OpenMax

OpenMax recalibrates the penultimate layer's activation vector (the logits) by fitting a Weibull distribution to the distances between correctly classified training samples and their class mean activation vectors. This statistical model captures the tail behavior of each known class, establishing a robust boundary for inclusion. During inference, the distance of a query sample to each class mean is evaluated against the fitted Weibull CDF to compute a recalibrated score, effectively shrinking the open space risk for each known class.

Unlike standard SoftMax, which forces a closed-world assumption by normalizing probabilities to sum to 1, OpenMax introduces an explicit unknown class index. The algorithm computes the probability of an input belonging to this unknown class based on the cumulative rejection scores from the Weibull models. If the recalibrated activation vector indicates the sample is far from all known class distributions, mass is shifted to this unknown index, allowing the model to output a calibrated probability that the emitter is novel or adversarial.

OpenMax employs a meta-recognition stage to determine the final classification. It sorts the top-k predicted classes and applies a sequential thresholding process using the Weibull CDF. If the probability of the top class falls below a calibrated rejection threshold, the sample is labeled as unknown. This mechanism avoids the brittle, arbitrary threshold selection common in simple anomaly detection by grounding the rejection logic in the statistical properties of the training data's tail distribution.

The algorithm operates on the Mean Activation Vector (MAV) for each known class, computed from the correctly classified training samples. For each class, a Weibull model is fit to the tail of the Euclidean distances between the MAV and the samples. This distance metric learning is crucial: it assumes that in a well-trained deep network, the activation space is semantically meaningful, and proximity to a class centroid correlates strongly with class membership, making it effective for open set emitter recognition.

OpenMax is designed as a lightweight post-processing layer that replaces the final SoftMax operation. It does not require retraining the backbone neural network or adding auxiliary output heads. The computational overhead involves only a few distance calculations and Weibull CDF lookups per inference, making it highly suitable for edge AI signal identification on SDRs and FPGAs where latency and power budgets are constrained. The Weibull parameters are pre-computed offline during a calibration phase.

The theoretical rigor of OpenMax comes from Extreme Value Theory (EVT), which models the probability of rare, extreme events. By fitting a Weibull distribution to the tail of the distance distribution for each class, OpenMax provides a principled statistical basis for rejecting outliers. This contrasts with heuristic methods that use raw SoftMax confidence scores, which are known to be poorly calibrated and often produce high probabilities for out-of-distribution inputs, making EVT essential for reliable open set emitter recognition.

The statistical foundation of OpenMax. EVT models the distribution of maxima in data, specifically the tail behavior where extreme events occur. OpenMax uses EVT to fit a Weibull distribution to the distances between a sample's activation vector and the mean activation vector of each known class. This calibration models the probability of a sample belonging to the tail of a class distribution, enabling a principled estimate of the likelihood that an input is an unknown class rather than a poorly matched known class.

A standard SoftMax layer forces a closed-world assumption by normalizing all logits into a probability distribution that sums to 1.0, guaranteeing a prediction even for nonsense inputs. OpenMax replaces this by:

• Fitting a Weibull model per class using EVT on the penultimate layer's activation vectors.
• Recalibrating the logits based on the probability of being an outlier.
• Introducing an explicit unknown class score. The output is a probability distribution over K+1 classes, where the +1 represents the rejection of all known categories.

The core problem OpenMax solves. Open Space Risk is the probability of labeling an unknown sample as a known class, quantified by the volume of feature space far from any training data that is nonetheless classified. OpenMax explicitly bounds this risk by using EVT to define a probability of inclusion for each class. If a sample falls in the tail beyond a calibrated threshold, its probability mass is shifted to the unknown class, effectively shrinking the decision boundary around the known training data.

The specific EVT technique used by OpenMax. For each known class, the algorithm computes the distances between all correctly classified training samples and their class mean activation vector. A Weibull distribution is fit to the tail of these distances. At inference, the distance of a new sample to each class mean is evaluated against the fitted Weibull CDF. A low CDF value indicates the sample is an outlier relative to that class, triggering a recalibration of the logit that reduces the false confidence of the SoftMax output.

OpenMax operates on the penultimate layer of a neural network, known as the embedding or activation vector space. This layer represents the high-level features extracted from the input before final classification. The algorithm relies on the assumption that in a well-trained network, samples from the same class cluster tightly in this space. The Mean Activation Vector (MAV) for each class serves as the prototype, and the distance to this MAV is the signal used for EVT calibration and outlier detection.

The broader classification paradigm that OpenMax implements. OSR requires a model to simultaneously:

• Accurately classify inputs belonging to known classes.
• Detect and reject inputs from unknown classes not seen during training. This contrasts with closed-set classification, which assumes all test classes were present in training. OSR is critical for deploying models in dynamic, real-world environments like spectrum surveillance and cognitive radio, where new emitters constantly appear.