Frequency Filtering Curriculum

• Frequency filtering curriculum is a method that structures learning by initially emphasizing low-frequency signals to capture global patterns before fine details.
• It employs techniques such as Fourier spectrum cropping, downsampling, and latent frequency modulation to accelerate convergence and enhance robustness.
• Adaptive scheduling progressively integrates higher frequency information, resulting in improved accuracy, efficiency, and generalization across various domains.

A frequency filtering curriculum refers to a family of machine learning and signal processing strategies in which the training or processing pathway is organized around frequency content—typically prioritizing the extraction, manipulation, or introduction of specific frequency bands first, followed by a progressive or staged inclusion of other, usually higher-frequency, information. Across domains such as computer vision, audio analysis, time series forecasting, and large-scale model training, frequency filtering curricula are employed to accelerate convergence, improve robustness, facilitate efficient computation, and enhance the generalization ability of learned models.

1. Conceptual Foundations and Motivations

Fundamentally, a frequency filtering curriculum leverages the observation that lower-frequency components of a signal usually encode global or broadly coherent structure, while higher-frequency components capture finer, local, or rapidly changing details. In the context of machine learning, especially for visual or auditory data, models tend to learn low-frequency (global) patterns earlier and more easily, whereas high-frequency (detailed) information is acquired later and may be more susceptible to noise or overfitting (2211.09703, 2507.03779). By explicitly structuring the learning schedule or data handling pipeline to reflect this order of complexity, frequency filtering curricula offer a principled pathway for both human-designed and automated learning procedures.

The rationale for such curricula is twofold: (1) to simplify the early optimization landscape by presenting smoother, less ambiguous data; (2) to provide a mechanism for gradually incorporating complexity, thereby improving both task-specific accuracy and stability across domains.

2. Methodological Realizations

2.1. Frequency-Prioritized Input Transformations

Frequency filtering curricula often begin at the data or input level. A common approach is to transform samples using a discrete Fourier transform (DFT) or similar operation, selectively retaining or attenuating frequency bands:

• Fourier spectrum cropping: Early training epochs present only the central (low-frequency) patch of each image’s spectrum; higher frequencies are reintroduced later (2211.09703, 2405.08768, 2507.03779).
• Downsampling: Reducing input resolution to discard high-frequency details, followed by later restoration of full resolution as learning progresses (2507.03779).

2.2. Dynamic Frequency Modulation in Latent Representations

Beyond the input, frequency filtering curricula are implemented within the network’s intermediate layers:

• Latent frequency masking: After applying a Fast Fourier Transform (FFT) to feature maps, instance-adaptive spatial masks are learned to emphasize transferable, low-frequency components and suppress non-conducive or high-frequency features (2203.12198).
• Learned or fractional filter bases: In convolutional architectures, filters may be constructed from basis functions (e.g., Gaussian derivatives), with the frequency-determining parameters learned as part of the model (2111.06660). Fractional derivatives enable smooth tuning of the filter’s frequency response according to data requirements.

2.3. Curriculum Schedules

The evolution from low- to high-frequency emphasis is governed either by fixed schedules (e.g., epoch-based stepwise increases of spectral inclusion width) or by adaptive search algorithms that maintain training performance while increasing frequency bandwidth (2211.09703). Such schedules may be coupled with other curriculum components such as gradually intensified data augmentation (2405.08768).

2.4. Frequency Filtering in Optimization Objectives

In some frameworks, frequency content is explicitly regulated via the loss function or the construction of synthetic data:

• Spectral filtering for dataset distillation: By applying spectral filter functions to the eigenvalues of feature-feature correlation matrices, objectives can emphasize global (low-frequency) or local (high-frequency) matching between synthetic and real datasets. “Curriculum Frequency Matching” (CFM) gradually adjusts filter parameters to transition from detail to global structure (2503.01212).
• Noise prior balancing in diffusion models: For generative video diffusion, frequency filtering curricula modify the spectral distribution of noise injects to maintain both low- and high-frequency variance, improving detail and semantic quality (2502.03496).

3. Practical Applications and Performance Benefits

3.1. Vision and Self-Supervised Model Training

Frequency filtering curricula are applied to accelerate the pre-training of large vision backbones (e.g., DINOv2/ViT) on datasets such as ImageNet-1K. By prioritizing coarse, low-frequency features (either via downsampling or Fourier cropping), convergence is reached more efficiently, with demonstrated reductions in pre-training time (up to 1.6× to 3.0× speedup) and FLOPs, while preserving or even enhancing robustness to real-world corruptions such as noise, blur, or occlusion (2211.09703, 2405.08768, 2507.03779).

3.2. Domain Generalization

Explicit frequency modulation in latent features improves domain generalization. By selectively enhancing or suppressing frequency bands, features with greater invariance and transferability across data domains are emphasized, resulting in superior performance on tasks such as person re-identification and image classification under domain shifts (2203.12198).

3.3. Multivariate Time Series Forecasting

In time series forecasting, frequency filtering curricula decompose each sequence into static (globally stable) and dynamic (cross-variable, window-specific) frequency components using modules that operate in the frequency domain. This approach improves forecasting accuracy and computational efficiency in datasets with pronounced periodic and trend structures (2505.04158).

3.4. Dataset Distillation

During dataset distillation, a curriculum that gradually shifts emphasis from high- to low-frequency correlation matching leads to synthetic datasets that simultaneously capture fine-grained local details and global structures, outperforming static (fixed-frequency) approaches across benchmarks like CIFAR-10/100 and ImageNet-1K (2503.01212).

3.5. Signal Extraction and Instantaneous Frequency Estimation

Frequency filtering curricula are also central to signal processing pipelines, including the design of digital filters for high-fidelity frequency estimation in noisy environments. Stage-wise filtering strategies (high-pass differentiation followed by optimized low-pass smoothing) ensure rapid and robust estimation of instantaneous frequencies in radar, wildlife acoustics, and communications (2307.00452).

4. Comparative Analyses and Theoretical Insights

Empirical results across domains indicate that frequency filtering curricula can improve convergence speed, robustness, and generalization relative to standard curricula or unstructured training. For example:

• Filtering out high-hardness (frequent misclassification) data during training yields larger accuracy gains than merely sorting instances by “difficulty” (1312.4986).
• In self-supervised vision settings, the curriculum’s two-stage structure (low-frequency first, then high-frequency plus Gaussian patching) preserves clean-set accuracy while increasing resistance to high-frequency-targeted corruptions (2507.03779).
• Theoretical justifications for various approaches derive from spectral analysis, information theory, and optimization landscape considerations. For example, lower-frequency biases in the early stages simplify optimization and prevent early overfitting to noise or spurious patterns (2211.09703, 2203.12198, 2111.06660). In dataset distillation, spectral filter functions clarify how various objectives balance between global and local representation (2503.01212).

5. Challenges, Extensions, and Future Directions

Several practical and theoretical challenges remain:

• Adaptive scheduling: Fixed-stage curricula may not adapt optimally for all datasets or architectures. Future directions include developing dynamic or convergence-driven transitions between frequency stages (2507.03779).
• Architectural considerations: In vision transformers, position embeddings and patchification complicate input resizing and frequency manipulation. Careful design is needed to maintain consistency across different input resolutions (2211.09703, 2405.08768).
• Generality and transfer: While most benefits are observed in visual and audio domains, analogues for other modalities (e.g., language, structured data) or tasks (e.g., reinforcement learning) are under-explored.
• Integration with other curricula: Frequency filtering can be combined with data augmentation, adversarial training, and curriculum dropout for finer control of robustness and generalization.

6. Representative Mathematical Tools and Scheduling Schemes

The following table summarizes common scheduling and filtering methods used in recent research:

Approach / Paper	Frequency Filtering Mechanism	Scheduling Strategy
EfficientTrain (2211.09703)	Fourier spectrum cropping of image inputs	Stepwise or greedy
FastDINOv2 (2507.03779)	Early low-frequency (downsampled) training, then full-res	Fixed percentage split
DFF (2203.12198)	FFT of latent features, attention masking in frequency	Adaptive per instance
CFM (2503.01212)	Eigenvalue-based spectral filtering in DD objectives	Cosine schedule for β
FilterTS (2505.04158)	Dynamic/static FFT-based filtering in time series	Merged per batch

Typical curriculum steps include frequency cropping:

$$X_{\text{c}} = \mathcal{F}^{-1} \circ \mathcal{C}_{B,B} \circ \mathcal{F}(X)$$

Scheduling of filter parameters as in CFM:

$$\beta_t = \beta \frac{1 + \cos\left(\frac{\pi t}{T}\right)}{2}$$

or selection of harder examples via instance hardness ranking:

$$IH(x_i, y_i) \approx 1 - \frac{1}{|\mathcal{L}|} \sum_{j=1}^{|\mathcal{L}|} p(y_i | x_i, t, g_j)$$

7. Broader Implications

The frequency filtering curriculum paradigm generalizes across supervised, self-supervised, and unsupervised settings, with successful deployments in standard vision and audio tasks, large-scale foundation model pretraining, time series forecasting, and data distillation. By enabling models to focus sequentially or adaptively on easier-to-harder or global-to-local patterns, these curricula facilitate scalable, robust, and generalizable learning with practical efficiency and improved real-world applicability.