Beyond the Simplex: Balanced Prototype Geometry for Scorer-Agnostic Open-Set Recognition
Authors: Mayank Sharma, Rohit Kumar Mourya
Organizations: Indian Institute of Technology Jodhpur
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 < C-1. Our analysis centers on balanced equal-norm codes: prototype configurations with equal lengths and zero sum, which exist for all 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.
Open-world object detection requires models to recognize known categories, reject unfamiliar objects, and incorporate new classes over time. This is especially challenging in scarce-data settings such as biomedical and scientific imaging, where rare categories may have only a few annotated examples and fine-grained classes differ by subtle morphology. Prototype-based detectors are natural for this regime, but they typically learn class prototypes as independent anchors, ignoring relational structure among classes. We propose class-geometry supervision (CGS), a general framework that constrains learned prototype or class-representation spaces to preserve visual or semantic class dissimilarities estimated from training data. CGS introduces a dissimilarity-preserving objective that aligns pairwise distances among learned class representations with a target class-geometry matrix while retaining the standard task loss. We instantiate the same objective across prototype recognition, few-shot biomedical object detection, open-set detection, novel-class insertion, and OWOD adaptation on COCO. Experiments show that CGS improves sample efficiency in recognition and ova detection, substantially strengthens novel-class insertion, and improves unknown recall on COCO while retaining much of the known-class detection performance. Ablations show that meaningful visual geometry provides the most reliable gains, while random geometry can help novel separation but is less consistent for few-shot detection. These results suggest that relational class geometry is an effective supervisory signal for building calibrated and extensible open-world detectors under limited supervision.
Medical imaging datasets are often characterized by extreme class imbalances, where rare pathologies are significantly underrepresented compared to common conditions. This imbalance poses a dual challenge for Open-Set Recognition (OSR): models must maintain high classification accuracy on known classes while reliably rejecting unknown samples unseen during training in the clinical settings. While recently proposed Deep Simplex Classifier (DSC)\cite{cevikalp2024reaching} and UnCertainty-aware Deep Simplex Classifier (UCDSC)\cite{Aditya_2026_WACV} successfully leverage Neural Collapse to ensure maximal inter-class separation, they rely on a uniform margin that does not account for the varying densities of medical classes. In this paper, we propose DMDSC an enhanced framework featuring a dynamic margin approach. Our approach automatically adapts class-specific margins based on label frequency, enforcing a higher penalty and tighter feature clustering for rare pathologies to counteract the effects of data imbalance. Extensive experiments conducted on diverse medical benchmarks on BloodMNIST\cite{medmnistv2}, OCTMNIST\cite{medmnistv2}, DermaMNIST\cite{medmnistv2}, and BreaKHis~\cite{spanhol2015dataset} datasets, demonstrate that our framework outperforms state-of-the-art methods.
Outlier Exposure (OE) is among the strongest training-based OOD detectors on standard benchmarks but exhibits scorer-dependent tradeoffs (e.g., strong on MSP, weak on KNN) and requires curated auxiliary data. We show why OE works: its features sit at the same geometric locus as real near-OOD data, with the boundary-adjacent quartile driving nearly all of OE's gain. OE is boundary calibration, not OOD coverage. GEODE (GEOmetry-preserving DEtection) replicates this calibration synthetically through an angle-adaptive norm loss in which targets scale per-sample with cosine similarity to the nearest class mean, preserving feature geometry where boundary structure matters. Four theorems grounded in neural collapse justify the design. GEODE works across all seven standard scorers on CIFAR-10 (near-OOD AUROC 89.0-92.3, far-OOD reaching 93.05; no catastrophic failure on any scorer). Since the OOD regime is unknown at deployment, this is the test that matters. GEODE outperforms vanilla CE at matched epoch counts. Combined with OE, GEODE reaches 95.0 MSP / 94.8 KNN on CIFAR-10 and beats OE on every scorer on CIFAR-100. The gains hold on WRN-28-10 (+4.5 Energy, 3 seeds). Unlike methods that push OOD into the classifier null space (e.g., PFS, 14.38 KNN AUROC, worse than random), GEODE's adaptive target preserves the geometry that distance-based scorers depend on.