Extreme Value Theory

EVT models the tail of the distance distribution by fitting a Weibull distribution to the largest distances between correct classifications and their class mean. This captures the statistical behavior of extreme values rather than the central tendency.

• Uses the Fisher-Tippett-Gnedenko theorem to justify tail modeling
• Fits only the top-k largest distances per class, ignoring the bulk of the distribution
• The Weibull parameters (scale, shape, location) define each class's rejection boundary

The OpenMax layer replaces standard SoftMax by using EVT-fitted Weibull models to recalibrate activation vectors. It estimates the probability that an input belongs to no known class.

• Computes Weibull CDF probability for each class's top-k distances
• Reduces known class scores proportionally to their tail probability
• Allocates the subtracted probability mass to a new unknown class pseudo-activation
• Enables a calibrated rejection threshold without retraining the base network

The choice of distance metric critically impacts EVT tail modeling. Euclidean distance is common, but statistically informed alternatives provide better separation.

• Mahalanobis distance accounts for class covariance structure, producing tighter in-distribution clusters
• Cosine distance focuses on angular separation, robust to magnitude variations in feature vectors
• Deep feature embedding quality directly determines EVT effectiveness—collapsed features invalidate tail assumptions

The number of extreme samples used for Weibull fitting—the tail size—is a critical hyperparameter. Too few samples yield high-variance estimates; too many violate the extreme value assumption.

• Typically set as a percentage of correctly classified training samples per class
• Cross-validated against a held-out set containing both known and unknown modulations
• Affects the trade-off between open set rejection rate and closed set accuracy

EVT-based open set recognition implements meta-recognition: the system reasons about its own decision confidence by analyzing the tail behavior of its internal representations.

• Quantifies epistemic uncertainty—what the model doesn't know due to limited exposure
• Provides a statistically principled threshold rather than an arbitrary confidence cutoff
• Generalizes beyond modulation classification to any deep learning task requiring novelty rejection

EVT methods assume the training data adequately represents the extreme behavior of known classes. Violations degrade rejection performance.

• Requires sufficient samples per class for reliable tail estimation—problematic in few-shot scenarios
• Assumes stationary feature distributions; distributional shift in deployment can invalidate fitted Weibull models
• The i.i.d. assumption of extreme values may not hold for temporally correlated signal streams

A deep learning layer that replaces the standard SoftMax function by recalibrating activation vectors using Extreme Value Theory. It fits a Weibull distribution to the distances between correct classifications and their class means, enabling the model to estimate the probability that an input belongs to an unknown class rather than forcing it into a known category.

The task of identifying input samples that differ significantly from the training data distribution. EVT models the tail behavior of in-distribution scores to set statistically principled rejection thresholds. Key approaches include:

• Density-based methods using class-conditional Gaussians
• Distance-based methods in embedding spaces
• EVT calibration of SoftMax confidence scores

The theoretical risk of labeling an unknown input as a known class. Quantified as the relative measure of the feature space far from any known training data that is nonetheless classified as known. EVT directly addresses this by characterizing the tails of the distance distribution, allowing the system to bound the probability of misclassifying a novel modulation scheme as a familiar one.

The technique of aligning a model's predicted probability with the actual likelihood of correctness. EVT-based calibration methods like temperature scaling adjust the output logits to ensure that low confidence scores reliably indicate unknown inputs. A well-calibrated model produces confidence scores that match empirical accuracy: a 70% confidence prediction should be correct 70% of the time.

The model's uncertainty arising from a lack of knowledge or data, which is reducible with more training samples. EVT helps separate epistemic uncertainty from aleatoric uncertainty (inherent noise) by modeling the distribution of prediction confidence. High epistemic uncertainty signals that an input falls in a sparsely sampled region of the feature space, making it a strong indicator for novelty detection.

A distance metric that accounts for the covariance structure of a class distribution, providing a more statistically informed measure for out-of-distribution detection than Euclidean distance. When combined with EVT, the Mahalanobis distance to the nearest class centroid is fitted to a Generalized Pareto Distribution, enabling rigorous statistical testing for whether a signal belongs to any known modulation scheme.