Multiclass Classification without Labels: Posterior Simplex Geometry unlocks hidden classes

In many real-world scenarios, reliable instance-level labels are unavailable—think medical imaging, astrophysics, or fraud detection. However, datasets often come from different environments with shifting class proportions (e.g., different patient populations or experimental cuts). The existing CWoLa principle handles binary classification (K=2) by training a classifier to distinguish two impure mixtures, recovering the optimal discriminator without any knowledge of mixture proportions.

Now, a team from the University of Milan (Prof. Vincenzo Piuri), the Niels Bohr Institute (Troels Petersen), and co-authors Raphaël Bonnet-Guerrini and Johann Ioannou-Nikolaides have mathematically extended this to multiclass (K>2) settings. They prove that the Bayes-optimal mixture classifier g* maps data points into a (K-1)-simplex embedded in posterior space. The simplex vertices correspond to latent classes linked by an unknown mixing matrix. Using only mixture identity labels, they propose two prior-free extraction methods: post-hoc simplex fitting (fitting a simplex to learned representations) and a bottleneck architecture that enforces a simplex structure during training.

Tested on MNIST (10 digits), CIFAR-10 (10 object classes), and Galaxy10 DECaLS (10 galaxy types), the method recovers latent classes and their proportions with accuracy rivaling fully supervised classifiers. For example, on CIFAR-10, the bottleneck method achieves classification accuracy within 3-5% of the oracle. This narrows the gap between weakly supervised and fully supervised learning, providing a mathematically grounded, scalable tool for multiclass discovery when labels are scarce or costly.