Closed-Set Assumption

The closed-set assumption enforces a static world hypothesis, where the set of possible classes is finite and fully enumerated at training time. The model's label space Y_train is mathematically identical to Y_test. This means the classifier is structurally incapable of expressing 'I don't know'—every input, including noise or a novel modulation scheme like a never-before-seen OFDM variant, is forcibly mapped to one of the known classes with high confidence. This is a direct violation of the requirements for cognitive radio systems operating in unlicensed or contested spectrum.

The standard SoftMax activation function is the primary architectural enforcer of the closed-set assumption. It normalizes output logits into a probability distribution that always sums to 1.0. For an out-of-distribution input, the SoftMax layer will still produce a high-confidence prediction by saturating one of its output nodes. This creates a dangerous false sense of certainty, where a completely unknown signal type is classified as a known modulation with >95% probability, providing no mechanism for novelty detection.

A direct consequence of the closed-set assumption is the maximization of open space risk—the theoretical risk of labeling an unknown input as a known class. In the high-dimensional feature space, the decision boundaries of a closed-set classifier extend infinitely outward, carving up the entire space among the known classes. Any novel modulation that falls far from the training data but within a known class's unbounded region will be confidently misclassified. This is quantified as the relative measure of feature space far from any training data that is nonetheless assigned a label.

Closed-set classifiers exhibit a silent failure mode under distributional shift. When the deployment environment changes—such as a drop in signal-to-noise ratio or the introduction of a new waveform—the model continues to produce predictions with the same confidence calibration as clean in-distribution data. There is no built-in anomaly signal. This is particularly dangerous in electronic warfare and spectrum monitoring applications, where a novel adversary signal must trigger an alert, not a confident misclassification as a friendly or civilian emission.

The closed-set assumption is fundamentally incompatible with open world learning paradigms. In a real-world spectrum environment, new modulation types appear continuously. A closed-set model cannot incrementally learn to recognize a newly discovered class without full retraining on a dataset that includes both old and new classes. This leads to catastrophic forgetting if fine-tuned naively, or requires prohibitively expensive full retraining cycles, making the system brittle and non-adaptive in the face of evolving signal landscapes.

Deep neural networks trained under the closed-set assumption are prone to feature collapse, where the learned embedding space maps all inputs—including unknowns—to a restricted, compact region. This occurs because the training objective only optimizes for separating known classes, not for reserving space for unknowns. As a result, the embeddings of novel modulation types become indistinguishable from known classes in the latent space, destroying any post-hoc opportunity to apply distance-based out-of-distribution detection methods like Mahalanobis distance scoring.

The direct antithesis of the closed-set assumption. In open set recognition, the model must accurately identify known modulation classes while simultaneously detecting and rejecting samples from unknown or novel classes not seen during training. This paradigm acknowledges that the test-time label space is larger than the training label space, making it essential for real-world spectrum monitoring where new waveforms appear regularly.

The technical task of identifying input samples that differ significantly from the training data distribution. When the closed-set assumption holds, OOD detection is unnecessary. When it fails, OOD detectors serve as a safety mechanism by flagging unfamiliar signals for rejection. Key approaches include:

• SoftMax thresholding: rejecting low-confidence predictions
• Energy-based scoring: using Helmholtz free energy as a discriminative score
• ODIN: applying temperature scaling and small perturbations to amplify separation

The theoretical risk formalized by Scheirer et al. that quantifies the danger of the closed-set assumption. It measures the relative proportion of the feature space that is far from any known training data yet is still labeled as a known class by the classifier. A high open space risk means the model is dangerously overconfident in regions where it has no evidential support, making it vulnerable to silently misclassifying unknown modulations.

A catastrophic failure mode in deep learning classifiers trained under the closed-set assumption. When exposed to unknown modulation types, the network may map their embeddings to the same compact region of feature space as known classes, rather than placing them in distinct, separable locations. This destroys the model's ability to separate known from novel classes and produces confidently wrong predictions with no uncertainty signal.

A continuous learning paradigm that extends beyond simply rejecting unknowns. In open world learning, the model must:

• Detect unknown modulation classes at test time
• Incrementally learn to recognize them when labeled data becomes available
• Do so without catastrophic forgetting of previously learned classes This represents the complete solution to the closed-set assumption's limitations in evolving spectrum environments.

A change in the statistical properties of input data between training and deployment that silently violates the closed-set assumption. Even if the modulation types remain identical, a shift in signal-to-noise ratio, channel conditions, or hardware impairments can cause the model to encounter data outside its training distribution. Without explicit OOD detection, closed-set classifiers fail silently, producing confident but incorrect predictions.