Low-Frequency Replacement Module

• Low-Frequency Replacement (LFR) Module is a technique that manipulates low-frequency signal components to enhance domain invariance and improve deep learning generalization.
• It employs methods such as Gaussian low-pass filtering, Fourier masking, and spectrum replacement to mitigate noise and reinforce global structure.
• Empirical results demonstrate significant gains in classification accuracy, reconstruction fidelity, and memory efficiency across applications like seismic inversion and few-shot learning.

Low-Frequency Replacement (LFR) Module refers to a class of algorithmic components that operate by identifying, extracting, or manipulating low-frequency content in signals—whether image, seismic, neural-network weights, or latent representations—to address fundamental limitations of generalization, reconstruction, or fidelity in deep learning systems. LFR modules encompass explicit frequency-domain operations (e.g., Fourier masking, low-pass filtering, spectrum truncation), signal spectrum replacement between domains, and corrective mechanisms that re-inject missing low-frequency information absent from model training or data acquisition. These techniques have been deployed across unsupervised domain adaptation, cross-domain few-shot learning, inversion of seismic data, neural operators for parametric PDEs, and latent-space denoising in generative models.

1. Motivating Principles and Frequency-Space Foundations

LFR modules universally stem from the observation that low-frequency components in signals (or model parameters) often encode domain-invariant, structural, or physically salient information (e.g., shapes, subsurface profiles, illumination, layerwise regularity), whereas high-frequency components capture noise, texture, domain-specific artifacts, or parametric volatility (Li et al., 2022, Hui et al., 10 Nov 2025). In neural architectures, this is reflected in the frequency principle: network fitting progresses from low to high frequencies, rendering low-frequency content easier to learn and generalize (Wang et al., 21 Jun 2025). In data distributions, domain gaps and few-shot generalization failures are dominated by biases in the low-frequency spectrum (Hui et al., 10 Nov 2025). In signal reconstruction and inverse problems (seismic or generative), missing or mismatched low-frequency information leads to reconstruction bias, cycle-skipping, or loss of global coherence (Cong et al., 2024, Hu et al., 2023).

2. Mathematical Formulations and Operational Variants

LFR implementations fall into several prototypical forms:

• Discrete Gaussian Low-Pass Filtering: In domain adaptation, convolutional feature maps are passed through a fixed Gaussian kernel $$G(x, y) = \frac{1}{2\pi \sigma^2} \exp\left(-\frac{x^2 + y^2}{2\sigma^2}\right)$$ (e.g., $m = 3, \sigma = 1$), effecting a linear shift toward low-frequency structure with no trainable parameters (Li et al., 2022).
• Fourier Masking and Spectrum Replacement: In cross-domain few-shot learning, images are represented by $\hat{X} = \mathcal{F}(X)$ (FFT per channel), and a binary mask $M_{\text{low}}$ selects frequencies within radius $r = \gamma \cdot \min(H,W)$, typically $\gamma \sim U(0,0.2)$. The low-frequency band of a source is replaced with that of a paired target, i.e.,

$$\hat{X}' = \hat{X}_{\text{low}}^{\text{tgt}} + \hat{X}_{\text{high}}^{\text{src}},$$

followed by inverse FFT to reconstruct the mixed-spectrum image $\widetilde{X}$ (Hui et al., 10 Nov 2025).

• Layerwise Fourier Reduction in Neural Operators: In parametric PDE solvers, each weight vector $W$ is truncated in the Fourier domain: only the first $p \ll N$ coefficients are generated by a per-layer hypernetwork; higher frequencies are zeroed. Reconstruction proceeds via

$$W_n^{(p)} = \frac{1}{N} \sum_{k=0}^{p-1} \hat{W}_k e^{2\pi i \frac{kn}{N}}.$$

This targets computational and statistical efficiency by filtering residual noise (Wang et al., 21 Jun 2025).

• Terminal Latent Correction in Diffusion Models: OMS/LFR modules introduce an additional inference step: a compact U-Net predicts the missing low-frequency ($v$-parameter) from pure Gaussian noise $x_T^{\mathcal{S}} \sim \mathcal{N}(0, I)$, reconstructing the proper terminal latent via

$$\tilde{x}_T^{\mathcal{T}} = \sqrt{\bar{\alpha}_T} \cdot \tilde{x}_0 + \sqrt{1 - \bar{\alpha}_T - \sigma_T^2} \cdot x_T^{\mathcal{S}} + \sigma_T \cdot \epsilon,$$

before running the standard denoising loop (Hu et al., 2023).

3. Module Architectures and Integration Patterns

ConvNet Integration:

• LFR modules are generally parameter-free layers (fixed kernel convolutions, e.g., depthwise Gaussian in PyTorch), slotted after feature extraction or downsampling, or at the final block prior to pooling/classification (Li et al., 2022).
• Typical utilization schemes include Insert-at-End (IE) and Replace Strided Layers (RSL). IE applies LFR after all convolutions, while RSL swaps strided convs for non-strided convs plus LFR, preserving anti-aliasing and Nyquist compliance.

Meta-learning Pipelines:

• LFR in FreqGRL is a pure FFT-based augmentation layer, applied to all input images during episode sampling. The mask is generated per-episode; pseudo-source images with target low-frequencies are co-trained alongside original source and target images in episodic classification loss (Hui et al., 10 Nov 2025).

Transformer and PINO Frameworks:

• LFR in seismic inversion wraps a fully window-based Transformer with shifted-window self-attention. 1D convolutions first lift input channels, followed by $N$ blocks alternating classic and shifted windows, ending with convolutional projection to output (Cong et al., 2024).
• LFR-PINO modularizes low-frequency spectrum generation per layer, with each hypernetwork producing only complex coefficients for the spectral low-frequency bands. No direct high-frequency learning occurs; the entire PINO stack operates with truncated spectra (Wang et al., 21 Jun 2025).

Diffusion Pipeline Augmentation:

• OMS/LFR modules train only the corrective network $\psi$ and keep all pre-trained sampling weights fixed. OMS is invoked once at inference prior to the denoising loop, supporting plug-and-play deployment for generative pipelines (Hu et al., 2023).

4. Empirical Benchmarks and Quantitative Impact

Classification and Detection:

Dataset/Task	Baseline	+LFR (IE/RSL)	Gain
Office-31 (ResNet-50)	76.1% (ft)	81.4–81.6%	+5.3%
VisDA-2017 (ResNet-101)	86.8% (CAN)	87.3–87.4%	+0.5%
Cityscapes→FoggyCityscapes	40.8 mAP	42.1 mAP	+1.3 mAP

Few-shot Learning (CUB 5-way 1-shot):

Scheme	Accuracy	Gain
Baseline	57.99%	—
+LFR (γ ∼ U(0,0.2))	64.06%	+6.07%

Seismic Data Reconstruction:

Model	MSE	SSIM	SNR (low-freq band)	Infer Time
1-D U-Net	1.22e-1	0.59	—	~23 s/shot
LFR Transformer	1.46e-2	0.89	+15 dB relative	~16 s/shot

PINO Error and Memory:

PDE Task	LFR-PINO $L_2$	Hyper-PINN $L_2$	Reduction (%)
Anti-derivative	0.00336	0.00486	–30.9
Advection	0.00621	0.01982	–68.7

Memory usage reductions between 28.6%–69.3% are reported in (Wang et al., 21 Jun 2025).

Diffusion Generative Metrics:

Metric	SD1.5 Raw	OMS	Impact
FID	12.52	14.74	+2.22
CLIP	0.2641	0.2645	≈ parity
ImageReward	0.1991	0.2289	+0.0298
PickScore	21.49	21.55	+0.06
Mean pixel dist	22.47	7.84	–14.63

OMS modules markedly spread output brightness and color, correcting low-frequency truncation.

5. Application Contexts and Deployment Strategies

Domain Adaptation/Generalization:

• LFR modules enforce domain invariance by focusing classifiers on low-frequency content, reducing cluster separation and feature mismatch as measured by MMD (Li et al., 2022, Hui et al., 10 Nov 2025).

Cross-Domain Few-Shot Training:

• In FreqGRL, LFR suppresses source-domain bias while enhancing target sensitivity, critical for tasks with severe label imbalance. It improves feature alignment and cross-domain transfer without adding model parameters.

Seismic Full-Waveform Inversion (FWI):

• LFR plug-in modules supply synthetic low-frequency traces used in the first stage of FWI. This mitigates cycle-skipping, produces robust low-wavenumber velocity models, and allows more accurate high-frequency inversion (Cong et al., 2024).

Physics-Informed Neural Operators:

• Layerwise LFR truncation allows pre-trained PINO models to generalize efficiently to new PDEs, maintain solution fidelity, and control memory, with retrainable top layers for downstream adaptation (Wang et al., 21 Jun 2025).

Latent-Space Correction in Diffusion Models:

• OMS/LFR modules restore proper low-frequency content at the terminal timestep of the denoising chain. This rectifies brightness bias, enhances coverage, and affords additional low-frequency style control via prompt manipulation (Hu et al., 2023).

6. Practical Guidelines and Implementation Notes

• LFR modules are computationally light: fixed filters or FFT/IDFT operations are performed once per batch or episode, amortized across data (Li et al., 2022, Hui et al., 10 Nov 2025).
• No learnable parameters are introduced in standard LFR modules; memory and computation are minimized except where a lightweight corrective net ($\psi$) is used (OMS) (Hu et al., 2023).
• In physical and scientific networks (PINO), only low-frequency spectral bands ($p/N=0.2–0.4$) should be retained; high-frequency bins can be monitored and collapsed further if underutilized (Wang et al., 21 Jun 2025).
• Gaussian and spectrum-based LFR modules preserve spatial resolution if padding and normalization are set appropriately.
• LFR can be fused with other adaptation mechanisms (MMD, RevGrad, CAN, etc.), yielding additive or synergistic gains on standard benchmarks (Li et al., 2022).
• For diffusion models, OMS modules should share the latent domain; otherwise, retrain only the corrective network in the new latent space (Hu et al., 2023).
• For seismic and scientific deployments, LFR modules can be integrated as an upstream pre-processing step with negligible latency, facilitating operational full-waveform inversion.

7. Limitations and Prospective Directions

• LFR relies on the assumption that low-frequency content is inherently more domain-invariant or physically regular; this may not hold for all tasks (e.g., texture-driven classification, cases with significant target high-frequency signature).
• Further investigation into multi-scale replacements, spatially adaptive or learnable low-pass filters, and task-specific spectral manipulation is warranted.
• Integrating spectrum replacement with adversarial or reinforcement signals could further improve feature disentanglement.
• In generative modeling, end-to-end differentiable spectrum correction or fusion with learned schedule modifications may yield additional improvements in fidelity and flexibility.

Low-Frequency Replacement modules constitute an orthogonal, plug-and-play class of techniques for controlling, correcting, and biasing deep learning models toward robust exploitation of essential low-frequency structure—a central mechanism in addressing domain shift, missing data, and stability in both discriminative and generative pipelines.