OpenMax

OpenMax fits a Weibull distribution to the distance between a sample's activation vector and each known class's mean activation vector. This models the probability of extreme values, providing a statistically rigorous threshold for rejecting unknown inputs. Unlike simple confidence scoring, this calibration accounts for the tail behavior of each class's feature space, preventing overconfident misclassifications of rogue emitters.

The algorithm computes a meta-recognition score by adjusting the penultimate layer's activation vector. For each known class, OpenMax reweights the logit based on the Weibull CDF probability that the sample belongs to the class's tail. A synthetic 'unknown' logit is created from the residual probability mass. This recalibrated vector is then passed through SoftMax, producing a probability distribution that explicitly includes an unknown class.

Traditional SoftMax forces classification into one of K known classes, achieving high closed-set accuracy but catastrophically failing on unknowns. OpenMax maintains comparable closed-set performance while adding robust open-set capability:

• Closed-set: Accuracy within 1-2% of standard SoftMax
• Open-set: Dramatically reduces false positive rate for unknown emitters
• AUROC: Significantly higher area under the ROC curve for novelty detection

OpenMax relies on the Mean Activation Vector (MAV) for each known class, computed from correctly classified training samples. At inference, the algorithm calculates the distance between the sample's activation vector and the top-ranked class's MAV. Samples far from all known MAVs trigger the rejection mechanism. This distance metric is typically Euclidean or cosine similarity, operating in the semantically rich latent space learned by the network.

The tailsize hyperparameter controls how many extreme distance values are used to fit the Weibull distribution per class. A smaller tailsize captures only the most extreme outliers, creating a tighter rejection boundary. A larger tailsize includes more data, smoothing the distribution. Tuning this parameter balances:

• Precision: Correctly rejecting true unknowns
• Recall: Correctly accepting known classes
• F1-Open: A specialized metric combining both for open-set evaluation

OpenMax is architecture-agnostic and operates as a drop-in replacement for the final classification layer. It can be integrated with any deep feature extractor, including:

• CNNs trained on spectrograms or raw IQ data
• Transformer encoders processing sequential signal embeddings
• Siamese networks generating discriminative feature vectors The only requirement is access to the penultimate activation layer and a labeled training set for MAV computation and Weibull fitting.

OpenMax replaces the traditional SoftMax layer with a calibrated rejection mechanism based on Extreme Value Theory (EVT). Instead of forcing a classification among known emitters, it estimates the probability that an input belongs to an unknown class. This is critical for spectrum surveillance where new devices constantly appear.

• Fits a Weibull distribution to the distance of correct classifications from their class mean
• Computes a recalibrated score vector that includes an explicit unknown class
• Rejects inputs when the unknown probability exceeds any known class probability

The core innovation of OpenMax is meta-recognition—the algorithm's ability to assess its own certainty. By analyzing the activation vector of the penultimate layer, OpenMax models the tail behavior of each known class. If an activation pattern falls in the extreme tail of all known distributions, it is flagged as novel.

• Uses EVT to model the distribution of top activation scores per class
• Determines an adaptive threshold for each class based on tail probability
• Provides a statistically grounded confidence score rather than an arbitrary cutoff

In cognitive radio and spectrum enforcement, OpenMax enables receivers to autonomously flag unauthorized or anomalous transmitters. When a new device enters the environment, the model does not misclassify it as a known friendly emitter—it correctly identifies it as unknown.

• Prevents spoofing attacks where adversaries mimic known device fingerprints
• Enables dynamic spectrum access by identifying rogue interferers
• Reduces false positives in electronic warfare threat libraries

OpenMax relies on fitting a Weibull distribution to the distances between correctly classified training samples and their class mean activation vectors. The Weibull is chosen because it effectively models extreme value behavior in bounded data.

• Computes per-class shape and scale parameters from training data
• Uses the fitted distribution to estimate the probability of inclusion for new inputs
• Recalibrates SoftMax scores by weighting them with the Weibull CDF

OpenMax is architecture-agnostic and can be appended to any deep neural network that produces a feature embedding before classification. In RF fingerprinting, this means it works with CNNs, ResNets, and Transformer backbones trained on IQ data or spectrograms.

• Attaches to the penultimate layer of a pre-trained classifier
• Requires no architectural changes to the backbone network
• Compatible with transfer learning and domain adaptation pipelines

Unlike generic anomaly detection methods such as One-Class SVM or Isolation Forest, OpenMax is tightly integrated with the discriminative features learned by a neural network. It leverages the semantic structure of the learned latent space rather than operating on raw signal statistics.

• Preserves the hierarchical feature representations learned by deep models
• Provides class-conditional rejection rather than binary novelty detection
• Scales efficiently with the number of known emitter classes

The statistical backbone of OpenMax that models the tail-end distribution of activation vectors. Instead of assuming a Gaussian distribution, EVT fits a Weibull distribution to the largest distances between correctly classified samples and their class mean activation vectors. This calibrated model determines the probability that a new input belongs to an unknown class, providing a mathematically rigorous rejection threshold rather than an arbitrary confidence cutoff.

A standard SoftMax layer forces a closed-world assumption by normalizing all outputs to sum to 1.0, guaranteeing a prediction even for nonsensical inputs. OpenMax replaces this with a recalibrated activation mechanism:

• Computes distances from each class's mean activation vector
• Uses EVT to estimate the probability of an input being "out-of-distribution"
• Adds an explicit unknown class dimension to the output vector
• Redistributes probability mass away from known classes when rejection is warranted

The broader classification paradigm that OpenMax implements. Unlike traditional closed-set classifiers that assume all test classes appear during training, OSR systems must:

• Correctly classify known emitter types
• Detect and reject unknown or rogue transmitters
• Operate in dynamic electromagnetic environments where new devices constantly appear This is critical for spectrum surveillance and cognitive radio applications where encountering previously unseen emitters is the norm, not the exception.

OpenMax operates on the penultimate layer of a neural network, where high-dimensional signal data has been compressed into a discriminative feature embedding. The algorithm analyzes the geometry of this latent space:

• Computes mean activation vectors for each known class
• Measures distances between activations and class centroids
• Fits EVT models to the distribution of these distances A well-structured embedding space where classes form tight, separable clusters is essential for reliable open set rejection.

The core mathematical operation in OpenMax involves fitting a Weibull distribution to the tail of the distance distribution for each known class. This process:

• Takes the η largest distances between correctly classified samples and their class mean
• Estimates Weibull scale (λ) and shape (κ) parameters
• Uses these parameters to compute the probability that a new input's distance belongs to the tail The result is a calibrated rejection score that adapts to the natural variance within each class rather than using a global threshold.

While both address unknown inputs, they differ fundamentally:

• Novelty detection is a binary decision: known or unknown
• OpenMax provides a multi-class output that includes both specific known-class probabilities and an explicit unknown probability This makes OpenMax more informative for spectrum monitoring, where operators need to know not just that something is new, but also the confidence that existing friendly emitters have been correctly identified alongside the rejection of unknowns.