Deep SAD

Unlike unsupervised Deep SVDD, Deep SAD leverages labeled anomalies to explicitly repel known outliers from the center c. The objective function combines two terms: minimizing the distance of normal samples to c (attraction) while maximizing the distance of labeled anomalies beyond a margin (repulsion). This dual-force mechanism creates a tighter, more discriminative boundary that reduces open space risk in the feature embedding.

The Deep SAD loss function is a weighted combination of an unsupervised SVDD loss for unlabeled data and a supervised anomaly repulsion loss for labeled outliers. The hyperparameter η (eta) controls the influence of labeled anomalies, balancing the model's sensitivity to known attack patterns against its ability to generalize to unseen anomaly types. This formulation prevents the model from overfitting to the few labeled examples while still benefiting from their guidance.

Anomaly scores are computed as the Euclidean distance from a sample's learned representation φ(x; W) to the hypersphere center c. Samples mapping inside the radius r are considered normal, while those outside are flagged as anomalous. The score is continuous, allowing operators to set application-specific thresholds for emitter rejection. This contrasts with SoftMax-based open set methods by providing a direct geometric interpretation of novelty.

Deep SAD is particularly effective in few-shot anomaly scenarios where only a handful of confirmed rogue emitter signatures are available. Even 1-5 labeled anomalies can significantly improve AUROC over purely unsupervised Deep SVDD. This makes it ideal for RF fingerprinting applications where collecting large datasets of adversary transmissions is impractical, but a small library of known threats exists from prior spectrum surveillance operations.

To avoid hypersphere collapse—where the network trivially maps all inputs to a constant—Deep SAD networks are pretrained as autoencoders. The encoder weights are then transferred to the Deep SAD model, and the center c is initialized as the mean of the encoded representations of the training data. This ensures the network learns a meaningful, non-degenerate feature embedding before the anomaly detection fine-tuning phase begins.

Deep SAD demonstrates superior robustness when the unlabeled training set contains unknown contamination—normal data inadvertently mixed with unidentified anomalies. The explicit repulsion signal from the few labeled anomalies acts as a regularizer, preventing the model from learning a hypersphere that inadvertently encloses hidden outliers. This property is critical for real-world emitter recognition where perfectly clean training sets are rare.

The direct unsupervised predecessor to Deep SAD. Deep SVDD trains a neural network to map all normal data into a minimal-volume hypersphere centered at a fixed point. Anomalies are identified as points falling outside this boundary. Deep SAD extends this by incorporating labeled anomalies to explicitly push them away from the center, creating a more discriminative boundary.

A classical kernel-based method that learns a decision boundary tightly enveloping the normal training data in a high-dimensional feature space. Unlike Deep SAD's neural network approach, One-Class SVM relies on the kernel trick to separate inliers from outliers. It serves as the traditional baseline that deep one-class methods aim to outperform on complex, high-dimensional data like images and RF signals.

A self-supervised representation learning framework that Deep SAD implicitly leverages. Contrastive methods pull positive pairs (semantically similar samples) together and push negative pairs apart in embedding space. Deep SAD's loss function performs a similar operation: it attracts normal samples to the hypersphere center while repelling known anomalies, creating a well-separated feature space for open set rejection.

The broader task of identifying inputs that differ fundamentally from the training distribution. Deep SAD is one architectural approach to this problem, using the distance to the hypersphere center as an anomaly score. Other OOD methods include:

• Energy-Based Models: Assign low energy to in-distribution data
• Mahalanobis Distance: Measure distance accounting for feature covariance
• Monte Carlo Dropout: Estimate epistemic uncertainty through stochastic inference

A formal concept quantifying the danger of labeling unknown samples as known classes. Deep SAD directly minimizes open space risk by defining a compact, closed boundary around normal data. The hypersphere radius creates a finite volume of acceptance; anything outside is rejected. This contrasts with SoftMax classifiers that partition the entire feature space, leaving unbounded regions where unknowns can be confidently misclassified.

The low-dimensional vector representation learned by the neural network backbone of Deep SAD. The quality of this embedding is critical: normal samples must cluster tightly around the center, while anomalies must be pushed to the periphery. Techniques like angular margin losses (ArcFace, CosFace) can be combined with Deep SAD to enforce maximum inter-class separation in this learned space.