FourierSampler: Frequency-Domain Sampling

• FourierSampler is a collection of methods that utilize the frequency domain to sample, resample, and guide inference in high-dimensional, structured signals.
• It separates low-frequency components that capture global structure from high-frequency details, improving decoding performance in models like diffusion-based language models.
• Its applications span computational imaging, kernel regression, group-theoretic sampling, and compressed sensing, with strong theoretical guarantees and adaptive error control.

FourierSampler refers to a spectrum of methods and algorithms that leverage frequency-domain techniques for sampling, resampling, or guiding inference and reconstruction in high-dimensional structured signals and neural models. These approaches appear in numerical analysis, computational imaging, group representation, statistical machine learning, and, most recently, the frequency-guided decoding of diffusion-based LLMs. The following sections comprehensively review core methodologies, use cases, theoretical frameworks, and empirical properties as detailed in recent literature.

1. Spectral Decomposition and Frequency-Domain Filtering in LLMs

FourierSampler in the context of non-autoregressive diffusion LLMs (dLLMs) denotes a spectral-guided generation algorithm that exploits the intrinsic frequency structure of hidden states. Empirical spectral analysis of the hidden-state matrix $H\in\mathbb{R}^{L\times D}$, via discrete Fourier transform (DFT) along the token dimension, reveals a partition between low-frequency components encoding global, structural information, and high-frequency components encoding local, detail-rich content. This motivates a frequency-domain sliding window mechanism in which, at each diffusion step $s$, hidden states are masked and filtered through a window $g(s)$ that linearly traverses the frequency spectrum from low to high as generation proceeds.

The filtered hidden vector $H'(s) = F_r^{-1}(F_r(H(s)) \odot g(s))$ is used to compute a normalized in-band energy $\ell_{s,t}$ per token, adaptively combined with the model's confidence $c_{s,t}$ via a dynamic weight $\beta_s$ controlled by the confidence variance trajectory. Token selection is then governed by maximizing $\tilde c_{s,t} = c_{s,t} + \beta_s \ell_{s,t}$. Early steps thus favor structure tokens (high energy in low frequencies), while later steps prioritize fine details (high energy in high frequencies). This structure-to-detail schedule achieves substantial empirical improvements over previous decoding approaches in dLLMs, e.g., delivering $+20.4\%$ relative gain on MBPP and consistently surpassing equivalently-sized autoregressive baselines (He et al., 30 Jan 2026).

2. Frequency-Guided Resampling, Interpolation, and Up-sampling

Fourier-domain sampling and resampling arise in classical and modern computational contexts:

• Image resampling in the Fourier domain: High-fidelity resampling of images and signals in Fourier space, employing interpolation via wrapped-sinc kernels, enables accurate transformations such as PSF adjustment, rescaling, and shearing. The gold-standard approach involves zero-padding followed by resampling using specially-designed compact-support interpolants, such as the 6-point, $C^2$-continuous piecewise-quintic kernel, to suppress ghost images and achieve error $<10^{-3}$ (Bernstein et al., 2014).
• Deep Fourier up-sampling: The FourierUp operator in deep learning reconstructs higher-resolution spatial features by expanding and manipulating their Fourier spectra, employing periodic or area-based frequency tiling followed by inverse FFT. It promotes global dependency and improved downstream performance compared to local spatial interpolants. FourierUp components include DFT, spectral dimension increase, learnable magnitude/phase adjustments, and IDFT, and can be integrated into CNNs for object detection, segmentation, and restoration tasks (Zhou et al., 2022).

These FourierSampler utilities are mathematically principled and computationally efficient, with analytical error bounds and minimal algorithmic complexity increases over native spatial-domain or uniform methods.

3. Universal and Deterministic Sampling Theorems

Statistically optimal signal reconstruction from partial samples has motivated the development of universal Fourier sampling frameworks:

• Universal sampling for band-limited, sparse, and multiband signals: The sample complexity of reconstructing a signal constrained by a given Fourier measure $\mu$ is dictated by its statistical dimension $s_{\mu,\epsilon}$—essentially the trace of a regularized kernel operator in the native function space. Sampling time points from a universal, heavy-tailed density proportional to the ridge leverage score achieves near-optimal rates $$q = O(s_{\mu,\epsilon} \log^2(s_{\mu,\epsilon}/\delta))$$ for $O(\epsilon)$ MSE, regardless of $\mu$. The reconstruction algorithm reduces to kernel ridge regression with these non-uniform samples (Avron et al., 2018).
• Explicit deterministic sampling on finite abelian groups: For $t$-sparse Fourier signals on $G=\mathbb{F}_p^r$, there exist explicit sampling sets $T_1$ and $T_2$, constructed from collection of small subgroups and sparse cosets, which enable deterministic, polynomial-time $\ell_1$-approximation algorithms. These achieve $O(p t^2 r^2)$ or $O(p t^2 r^3 \log p)$ sample complexity for exact or approximate recovery (Morotti, 2015).
• Gauss-Legendre sampling on SO(3): For signals band-limited at degree $L$ on the rotation group, sampling at carefully chosen Gauss-Legendre nodes in $\beta$ and equiangular points in the other Euler angles achieves sampling efficiency $\eta(L)\to 1/3$, twice as high as prior schemes, with stable, inversion- and round-trip-accurate Wigner-D forward/inverse transforms (Khalid et al., 2015).

A summary table is shown below for representative FourierSampler theoretical guarantees:

Context	Sampling Complexity	Error Guarantee
Universal time-domain	$O(s_{\mu,\epsilon} \log^2 s)$	$O(\epsilon)$ MSE
Finite abelian group	$O(p t^2 r^2)$ or $O(\cdot \log p)$	$(1+\frac{3}{2})\epsilon$ $\ell_1$-error
SO(3), bandlimit $L$	$O(L^3)$	Exact FT/IFT, $\sim 10^{-14}$ rel. error

4. Adaptive and Randomized Spectral Sampling in Machine Learning

FourierSampler methodologies are foundational in the machine learning literature for efficient kernel approximation and compressed sensing:

• Adaptive random Fourier feature sampling: In high-dimensional kernel regression/classification, optimal convergence is realized via resampling Fourier frequencies adaptively in proportion to learned spectral magnitudes. The iterative algorithm combines random walks in frequency space, empirical amplitude estimation, and importance resampling, converging to the minimax density $p_*\propto |\hat f(\omega)|$. Generalization error achieves the rate $(1+o(1))C_{p_*}/m$ for $m$ features; in practice, adaptive resampling offers superior accuracy and consistency to uniform and Metropolis Monte Carlo baselines (Huang et al., 3 Sep 2025).
• Compressed Fourier single-pixel imaging: Variable-density, Gaussian random sampling of Fourier coefficients (probability $p(k)\propto \exp(-\hat{k}^2/2\sigma^2)$) provides near-theoretically optimal acquisition for imaging tasks, reconciling energy concentration in natural images. With a sampling ratio $n$, the optimal standard deviation is $\sigma \simeq n$, offering sharp reconstructions (SSIM $>0.83$ at $10\%$ sampling for $256\times256$ images) after $\ell_1$-sparse wavelet recovery (Qiu et al., 2021).

These approaches exploit the empirical and theoretical concentration of signal power in low-frequency bands, aligning measurement resources for maximal information extraction and recovery.

5. FourierSampler in Semiclassical and Group-theoretic Contexts

FourierSampler is also integral to semiclassical analysis and group representation:

• Semiclassical sampling: For functions with Fourier content up to $O(h^{-1})$, optimal discretization rates are guided by the wavefront set under the action of Fourier integral operators (FIOs), with Nyquist step $s < \pi/B$ in each coordinate. The volume of the phase-space support provides a Landau-type lower bound on the number of stable samples. Sampling-induced aliasing is mapped through the canonical relation of the FIO, and optimal sampling lattices can be non-uniform or sheared, as in Radon transform applications (Stefanov, 2018).
• Half-infinite and step sampling: The half-infinite Dirac comb leads to succinct Fourier domain representations (e.g., $S_{h6}(f) = 1/(2j\sin(\pi f T))$), enabling analytically concise convolution-based representations of discretization effects. This approach provides both theoretical and computational benefits over the classical infinite comb summation (Li, 2021).

6. Algorithmic Patterns, Error Analysis, and Implementation Trade-offs

FourierSampler design typically follows a pattern: spectral band identification (analytic or learned), selection of optimal or near-optimal sampling or filtering schedules, and controlled inversion or reconstruction. Error is quantifiable in terms of total variation, Wasserstein-1, $\ell_1$, or mean-squared bounds, scaling with grid spacing, statistical dimension, or signal sparsity.

• Analytical error guarantees: For piecewise-linear interpolation in band-limited density models, total variation and Wasserstein errors scale as $O(K^{-2})$ with kernel-determined leading constants, and grid step $\Delta = 2/K$ (Fuente et al., 9 May 2025).
• Computational costs: Classical approaches incur $O(N^4)$ cost for exact resampling, mitigated to $O(K \log K + S)$ in FFT-based and lookup-based algorithms, and $O(L^4)$ for fast group-theoretic sampling. Memory and compute trade-offs depend on zero-padding, kernel design, and sampling pattern efficiency. Spectral methods demand global transforms (FFT/IFFT), learnable parametric corrections in deep models, and well-designed interpolation or zero-padding for ghost suppression.

A plausible implication is that future FourierSampler algorithms will increasingly blend adaptive, data-driven spectral analysis (e.g., in neural network hidden representations) with classical and deterministic optimality principles established in group signal sampling, enabling high-accuracy, efficient learning and reconstruction pipelines across modalities.

7. Limitations and Prospects

Despite comprehensive empirical validation and mathematical guarantees, current FourierSampler designs can entail nontrivial computational overheads (FFT per step in neural models, global matrix assembly in kernel methods), which may limit scalability for extremely large inputs or models. Block size choices, window scheduling, and parameter tuning (e.g., $\rho$ for filter width, $\beta$ calibration) can materially affect empirical performance (He et al., 30 Jan 2026). There remains open territory in designing hardware-accelerated spectral architectures, developing adaptive or context-aware schedule schemes (e.g., for variable sequence length), and integrating explicit spectral regularization in the training of generative models. Extensions to hybrid AR/diffusion model regimes, spectral sampling on non-abelian groups beyond SO(3), and explicit control of aliasing in FIO-discretized forward models represent fertile ground for ongoing research.