Angular Margin Loss

The foundational angular margin loss that introduced a multiplicative angular margin to the standard SoftMax formulation. By projecting features onto a hypersphere and multiplying the angle by a margin parameter m, SphereFace forces the decision boundary to be more stringent, requiring features to be closer to their class center. This formulation directly optimizes for angular separability rather than Euclidean distance, making learned features more discriminative for open-set tasks. The key limitation is training instability at high margin values due to the non-monotonic nature of the multiplicative margin, which can cause the loss surface to become difficult to optimize without careful annealing strategies.

Reformulates the angular margin as an additive cosine margin rather than a multiplicative one, dramatically improving training stability. CosFace normalizes both feature vectors and weight vectors, then subtracts a margin m directly from the cosine of the angle before scaling with a temperature parameter s. This additive approach creates a linear decision margin in cosine space that is easier to optimize. Key properties include:

• Feature normalization ensures all samples lie on a hypersphere of radius s
• Additive margin avoids the vanishing gradient issues of multiplicative approaches
• Scale parameter s controls the radius of the hypersphere, typically set between 30 and 64
• Directly applicable to open-set emitter recognition where unknown transmitters must fall below a cosine similarity threshold

The most widely adopted angular margin loss that applies an additive margin directly to the angle rather than the cosine. ArcFace computes the arccosine of the logit, adds a constant margin m to the target angle, then takes the cosine. This yields a constant linear angular margin throughout the entire angle range, providing exact geodesic distance constraints on the hypersphere. Advantages for RF fingerprinting include:

• Exact angular interpretation: The margin directly corresponds to a minimum angular separation in radians
• Superior convergence: The linear angular penalty produces cleaner gradient signals
• Robustness to label noise: The constant margin is less sensitive to mislabeled training samples
• Widely validated on large-scale face recognition benchmarks with millions of identities

An automated hyperparameter-free variant that dynamically adjusts the scale parameter s during training based on the observed distribution of cosine similarities. AdaCos eliminates the need for manual tuning of the scale and margin parameters, which is critical when deploying angular margin loss across diverse emitter recognition tasks with varying numbers of device classes. The method analyzes the expected and observed probability distributions of the SoftMax output to compute an optimal scale at each training step. This self-calibrating property makes it particularly suitable for few-shot device enrollment scenarios where hyperparameter sweeps are impractical due to limited training data per emitter.

Extends angular margin loss by explicitly emphasizing hard negative classes during training. MV-SoftMax identifies the non-target class with the highest cosine similarity to a sample and applies an additional penalty to push the feature vector away from that specific class center. This targeted repulsion mechanism is valuable for open-set emitter recognition where certain device classes may have nearly identical hardware impairments, creating confusable class pairs. By focusing the margin on the most challenging negative class rather than applying a uniform margin, MV-SoftMax achieves finer separation between similar transmitters without unnecessarily distorting the embedding space for well-separated classes.

Introduces a curriculum learning strategy to angular margin training, where the margin emphasis shifts from easy positive samples early in training to hard negative samples later. In early epochs, CurricularFace behaves like standard CosFace to establish basic class structure. As training progresses, it adaptively increases the importance of difficult negative samples, mimicking the effect of hard negative mining. This prevents the network from collapsing in early training due to overly aggressive margins while still achieving tight class boundaries at convergence. For RF fingerprinting, this approach handles the wide variance in signal quality across different collection environments by gradually increasing discriminative pressure.

Angular margin losses rely on L2 normalization of both feature embeddings and classifier weight vectors, projecting them onto a unit hypersphere. This eliminates radial variance, forcing the network to discriminate solely based on angular proximity. The radius of the hypersphere is controlled by a learnable scale parameter s, which rescales the cosine logits to facilitate convergence. Normalization is critical for open set rejection, as unknown samples naturally fall into low-density angular regions.

Angular margin losses are inherently suited for open set emitter recognition because they enforce compact, well-separated class clusters in angular space. Unknown transmitters produce embeddings that fall into the angular gaps between known class prototypes. Rejection is performed by thresholding the maximum cosine similarity against all known class anchors. This geometric property provides a principled basis for distinguishing known from unknown devices without requiring explicit outlier exposure training.

Advanced angular margin formulations that adapt the margin magnitude based on feature quality or sample difficulty. MagFace links the margin to the embedding norm, encouraging high-quality samples to have larger norms and tighter angular constraints. AdaFace dynamically adjusts the margin based on image quality or signal-to-noise ratio, applying stronger margins to clean samples and relaxing constraints for degraded inputs. These adaptive approaches improve robustness in variable RF channel conditions.