# Beyond the Simplex: Balanced Prototype Geometry for Scorer-Agnostic Open-Set Recognition

arXiv: https://arxiv.org/abs/2606.01883 · HTML: https://mlsanity.com/paper/2606.01883/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 32 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 25 of 25)

- [Deep Anomaly Detection with Outlier Exposure](https://mlsanity.com/paper/1812.04606/) (2018)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Towards Open Set Deep Networks](https://mlsanity.com/paper/1511.06233/) (2015)
- [Recent Advances in Open Set Recognition: A Survey](https://mlsanity.com/paper/1811.08581/) (2018)
- [A Geometric Analysis of Neural Collapse with Unconstrained Features](https://mlsanity.com/paper/2105.02375/) (2021)
- [Towards Open World Recognition](https://mlsanity.com/paper/1412.5687/) (2014)
- [Reducing Network Agnostophobia](https://mlsanity.com/paper/1811.04110/) (2018)
- [Classification-Reconstruction Learning for Open-Set Recognition](https://mlsanity.com/paper/1812.04246/) (2018)
- [C2AE: Class Conditioned Auto-Encoder for Open-set Recognition](https://mlsanity.com/paper/1904.01198/) (2019)
- [Robust Classification with Convolutional Prototype Learning](https://mlsanity.com/paper/1805.03438/) (2018)
- [Learning Open Set Network with Discriminative Reciprocal Points](https://mlsanity.com/paper/2011.00178/) (2020)
- [DOC: Deep Open Classification of Text Documents](https://mlsanity.com/paper/1709.08716/) (2017)
- [Imbalance Trouble: Revisiting Neural-Collapse Geometry](https://mlsanity.com/paper/2208.05512/) (2022)
- [The Extreme Value Machine](https://mlsanity.com/paper/1506.06112/) (2015)
- [Neural Collapse with Normalized Features: A Geometric Analysis over the Riemannian Manifold](https://mlsanity.com/paper/2209.09211/) (2022)
- [Difficulty-Aware Simulator for Open Set Recognition](https://mlsanity.com/paper/2207.10024/) (2022)
- [Uncertainty-inspired Open Set Learning for Retinal Anomaly Identification](https://mlsanity.com/paper/2304.03981/) (2023)
- [Open-Set Recognition with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/2006.02003/) (2020)
- [Engineering the Neural Collapse Geometry of Supervised-Contrastive Loss](https://mlsanity.com/paper/2310.00893/) (2023)
- [A Survey on Open-Set Image Recognition](https://mlsanity.com/paper/2312.15571/) (2023)
- [Revealing the Two Sides of Data Augmentation: An Asymmetric Distillation-based Win-Win Solution for Open-Set Recognition](https://mlsanity.com/paper/2404.19527/) (2024)
- [Learning Large Margin Sparse Embeddings for Open Set Medical Diagnosis](https://mlsanity.com/paper/2307.04541/) (2023)
- [UCDSC: Open Set UnCertainty aware Deep Simplex Classifier for Medical Image Datasets](https://mlsanity.com/paper/2511.08196/) (2025)
- [DMDSC: A Dynamic-Margin Deep Simplex Classifier for Open-Set Recognition on Medical Image Datasets](https://mlsanity.com/paper/2605.00675/) (2026)

## Abstract (arXiv)

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.