Title: Moving Past the Simplex: Equilibrium Prototype Structures for Scorer-Independent Open-Set Recognition

Original: arXiv:2606.01883v1 Announce Type: new Abstract: Open-set recognition (OSR) requires a classifier to reject inputs from unseen classes which is essential in safety-critical settings such as medical imaging. Simplex based methods, which fix class prototypes at the vertices of a regular simplex and then reject via a distance-ratio score, perform well empirically but lack theoretical justification, and existing analysis applies only when the embedding dimension d is at least C-1, which is the regime in which a regular simplex exists. We give a theoretical account of simplex-ratio OSR that holds in every embedding dimension, including d = 2 and include the regular simplex as a special case. For these codes we show that an auxiliary squared ratio score has sublevel sets that are exact unions of Euclidean balls, which in turn bracket the acceptance region of the operational score; and we prove a sharp dichotomy: the prototypes attain one-distance symmetry, behaving like a regular simplex, if and only if d >= C-1, with controlled degradation governed by an explicit defect parameter below that threshold. We further show the false-acceptance rate decays exponentially in d under natural isotropy assumptions, and that the operational score is globally Lipschitz with compact acceptance regions. Empirically, we study balanced prototype geometry as both an analytic tool and a representation-learning prior, rather than as a stand-alone state-of-the-art detector. Across CIFAR and MedMNIST open-set splits, the geometry provides useful structure, but OSR performance remains strongly dependent on the scoring rule: raw ratio scores typically underperform nearest-neighbor and logit-based alternatives.

Rewrite: Open-set recognition (OSR) is a critical capability for classifiers operating in high-stakes environments like medical diagnostics, as it enables the system to identify and reject samples belonging to unknown classes. While methods utilizing regular simplexes—where class prototypes are positioned at the vertices of a regular simplex and rejection decisions are made using a distance-ratio metric—have demonstrated strong empirical results, they often suffer from a lack of theoretical grounding. Previous theoretical analyses were limited to embedding dimensions $d \ge C-1$, the specific range required for a regular simplex to exist. This work presents a comprehensive theoretical framework for simplex-ratio OSR that is valid across all embedding dimensions, including low-dimensional cases such as $d=2$, with the regular simplex serving as a particular instance.

We demonstrate that for these configurations, the sublevel sets of an auxiliary squared ratio score correspond precisely to unions of Euclidean balls. These sets effectively bound the acceptance region defined by the operational score. Furthermore, we establish a distinct dichotomy: prototypes exhibit one-distance symmetry, mimicking a regular simplex, if and only if the dimension $d \ge C-1$. In lower dimensions, any deviation from this symmetry is managed by an explicit defect parameter. Additionally, we prove that under standard isotropy assumptions, the false-acceptance rate diminishes exponentially as the dimension $d$ increases, and that the operational score is globally Lipschitz with compact acceptance regions.

From an empirical perspective, we investigate balanced prototype geometry not as a standalone state-of-the-art detector, but as both an analytical instrument and a prior for representation learning. Evaluations on open-set splits from CIFAR and MedMNIST reveal that while this geometric structure offers valuable organizational properties, OSR efficacy is heavily influenced by the choice of scoring mechanism. Specifically, raw ratio scores generally lag behind nearest-neighbor and logit-based approaches.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC