Variational Fourier-domain Filtering

• Variational Fourier-domain filtering is a method that combines Fourier spectral analysis with variational optimization to achieve optimal signal reconstruction and denoising.
• The approach generalizes classical filters by incorporating fractional derivatives, graph-based methods, and quantum-inspired algorithms to enhance computational efficiency.
• Its applications include image deblurring, deep generative modeling, robust Gaussian process inference, and scalable kernel learning on irregular domains.

Variational Fourier-domain filtering encompasses a family of methods that integrate Fourier spectral analysis and variational optimization for signal estimation, denoising, generative modeling, and kernel learning. These approaches formulate signal reconstruction, filtering, or inference problems as the minimization of an objective functional or evidence lower bound (ELBO), incorporating constraints or losses directly in the (generalized) Fourier domain. The resulting frameworks unify classical signal-processing filters, penalized regression, and modern deep generative architectures. Applications include spectral denoising, image deblurring, graph signal processing, robust Gaussian process inference, and the construction of efficient representations in quantum algorithms.

1. Variational Formulation in the Fourier Domain

Variational Fourier-domain filtering is grounded in the principle of encoding data fidelity and spectral smoothness via an energy functional defined over the Fourier representation of the target function. A canonical supervised learning example considers real-valued functions $u:\mathbb{R}^d\to\mathbb{R}$ whose Fourier transform $\hat u(\xi)$ minimizes a frequency-weighted $L^2$ energy while satisfying prescribed (generally interpolation) constraints on isolated points $\{x_i, y_i\}$:

$$\min_{\hat u\in L^1\cap L^2} \; J[\hat u] = \frac12 \int_{\mathbb{R}^d} A(\xi) |\hat u(\xi)|^2 \, d\xi$$

subject to

$$P_X\hat u = ( \int \hat u(\xi) e^{2\pi i \xi\cdot x_1} d\xi, ..., \int \hat u(\xi) e^{2\pi i\xi\cdot x_n} d\xi )^T = Y$$

where $A(\xi)$ is a positive spectral weighting, e.g., $A(\xi) = (1+|\xi|^2)^s$ reflecting Sobolev smoothness of order $s$ (Luo et al., 2020). The inverse weighting $1/A(\xi)$ induces an optimal, data-fit preserving, spectral filter concentrating energy in low-frequency bands. Well-posedness requires $s > d/2$ for point-evaluation and finite $H^s$ norm. The solution admits a closed Fourier-multiplier form:

$$\hat u^*(\xi) = (1+|\xi|^2)^{-s} \sum_{i=1}^n \lambda_i e^{2\pi i \xi \cdot x_i}$$

with Lagrange multipliers $\lambda$ solving a system built from the associated Green's kernel.

In the context of spectral denoising or penalized regression, this formulation generalizes to any scenario where data-fidelity in the spatial domain is traded off against regularized spectral energy, producing a family of low-pass or band-limited variational filters.

2. Fractional and Structured Filtering via Fourier Variational Methods

Fractional-order variational Fourier-domain filtering extends classical quadratic (Tikhonov) regularization to higher or fractional Sobolev penalties, parameterizing filter smoothness continuously. For a one-dimensional signal $u(t)$, the variational problem is:

$$J[f] = \int_{\mathbb{R}} |f(t) - u(t)|^2 \, dt + \lambda\int_{\mathbb{R}} |D^{\alpha}f(t)|^2 \, dt$$

where $D^{\alpha}$ denotes the Riemann–Liouville fractional derivative of order $\alpha>0$ (Lemes et al., 15 Nov 2025). Transforming into the Fourier domain, taking $$\mathcal{F}\{D^{\alpha}f\} = (i\omega)^\alpha \hat f(\omega)$$, one obtains a closed-form frequency response:

$$\hat f^*(\omega) = \frac{\hat u(\omega)}{1+\lambda \omega^{2\alpha}}$$

$$\hat h(\omega) = \frac{1}{1 + \lambda \omega^{2\alpha}}$$

This spectral transfer function interpolates between ideal low-pass (large $\lambda$) and identity (small $\lambda$); the fractional order $\alpha$ smoothly interpolates filter sharpness. Optimal $(\alpha, \lambda)$ can be selected by minimizing the Shannon entropy of the filtered intensity, balancing noise suppression and feature retention.

These fractional variational Fourier-domain filters are computationally efficient ($O(N\log N)$ per channel with DFTs), robust, and flexible, supporting multivariate and multidimensional signals with direct extension to images (per-channel processing with the fractional Laplacian $(-\Delta)^{\alpha/2}$).

3. Variational Fourier-domain Filtering in Deep Generative Modeling

Within deep generative models, variational Fourier-domain filtering is applied to enforce sharp reconstruction and perceptual quality. Standard VAEs trained with Gaussian likelihoods produce blurry samples due to the averaging effect of squared-error losses and the Kullback–Leibler prior term. Augmenting the VAE ELBO with a Fourier-domain loss overcomes this limitation by directly penalizing deficits in high-frequency (detail-rich) spectral components (Dalal, 2024).

A state-of-the-art approach employs a short-time Fourier transform (STFT) computed over local image patches with a Hann window. Within this framework, for each local STFT spectrogram coefficient (indexed by frequency $(u,v)$ and spatial location $(m_0, n_0)$):

• Amplitude $A_{i,j}$ and phase $P_{i,j}$ are extracted for both the input and reconstruction
• A composite amplitude-phase $L_1$ loss is imposed, with phase discrepancies emphasized (scaled by $\lambda > 1$)
• A frequency-dependent weighting $W_j$ penalizes loss mismatch more strongly at higher spectral radii (fine details)

The full loss combines this STFT-based loss with pixelwise perceptual loss (1-SSIM) and the usual ELBO KL regularization:

$$\mathcal{L} = \beta\, D_{KL}(q_\phi(z|x)\,\|\,p(z)) + \lambda_{freq}\,L_{freq} + [1 - \mathrm{SSIM}(x, \hat{x})]$$

This local, phase-sensitive, frequency-weighted filtering loss forces the model to match both the amplitude and phase structure of the data in the spectral bands responsible for sharpness, outperforming global DFT or pixelwise-only losses in quantitative and perceptual metrics (e.g., PSNR, SSIM on MNIST).

4. Variational Fourier-domain Filtering on Graph Domains

Generalizing from Euclidean to irregular domains, variational Fourier-domain filtering is constructed on graphs by replacing the standard Fourier basis with graph Fourier transforms (GFT) derived from the eigendecomposition of a positive semi-definite variation operator $V$ such as the combinatorial or normalized Laplacian (Pavez et al., 2022). This yields a spectral representation:

$$V = U \Lambda U^T, \quad \hat f = U^T M f, \quad f = M^{-1} U \hat f$$

Filter banks are defined by spectral multipliers—diagonal operators $h(\lambda)$ acting on graph frequencies. Variational formulations appear in smoothing, downsampling, and critical sampling via filterbanks modeled on the spectrum:

• Perfect reconstruction and cross-alias-free conditions are formulated directly in the spectral domain with constraints on the sum and "folding" of filter multipliers
• The distributive structure of the GFT allows both spectral (global) and vertex-domain (localized polynomial) implementations
• Vertex partitions for downsampling can be optimized via spectral relaxations of balanced max-cut problems, ensuring numerical stability and statistical efficiency

For graph signals, this approach delivers scalable, well-conditioned, and interpretable filtering analogous to classical Fourier-domain variational approaches.

5. Variational Fourier Features and Kernel Learning

Variational Fourier-domain techniques also underlie modern kernel-learning frameworks, such as the harmonizable mixture kernel (HMK) for Gaussian processes. Here, the kernel admits a spectral (Fourier-nonlocalized) representation via a generalized spectral density $S(\omega, \omega')$ and is parameterized as a mixture of locally stationary modulations (Shen et al., 2018):

$$k(x, x') = \iint e^{2\pi i (\omega^T x - \omega'^T x')} S(\omega, \omega') \, d\omega \, d\omega'$$

Sparse variational inference leverages inducing variables at selected frequencies (variational Fourier features). The ELBO is maximized by variationally learning (i) the feature distribution in the spectral domain, (ii) kernel mixture parameters, and (iii) hyperparameters, yielding scalable, non-stationary kernels and filters. The streaming interpretation treats the inducing coefficients as a fixed state, with sequential Bayesian updates on observation arrival.

This approach unifies expressive non-stationary kernel modeling and spectral filtering, supporting streaming and high-dimensional applications with manageable computational complexity ($O(M^3 + NM^2)$ for $M$ features and $N$ data).

6. Fourier-domain Filtering in Variational Quantum Algorithms

In variational quantum circuits, Fourier expansion and truncation serve as a form of spectral filtering for loss function surrogates. For a Clifford+Pauli circuit $U(\theta)$, the observable expectation $$L(\theta) = \langle 0|U(\theta)^{\dagger} H U(\theta)|0\rangle$$ admits a full multinomial Fourier expansion. Truncation at degree $m$ filters out higher-frequency monomials, providing computationally efficient low-degree surrogates with quantified $L^2$ error (Nemkov et al., 2023):

• Classical algorithms propagate observables through circuit generators, recursively "branching" via commutation relations to enumerate Fourier monomials
• Truncating at level $m$ yields a low-degree surrogate $L^{(m)}(\theta)$ whose approximation error $\|\cdot\|_2$ is precisely bounded in terms of retained coefficient norms
• For local circuit ansätze, most coefficients are concentrated at low degrees, so filtering drastically reduces evaluation cost with minimal loss

This illustrates a computational-spectral perspective on variational filtering in the Fourier domain, applicable to optimizing quantum generative and inference algorithms.

7. Context, Significance, and Extensions

Variational Fourier-domain filtering is characterized by several unifying features:

• The variational principle provides a systematic, theoretical basis for the derivation of spectral filters, guaranteeing optimality with respect to the specified functional
• Closed-form solutions are frequently available via spectral multipliers, ensuring computational efficiency
• Weightings in the spectral domain encode application-specific priors: e.g., high-order/fractional derivatives for sharp feature preservation or regularization, graph Laplacians for structural smoothness, and phase-sensitive losses for perceptual quality
• The connection to kernel methods and probabilistic graphical models offers statistical interpretability and Bayesian extensions

Prominent open directions include the learning of spectral weights and phase-amplitude factors, multi-scale or multi-resolution spectral supervision, adaptation to manifold or non-Euclidean domains, and the development of deep architectures explicitly integrating variational Fourier-domain layers or loss components.

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Key References:

• Short-Time Fourier Transform for deblurring Variational Autoencoders (Dalal, 2024)
• Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning (Luo et al., 2020)
• A Fractional Variational Approach to Spectral Filtering Using the Fourier Transform (Lemes et al., 15 Nov 2025)
• Two Channel Filter Banks on Arbitrary Graphs with Positive Semi Definite Variation Operators (Pavez et al., 2022)
• Harmonizable mixture kernels with variational Fourier features (Shen et al., 2018)
• Fourier expansion in variational quantum algorithms (Nemkov et al., 2023)