that “KAN-AE” is best understood as a design space organized around edge-function parameterization rather than as a single model class with uniform behavior.

6. Limitations, open questions, and future directions

The existing KAN-AE literature remains heterogeneous and in several domains explicitly preliminary. The KAE representation-learning model is shallow and deterministic, with no KL regularization or probabilistic modeling; its authors propose combining it with VAE objectives, scaling to deeper architectures with KAN-based CNN layers and residual connections, exploring alternative activations, and applying it to more challenging generation tasks (Yu et al., 2024). The asset-pricing study notes that KAN best practices for depth, spline parameterization, regularization, and initialization are not yet standardized, and that identifiability remains limited to rotations and scalings, as in latent factor models generally (Wang et al., 2024).

The image-reconstruction study identifies a separate set of limitations: AE-KAN parameter counts are orders of magnitude higher than those of CNN baselines, scalability to larger images and datasets is not evaluated, training stability and hyperparameter sensitivity are not deeply analyzed, and interpretability is asserted rather than quantified (Moradi et al., 2024). The channel-coding study is limited to $$f(x_1, x_2, \ldots, x_d)=\sum_{k=1}^{2d+1}\Phi_k\left(\sum_{j=1}^{d}\phi_{k,j}(x_j)\right).$$45 codes, AWGN and flat Rayleigh channels, perfect CSI for Rayleigh equalization, and a specific GPU measurement setup; it explicitly leaves sensitivity analysis of the non-linearity metric and its weighting, larger block sizes, and broader robustness evaluation for future work (Perre et al., 4 Jan 2026). The fault-detection study is restricted to single time-step inputs, emphasizes detection rather than diagnosis, and does not systematically evaluate interpretability despite stressing the transparency potential of structured edge functions (Villagómez et al., 4 Aug 2025).

A broader review of scientific machine learning with KANs identifies the most persistent open problems as computational efficiency, theoretical guarantees, hyperparameter tuning, and algorithm complexity, together with robustness, scalability, and physical consistency (Faroughi et al., 30 Jul 2025). For KAN-AEs specifically, the literature therefore points toward four unsettled research directions: principled basis selection, scalable implementations for high-dimensional data, probabilistic and physics-informed extensions, and interpretability methods that move beyond visual plausibility to formal analysis.