FLARE Loss: Frequency & Boundary Reliability

• FLARE Loss is a composite loss function that combines frequency-domain analysis with local-boundary awareness to address spectral bias and boundary ambiguity.
• It employs techniques like dynamic spectral weighting, wavelet decompositions, and edge map extraction to boost performance in domains such as image synthesis, depth estimation, and solar flare forecasting.
• Empirical results demonstrate improvements in metrics like PSNR, SSIM, mIoU, and TSS, underscoring its effectiveness in preserving fine details and accurately penalizing boundary errors.

Frequency & Local-Boundary-Aware Reliability Loss (FLARE Loss) refers to a composite loss formulation that integrates frequency-domain sensitivity and local boundary awareness to improve predictive reliability, particularly in tasks suffering from class imbalance, boundary ambiguity, or poor high-frequency response. This concept has origins in several research domains including image reconstruction, interface dynamics, monocular depth estimation, federated learning, and solar flare forecasting.

1. Conceptual Foundations

FLARE Loss builds upon the limitations of traditional spatial-domain loss functions such as mean squared error (MSE) or cross-entropy, which treat all discrepancies uniformly and often fail to address spectral bias or the significance of errors near boundaries. By introducing frequency-aware components—such as dynamic weighting of hard-to-synthesize frequency bands (Jiang et al., 2020), wavelet-based decompositions (Prantl et al., 2022), and adaptive filters (Chen et al., 2022)—the loss function shifts emphasis towards regions and components most critical for perceptual or safety outcomes. Local-boundary awareness further augments these capabilities by giving higher correction weight at edges or regions of high semantic entropy, as seen in boundary detection tasks and model reliability evaluation.

2. Mathematical Formulation

The mathematical definition of FLARE Loss varies by application but generally involves two terms:

• A frequency-based loss, where predictions and targets are mapped into a frequency domain (e.g., via discrete Fourier or wavelet transform) and compared using attention weights that emphasize difficult frequency components.
• A local-boundary reliability term, in which reliability weights derived from edge/map detectors modulate the loss to focus on pixels or voxels near significant boundaries.

A representative composite formula is:

$$\mathrm{FLARE} = \lambda_1 \cdot \mathrm{FFL} + \lambda_2 \cdot \mathrm{LBR}$$

where:

• $\mathrm{FFL}$ is focal frequency loss (e.g., weighted DFT or wavelet difference as in (Jiang et al., 2020, Prantl et al., 2022)),
• $\mathrm{LBR}$ is local boundary reliability, typically of the form:

$$\mathrm{LBR} = \frac{1}{N} \sum_{x,y} R(x, y) \cdot \left\| I_{\text{real}}(x, y) - I_{\text{fake}}(x, y) \right\|^2$$

with $R(x, y)$ derived from local gradients, edge detectors, or specialized networks (Borse et al., 2021, Chen et al., 2022).

For classification problems under imbalance, FLARE Loss extends these ideas by balancing terms using class frequency and sample "influence", also incorporating metrics such as Brier Skill Score (BSS) for probabilistic reliability (Takagi et al., 12 Sep 2025). The most general form becomes:

$$L = \left( L'_{\mathrm{CE}} + L^{\mathrm{IB}}_{\mathrm{CE}} \right) + \lambda_{\mathrm{BSS}} \left( L'_{\mathrm{BSS}} + L^{\mathrm{IB}}_{\mathrm{BSS}} \right)$$

where $L^{\mathrm{IB}}_{\mathrm{CE}}$ and $L^{\mathrm{IB}}_{\mathrm{BSS}}$ penalize near-boundary samples based on their influence and class frequency.

3. Boundary and Frequency Synergy

FLARE Loss's dual focus addresses two common weaknesses in deep learning models:

• Spectral bias: Networks preferentially learn low-frequency components, often generating outputs that lack fine texture or edge fidelity (Jiang et al., 2020, Ko et al., 19 Aug 2025).
• Boundary ambiguity: Errors near boundaries are more impactful but harder to localize and penalize effectively (Borse et al., 2021, Chen et al., 2022).

By fusing frequency adaptivity (via dynamic spectral weighting, wavelet decomposition, or modulated basis functions) with boundary reliability (via edge maps, ambiguity masking, or spatial transformation networks), FLARE Loss ensures both global high-frequency detail and local spatial fidelity.

4. Practical Implementations Across Domains

Image Reconstruction/Synthesis

Use 2D DFT or wavelet transforms to compare frequency spectra and dynamically weight error terms by their difficulty (Jiang et al., 2020, Prantl et al., 2022). Complementary boundary-aware losses use edge detectors or transformation matrices to assign higher penalty near object contours (Borse et al., 2021).

Self-Supervised Depth Estimation

FLARE-inspired methods combine ambiguity masking to remove unreliable supervisory signals at image boundaries and apply frequency-adaptive gaussian blurring to soften harsh penalties in high-frequency regions. This improves depth edge sharpness and loss fairness (Chen et al., 2022).

Solar Flare Forecasting

FLARE Loss incorporates class-frequency-based and influence-based weighting in multi-class predictions, ensuring better detection of rare, critical events and preventing overconfident boundary decisions. Enhanced metrics such as the Gandin-Murphy-Gerrity score and TSS demonstrate improved reliability and performance (Takagi et al., 12 Sep 2025).

Federated Learning

FLARE mechanisms provide reliability against adversarial concept drift (Chow et al., 2023) and label-flipping attacks (Liu et al., 16 Jul 2025), via neuron-wise or output-layer analysis, combined with density-based clustering (HDBSCAN) to identify and filter malicious updates by their effect on model outputs in critical regions.

5. Evaluation Metrics and Empirical Impact

FLARE Loss has been shown to outperform traditional losses across several domains:

• Quantitative gains in PSNR, SSIM, FID, and Inception Score for image synthesis (Jiang et al., 2020).
• Increased mIoU and mean boundary accuracy in segmentation (Borse et al., 2021).
• Lower absolute and squared relative errors, improved edge accuracy in depth estimation (Chen et al., 2022).
• Superior composite skill scores (CSS), true skill statistics (TSS), and Brier Skill Scores in solar flare prediction (Takagi et al., 12 Sep 2025, Pandey et al., 21 Aug 2024).
• Enhanced safety and recall under threat models in federated autonomous systems (Liu et al., 16 Jul 2025).

6. Limitations and Implementation Challenges

Key challenges in deploying FLARE Loss include:

• Choosing and tuning the relative weighting hyperparameters $(\lambda_1, \lambda_2)$ so that neither component dominates (Jiang et al., 2020, Takagi et al., 12 Sep 2025).
• Ensuring computational efficiency and differentiability of local-boundary extraction methods, especially in real-time or resource-constrained settings (Borse et al., 2021).
• Stability concerns when integrating highly localized or noise-sensitive boundary terms, which may lead to training instabilities or overfitting (Chen et al., 2022).
• Calibration of class-frequency and influence-based weights, which may require additional tuning phases during training (Takagi et al., 12 Sep 2025).

7. Future Directions and Extensions

A plausible implication is that further research will refine the interplay between frequency attention and boundary reliability. Extensions may include learnable or task-adaptive boundary maps, integration with attention-based architectures, and direct incorporation into federated or edge learning protocols for robustness against dynamic drift or adversarial attacks (Chow et al., 2023, Liu et al., 16 Jul 2025). Advanced frequency-locality mechanisms, such as multi-scale wavelet encodings or modulated activation functions, are increasingly being used to overcome spectral bias and deliver finer-grained control over model outputs (Ko et al., 19 Aug 2025).

Summary Table: Representative FLARE Loss Applications

Task Domain	Frequency Component	Boundary Component
Image synthesis (Jiang et al., 2020)	Focal frequency weighting	Edge-based penalty
Segmentation (Borse et al., 2021)	Spectral similarity	Inverse-transform match
Depth estimation (Chen et al., 2022)	Adaptive blurring	Ambiguity masking
Solar flare forecasting (Takagi et al., 12 Sep 2025)	Weighted skill scores	Influence balance
Federated learning (Liu et al., 16 Jul 2025)	Output neuron analysis	Label distance/safety

FLARE Loss offers a principled framework for advancing model reliability and fidelity by integrating frequency-domain adaptivity with local-boundary awareness, tailored for complex domains where both global detail preservation and local error correction are essential.