Frequency-Windowed Kolmogorov-Arnold Network

• Frequency-windowed Kolmogorov-Arnold Network (FW-KAN) is a paradigm that applies spectral windowing to learnable univariate functions for multiresolution data modeling.
• It employs various spectral representations such as wavelet, Fourier, and spline-based methods to balance model capacity with computational efficiency.
• FW-KAN offers enhanced interpretability by enabling visualization of localized edge functions, aiding analysis in time-series, imaging, and other frequency-sensitive applications.

Frequency-Windowed Kolmogorov-Arnold Network (FW-KAN) denotes a Kolmogorov-Arnold Network-derived modeling paradigm in which learnable univariate edge functions, their inputs, or both are structured by spectral windows, multiresolution bases, or explicit frequency-band decompositions. In current arXiv literature, FW-KAN is not a standardized single architecture name; it is more accurately understood as a synthesized umbrella for several closely related constructions, including wavelet-based KANs, Fourier-based KANs, and time-series KAN systems built around frequency selection or frequency decomposition (Ji et al., 2024, Bozorgasl et al., 2024, Huang et al., 10 Feb 2025).

1. Foundational relation to Kolmogorov-Arnold Networks

The conceptual substrate of FW-KAN is the Kolmogorov-Arnold representation principle. A classical form stated in the KAN survey is

$$f(x_1, x_2, \ldots, x_n) = \sum_{i=1}^{2n+1} g_i \left( \sum_{j=1}^n h_{ij}(x_j) \right),$$

with continuous univariate functions $g_i$ and $h_{ij}$ (Ji et al., 2024). KAN operationalizes this viewpoint by replacing scalar edge weights with learnable univariate functions and letting nodes perform summation only. In the original KAN formulation, a layer is written as

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

so the nonlinearity resides on edges rather than at nodes (Liu et al., 2024).

In canonical spline-based KAN, each edge function is parameterized as

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

where $b(x)=\mathrm{silu}(x)$ and $B_i$ are B-spline basis functions (Liu et al., 2024). This architecture separates what that paper calls the external degrees of freedom—depth, width, and connectivity—from the internal degrees of freedom carried by the basis coefficients on each edge (Liu et al., 2024). That separation is central for FW-KAN, because frequency windowing can be imposed either on the basis expansion itself, on the transformed inputs delivered to those edge functions, or on the allocation of model capacity across bands.

This also explains why FW-KAN is better regarded as a family of frequency-aware KAN parameterizations than as a single fixed template. The KAN core remains the same: multivariate structure is assembled from sums and compositions of learned univariate maps. Frequency windowing modifies how those univariate maps are represented, where their support is concentrated, and which spectral components are emphasized.

2. Spectral and windowing mechanisms within the KAN literature

Even without an explicit Fourier stage, vanilla KAN already has a localized basis structure. The survey emphasizes that KAN commonly uses B-splines, and that each basis function has local support around knot intervals; the edge function

$\Phi(x) = \sum_m c_m B_m(x)$

therefore acts through compactly supported input-domain windows (Ji et al., 2024). Dynamic grid updates further align spline support with the activation range, so standard KAN already contains a form of adaptive localization.

Wavelet-based KAN makes the spectral interpretation explicit. Wav-KAN instantiates each edge function with wavelet bases, using scale and translation parameters as the primary controls:

$$\psi_{a,b}(x) = \frac{1}{\sqrt{a}}\,\psi\!\left(\frac{x-b}{a}\right).$$

Its layer mapping takes the form

$$x_i^{(l+1)} = \sum_{j=1}^n w_{i,j}\,\psi\!\left(\frac{x_j^{(l)} - \tau_{i,j}}{s_{i,j}}\right),$$

with learnable amplitude, scale, and translation on each edge (Bozorgasl et al., 2024). Because large scales capture low-frequency structure and small scales capture high-frequency localized structure, the wavelet edge itself functions as a frequency window.

Fourier-based KAN variants move from localized wavelet atoms to explicit sinusoidal spectral bases. Kolmogorov-Arnold Fourier Networks define a Random Fourier Feature map

$g_i$0

and combine it with a hybrid activation

$g_i$1

where $g_i$2 and $g_i$3 are learnable channel-wise scaling factors and the Fourier term is introduced as a high-frequency complement to GELU (Zhang et al., 9 Feb 2025). In this setting, the frequency distribution of the sampled Fourier basis and the learned mixing coefficients jointly determine the effective spectral window.

These mechanisms show that frequency windowing in the KAN literature is not restricted to one representation. Compactly supported splines, wavelet scales and translations, and Fourier-frequency distributions all implement different forms of localization or band selection.

3. Representative realizations resembling FW-KAN

Representative model	Frequency-windowing mechanism	Role of KAN
Wav-KAN	Wavelet scale and translation; CWT/DWT multiresolution	Edge functions are wavelet operators
KAF	Trainable RFF basis and hybrid GELU-Fourier activation	KA-style inner functions with merged linear outer mapping
KFS	FFT-domain top-$g_i$4 dominant frequency selection	KAN processes filtered series and mixing features
TimeKAN	Cascaded frequency decomposition into residual bands	Separate KAN block per band with band-dependent order

Wav-KAN is the most direct precursor of FW-KAN. It replaces spline activations with wavelet functions and explicitly states that the architecture captures both high-frequency and low-frequency components efficiently; its continuous and discrete wavelet constructions supply multiresolution, time-frequency localized edge operators (Bozorgasl et al., 2024). In a different direction, KAF addresses two stated limitations of vanilla KAN—parameter explosion and high-frequency feature capture—by replacing spline bases with trainable Random Fourier Features and by using an adaptive hybrid GELU-Fourier activation that progressively enhances frequency representation during training (Zhang et al., 9 Feb 2025).

KFS and TimeKAN are closer to the phrase “frequency-windowed” in an operational sense because they define windows or bands before KAN-based nonlinear learning. KFS introduces a FreK module that computes an FFT, sorts spectral energy, chooses the smallest $g_i$5 satisfying

$g_i$6

keeps those dominant frequencies, and reconstructs the denoised sequence by inverse FFT; KAN then models the filtered signal and participates again in feature mixing (Wu et al., 1 Aug 2025). TimeKAN constructs a hierarchy of smoothed sequences, recovers band components by

$g_i$7

and assigns a separate KAN to each band, with Chebyshev order increasing toward higher-frequency components (Huang et al., 10 Feb 2025).

Taken together, these architectures substantiate FW-KAN as a real design pattern, even though the exact label is not standardized. Some realizations window the basis functions themselves; others window the input spectrum and then apply KAN; others decompose the signal into bands and allocate a distinct KAN branch to each band.

4. Unified mathematical view and design space

A plausible unifying formalization of FW-KAN is a KAN layer in which each edge function is expanded in a frequency-localized dictionary. One synthesis in the KAN survey literature writes such an edge as

$g_i$8

with layer output

$g_i$9

where the basis functions $h_{ij}$0 may be wavelets, localized kernels, or other input- or frequency-localized atoms (Ji et al., 2024). This suggests an FW-KAN is characterized less by one canonical formula than by the choice of windowed univariate basis inside the standard KAN summation template.

A second axis is input-side spectral preprocessing. KFS applies hard spectral masking by dominant-energy selection in the FFT domain before KAN representation learning, while TimeKAN constructs a residual band decomposition through cascaded smoothing and frequency-preserving upsampling (Wu et al., 1 Aug 2025, Huang et al., 10 Feb 2025). This suggests a distinction between basis-windowed FW-KANs, in which the edge function itself carries the window, and input-windowed FW-KANs, in which the input to an otherwise ordinary KAN is first band-limited or band-separated.

A third axis is capacity assignment across bands. TimeKAN makes this explicit through a multi-order KAN rule

$h_{ij}$1

so higher-frequency bands receive higher-order ChebyshevKAN blocks and lower-frequency bands receive simpler ones (Huang et al., 10 Feb 2025). A plausible implication is that FW-KAN should not be identified solely with spectral filtering; it also includes band-dependent model complexity, because heterogeneous information density across frequencies motivates heterogeneous nonlinear approximation budgets.

Under this unified view, FW-KAN comprises three broad constructions: frequency-localized edge parameterizations, explicit spectral windowing of inputs, and band-aware allocation of KAN order, width, or branch structure.

5. Optimization, loss design, and computational trade-offs

Training behavior depends on which windowing mechanism is used. Wav-KAN employs standard backpropagation through smooth wavelet functions, applies batch normalization after the Wav-KAN layer, and in its MNIST experiments uses AdamW with learning rate $h_{ij}$2 and weight decay $h_{ij}$3 (Bozorgasl et al., 2024). KAF likewise remains end-to-end differentiable through trainable RFF parameters, and initializes the hybrid activation with $h_{ij}$4 and $h_{ij}$5, with $h_{ij}$6, so the Fourier component is initially weak and grows during training (Zhang et al., 9 Feb 2025).

When frequency selection is imposed as a hard operation, differentiability becomes partial rather than absent. KFS states that FFT and inverse FFT are linear and differentiable, whereas the sorting and discrete top-$h_{ij}$7 index selection in FreK are non-differentiable; gradients therefore propagate through the transforms and the downstream KAN blocks, but not through the combinatorial choice of which frequencies are kept (Wu et al., 1 Aug 2025). Its objective explicitly combines time-domain and frequency-domain supervision,

$h_{ij}$8

where $h_{ij}$9 aligns dominant spectral components and $$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$0 is pointwise time-domain error (Wu et al., 1 Aug 2025). By contrast, TimeKAN uses a structurally enforced decomposition and optimizes a standard $$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$1 forecasting loss without an auxiliary spectral penalty (Huang et al., 10 Feb 2025).

Efficiency considerations are a recurrent motivation for frequency-aware KAN variants. Wav-KAN reports parameter scaling $$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$2 because each edge carries weight, translation, and scale, whereas spline KAN scales as $$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$3 with grid-dependent overhead (Bozorgasl et al., 2024). KAF explicitly targets parameter explosion by merging KAN’s dual-matrix structure and yields per-layer parameter complexity that is asymptotically closer to an MLP when the number of Fourier features is small or fixed (Zhang et al., 9 Feb 2025). TimeKAN further reduces cost through variate-independent processing, depthwise convolution, and FFT-dominated complexity $$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$4 for multivariate time series (Huang et al., 10 Feb 2025). These results indicate that FW-KAN is often motivated not only by spectral fidelity but also by a search for more tractable replacements for dense spline parameterization.

6. Applications, interpretability, and open questions

The principal application domains are those in which informative structure is organized by scale or frequency. The survey literature highlights Wav-KAN and SpectralKAN for hyperspectral image analysis, time-series KAN variants for temporally structured data, and rational- or wavelet-based variants for richer function classes; it explicitly characterizes Wav-KAN as capturing multi-scale spatial-spectral patterns and SpectralKAN as separating spatial and spectral encoding (Ji et al., 2024). KAF extends the spectral line further into vision, NLP, audio processing, and differential equation-solving tasks, while KFS and TimeKAN address long-term time series forecasting by combining spectral filtering or decomposition with KAN-based representation learning (Zhang et al., 9 Feb 2025, Wu et al., 1 Aug 2025, Huang et al., 10 Feb 2025).

Interpretability remains one of the major inherited attractions. In base KAN, each edge carries a directly visualizable univariate function rather than a scalar weight (Liu et al., 2024). Wav-KAN adds semantically meaningful wavelet parameters—amplitude, scale, and translation—so an edge can be read as a localized detector of structures at a particular scale and location (Bozorgasl et al., 2024). KFS and TimeKAN preserve an additional bandwise semantics because nonlinear learning occurs after dominant-frequency selection or within explicitly reconstructed residual bands (Wu et al., 1 Aug 2025, Huang et al., 10 Feb 2025). This suggests that FW-KAN can support both function-level interpretability and spectrum-level interpretability, although the latter is clearest when the frequency decomposition is explicit.

A common misconception is that FW-KAN refers to a single canonical model. The current literature instead presents several partially overlapping realizations: wavelet edge functions, Fourier-feature activations, hard spectral masks followed by KAN blocks, and cascaded bandwise KAN pipelines. Another misconception is that frequency windowing must be Fourier masking. Wavelet scale-translation atoms and residual multiresolution decompositions also function as frequency windows, albeit with different localization trade-offs (Bozorgasl et al., 2024, Huang et al., 10 Feb 2025).

Open issues are correspondingly heterogeneous. The survey literature and later variants repeatedly point to computational complexity, high-dimensional scaling, and the challenge of modeling very high-frequency details with spline-only parameterizations, motivating wavelet, rational, and Fourier alternatives (Ji et al., 2024, Zhang et al., 9 Feb 2025). TimeKAN also exposes an unresolved design problem: band boundaries in cascaded decomposition are fixed by moving-average and decimation choices rather than learned directly, while KFS uses a hard top-$$x_{l+1,j} = \sum_{i=1}^{n_l} \phi_{l,j,i}(x_{l,i}),$$5 rule that is deterministic rather than differentiable (Wu et al., 1 Aug 2025, Huang et al., 10 Feb 2025). This suggests that future FW-KAN research will likely center on adaptive band formation, soft or learnable spectral masks, cross-band interactions, and scalable implementations that preserve KAN’s edge-function interpretability without reintroducing the parameter explosion that motivated these spectral variants in the first place.