Extreme Value Theory (EVT)

The foundational theorem of EVT states that the distribution of block maxima converges to one of three limiting forms: the Gumbel, Fréchet, or Weibull distributions. These are unified under the Generalized Extreme Value (GEV) distribution. In machine learning, this theorem justifies using EVT to model the tail of a classifier's activation or distance scores, providing a theoretically sound basis for setting rejection thresholds in open set recognition.

Rather than modeling only block maxima, the Peaks-Over-Threshold (POT) approach models all observations that exceed a sufficiently high threshold u. The Balkema-de Haan-Pickands theorem proves that these exceedances follow a Generalized Pareto Distribution (GPD). In novelty detection, this method is used to fit a GPD to the tail of a model's reconstruction error or negative log-likelihood scores, enabling precise calibration of the probability that a new sample is an outlier.

The OpenMax algorithm replaces the standard SoftMax layer with a calibrated rejection mechanism. For each known class, a Weibull distribution is fit to the tail of the top activation scores from correctly classified training samples. At inference, the algorithm computes a recalibrated probability that an input belongs to an unknown class by evaluating how extreme its activation is relative to these per-class tail models. This directly applies EVT to reject adversarial or novel emitter signatures.

Selecting the threshold u for the POT method is critical. The Mean Excess Function (MEF) plots the average excess over a threshold against the threshold value. A linear upward trend in the MEF indicates a heavy-tailed distribution suitable for GPD fitting. In deep learning signal identification, the MEF is used to empirically determine the optimal score boundary where a model's confidence transitions from representing known emitters to unknown or anomalous ones.

The tail index (ξ) is the shape parameter of the GEV and GPD that determines the heaviness of a distribution's tail. A positive ξ indicates a heavy-tailed Fréchet domain, common in deep neural network activation landscapes. Estimating ξ using the Hill estimator or maximum likelihood methods allows practitioners to quantify the risk of extreme events. A larger tail index signals a higher probability of encountering highly anomalous, high-magnitude outlier scores during inference.

Meta-recognition applies EVT to the output of a recognition system itself to predict its own failure. By fitting a Weibull or GPD to the distribution of a model's confidence scores, the system can determine if a given prediction is statistically extreme relative to its training experience. This provides a principled, distribution-aware alternative to arbitrary confidence thresholds, enabling a model to say 'I don't know' when encountering a signal from a transmitter class never seen during training.

A classification paradigm that goes beyond closed-world assumptions by explicitly rejecting unknown emitter classes not seen during training. Unlike standard SoftMax classifiers that force a prediction, open set systems use EVT-calibrated thresholds to declare a signal as 'unknown.'

• Key mechanism: Replaces the standard SoftMax layer with a rejection model
• EVT role: Models the tail of the distribution of known-class activation scores
• Application: Spectrum surveillance where new transmitters appear regularly

The process of mapping high-dimensional IQ samples or spectrograms into a compact, discriminative vector space where device identity is the primary organizing factor. EVT operates on these embeddings to define the boundary of known classes.

• Goal: Maximize inter-class distance while minimizing intra-class variance
• Training: Often uses triplet loss or contrastive learning objectives
• EVT integration: The distance from a sample to its class centroid in embedding space feeds the extreme value model

A continuous probability distribution central to EVT-based novelty detection. The Weibull distribution models the tail behavior of distances or activation scores, providing a principled statistical framework for setting rejection thresholds.

• Parameters: Shape (k) and scale (λ) control tail heaviness
• Fit process: Applied to the largest distances from class means in the training set
• Interpretation: A low Weibull CDF score indicates a sample is far from any known distribution

A self-supervised training paradigm that teaches a network to produce robust feature embeddings by pulling similar signal samples together and pushing dissimilar ones apart. This creates the well-separated clusters that make EVT-based rejection boundaries effective.

• Positive pairs: Different augmented views of the same transmitter's signal
• Negative pairs: Signals from different devices or noise
• Benefit: Learns discriminative features without requiring exhaustive class labels

A twin-branch neural architecture designed to learn similarity functions between pairs of signal samples. By comparing a query signal against enrolled device templates, Siamese networks generate similarity scores that EVT can model for verification.

• Architecture: Two identical subnetworks with shared weights
• Output: A distance metric indicating whether two signals originated from the same device
• EVT synergy: The distribution of genuine-pair distances is modeled to set verification thresholds