Objectosphere Loss

The loss function operates on two distinct fronts. For known classes, it applies a standard cross-entropy loss combined with a penalty that pushes the L2-norm of the feature vector toward a large, pre-defined minimum value. For unknown samples, a separate term minimizes the feature norm, driving it toward zero. This creates a magnitude gap in the embedding space, where known signals produce high-energy representations and unknown signals collapse to the origin.

By structuring the feature space around magnitude, Objectosphere loss enables a simple, computationally cheap rejection rule. A single distance threshold on the feature norm is established post-training. Any input whose feature vector magnitude falls below this threshold is rejected as unknown. This avoids the need for complex density estimation or recalibration layers like OpenMax, making it ideal for real-time spectrum monitoring on edge hardware.

A common failure mode in open-set recognition is feature collapse, where the embeddings of all inputs—both known and unknown—map to a confined region. Objectosphere loss directly combats this. The explicit repulsion of unknown features from the known class manifolds ensures that the network does not learn a degenerate solution. The result is a well-separated embedding where known classes form distinct, high-magnitude clusters far from the low-magnitude unknown region.

Objectosphere loss is often compared to Entropic Open-Set loss, which forces unknown samples to have high-entropy (uniform) SoftMax outputs. Objectosphere is a complementary approach that operates directly on the feature layer rather than the logit layer. While Entropic loss ensures uncertain classification probabilities, Objectosphere ensures a measurable geometric separation. The two can be combined to create a robust system that rejects unknowns using both probability distribution and feature magnitude cues.

To effectively minimize the feature norm for unknowns, the model must be exposed to representative outlier data during training. This auxiliary dataset does not need to be exhaustive of all possible unknown modulations but should be diverse enough to teach the network the concept of 'otherness.' The loss function leverages this exposure to carve out a hollow sphere in the embedding space: a central void where all non-target signal types are projected, surrounded by a high-magnitude shell of known classes.

The theoretical foundation of Objectosphere loss is the minimization of open space risk—the risk of labeling an unknown input as known. By bounding the known classes to a high-magnitude hypersphere shell and confining unknowns to the interior, the function mathematically limits the volume of feature space that is both far from known data and classified as known. This provides a more formal guarantee of rejection capability compared to purely probabilistic methods.