Reciprocal Point Learning

Each known class is represented by a learned reciprocal point in the embedding space, positioned such that known samples are constrained to lie within a bounded region opposite to this point. The classifier measures the maximum distance between a query sample's embedding and all reciprocal points. Known classes produce large distances to their corresponding reciprocal point, while unknown samples fall closer to multiple reciprocal points, enabling rejection.

• Contrast with prototype learning: Prototypes attract embeddings; reciprocal points repel them
• Open space risk minimization: The reciprocal constraint creates a compact, bounded region for each known class
• Distance metric: Typically uses Euclidean or cosine distance in the learned embedding space

The training objective combines a discriminative loss for known class separation with a reciprocal constraint that bounds the embedding space. The loss enforces that each known sample's embedding has a large distance to its class reciprocal point while maintaining a margin from other class reciprocal points. This dual objective creates a structured embedding geometry where unknown samples naturally fall into the inter-class open space.

• Margin enforcement: A hyperparameter controls the minimum distance between known embeddings and their reciprocal point
• Inter-class separation: Additional terms ensure reciprocal points of different classes are well-separated
• Gradient behavior: The reciprocal constraint provides stable gradients that prevent feature collapse

At inference time, the model computes the distance from the query embedding to every reciprocal point. The predicted class is the one with the maximum distance to its reciprocal point. If this maximum distance falls below a calibrated threshold, the sample is rejected as unknown. This threshold can be tuned using a validation set of known and auxiliary outlier samples.

• Threshold calibration: Set using the distribution of maximum distances on known validation samples
• Score function: The raw rejection score is the maximum distance across all reciprocal points
• AUROC optimization: The threshold can be selected to maximize the area under the ROC curve for novelty detection

Unlike OpenMax, which recalibrates SoftMax probabilities using Extreme Value Theory on class-mean distances, RPL fundamentally restructures the embedding geometry. OpenMax operates as a post-hoc layer on a conventionally trained network, while RPL's reciprocal constraint is baked into the training objective, producing a more principled open space.

• Training integration: RPL modifies the feature learning process; OpenMax is applied after training
• Weibull fitting: OpenMax requires fitting a Weibull distribution per class; RPL uses a simpler distance threshold
• Scalability: RPL scales more gracefully to large numbers of known classes due to its geometric rather than statistical approach

A critical advantage of RPL is its inherent resistance to feature collapse, where all embeddings map to a restricted region. Because the reciprocal constraint explicitly pushes known embeddings away from their reciprocal points, the network cannot collapse all representations to a single cluster. The repulsive geometry maintains an expansive embedding space where unknown samples have room to fall outside known class boundaries.

• Collapse resistance: The reciprocal loss term provides a repulsive force that prevents embedding contraction
• Open space preservation: The learned geometry naturally reserves space for unknown classes
• Training stability: The reciprocal constraint acts as a regularizer that improves convergence

In automatic modulation classification, RPL addresses the open set challenge where new modulation schemes appear in dynamic spectrum environments. Known modulations like QPSK, 16-QAM, and 64-QAM are each assigned a reciprocal point. When a novel modulation such as 256-QAM or a jammer waveform appears, its embedding falls closer to multiple reciprocal points, triggering rejection.

• Dynamic spectrum adaptation: Enables cognitive radios to flag unfamiliar waveforms for human analysis
• Electronic warfare applications: Detects adversarial signals using unknown modulation schemes
• Benchmark performance: RPL has demonstrated superior open set classification rates compared to SoftMax baselines on RadioML datasets